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package openpgp
const testKey1KeyId uint64 = 0xA34D7E18C20C31BB
const testKey3KeyId uint64 = 0x338934250CCC0360
const testKeyP256KeyId uint64 = 0xd44a2c495918513e
const signedInput = "Signed message\nline 2\nline 3\n"
const signedTextInput = "Signed message\r\nline 2\r\nline 3\r\n"
const recipientUnspecifiedHex = "848c0300000000000000000103ff62d4d578d03cf40c3da998dfe216c074fa6ddec5e31c197c9666ba292830d91d18716a80f699f9d897389a90e6d62d0238f5f07a5248073c0f24920e4bc4a30c2d17ee4e0cae7c3d4aaa4e8dced50e3010a80ee692175fa0385f62ecca4b56ee6e9980aa3ec51b61b077096ac9e800edaf161268593eedb6cc7027ff5cb32745d250010d407a6221ae22ef18469b444f2822478c4d190b24d36371a95cb40087cdd42d9399c3d06a53c0673349bfb607927f20d1e122bde1e2bf3aa6cae6edf489629bcaa0689539ae3b718914d88ededc3b"
const detachedSignatureHex = "889c04000102000605024d449cd1000a0910a34d7e18c20c31bb167603ff57718d09f28a519fdc7b5a68b6a3336da04df85e38c5cd5d5bd2092fa4629848a33d85b1729402a2aab39c3ac19f9d573f773cc62c264dc924c067a79dfd8a863ae06c7c8686120760749f5fd9b1e03a64d20a7df3446ddc8f0aeadeaeba7cbaee5c1e366d65b6a0c6cc749bcb912d2f15013f812795c2e29eb7f7b77f39ce77"
const detachedSignatureTextHex = "889c04010102000605024d449d21000a0910a34d7e18c20c31bbc8c60400a24fbef7342603a41cb1165767bd18985d015fb72fe05db42db36cfb2f1d455967f1e491194fbf6cf88146222b23bf6ffbd50d17598d976a0417d3192ff9cc0034fd00f287b02e90418bbefe609484b09231e4e7a5f3562e199bf39909ab5276c4d37382fe088f6b5c3426fc1052865da8b3ab158672d58b6264b10823dc4b39"
const detachedSignatureDSAHex = "884604001102000605024d6c4eac000a0910338934250ccc0360f18d00a087d743d6405ed7b87755476629600b8b694a39e900a0abff8126f46faf1547c1743c37b21b4ea15b8f83"
const detachedSignatureP256Hex = "885e0400130a0006050256e5bb00000a0910d44a2c495918513edef001009841a4f792beb0befccb35c8838a6a87d9b936beaa86db6745ddc7b045eee0cf00fd1ac1f78306b17e965935dd3f8bae4587a76587e4af231efe19cc4011a8434817"
// The plaintext is https://www.gutenberg.org/cache/epub/1080/pg1080.txt
const modestProposalSha512 = "lbbrB1+WP3T9AaC9OQqBdOcCjgeEQadlulXsNPgVx0tyqPzDHwUugZ2gE7V0ESKAw6kAVfgkcuvfgxAAGaeHtw=="
const testKeys1And2Hex = "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"
const testKeys1And2PrivateHex = "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"
const dsaElGamalTestKeysHex = "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"
const ed25519wX25519Key = "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"
const signedMessageHex = "a3019bc0cbccc0c4b8d8b74ee2108fe16ec6d3ca490cbe362d3f8333d3f352531472538b8b13d353b97232f352158c20943157c71c16064626063656269052062e4e01987e9b6fccff4b7df3a34c534b23e679cbec3bc0f8f6e64dfb4b55fe3f8efa9ce110ddb5cd79faf1d753c51aecfa669f7e7aa043436596cccc3359cb7dd6bbe9ecaa69e5989d9e57209571edc0b2fa7f57b9b79a64ee6e99ce1371395fee92fec2796f7b15a77c386ff668ee27f6d38f0baa6c438b561657377bf6acff3c5947befd7bf4c196252f1d6e5c524d0300"
const signedTextMessageHex = "a3019bc0cbccc8c4b8d8b74ee2108fe16ec6d36a250cbece0c178233d3f352531472538b8b13d35379b97232f352158ca0b4312f57c71c1646462606365626906a062e4e019811591798ff99bf8afee860b0d8a8c2a85c3387e3bcf0bb3b17987f2bbcfab2aa526d930cbfd3d98757184df3995c9f3e7790e36e3e9779f06089d4c64e9e47dd6202cb6e9bc73c5d11bb59fbaf89d22d8dc7cf199ddf17af96e77c5f65f9bbed56f427bd8db7af37f6c9984bf9385efaf5f184f986fb3e6adb0ecfe35bbf92d16a7aa2a344fb0bc52fb7624f0200"
const signedEncryptedMessageHex = "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"
const verifiedSignatureEncryptedMessageHex = "c2b304000108000605026048f6d600210910a34d7e18c20c31bb1621045fb74b1d03b1e3cb31bc2f8aa34d7e18c20c31bb9a3b0400a32ddac1af259c1b0abab0041327ea04970944401978fb647dd1cf9aba4f164e43f0d8a9389501886474bdd4a6e77f6aea945c07dfbf87743835b44cc2c39a1f9aeecfa83135abc92e18e50396f2e6a06c44e0188b0081effbfb4160d28f118d4ff73dd199a102e47cffd8c7ff2bacd83ae72b5820c021a486766dd587b5da61"
const unverifiedSignatureEncryptedMessageHex = "c2b304000108000605026048f6d600210910a34d7e18c20c31bb1621045fb74b1d03b1e3cb31bc2f8aa34d7e18c20c31bb9a3b0400a32ddac1af259c1b0abab0041327ea04970944401978fb647dd1cf9aba4f164e43f0d8a9389501886474bdd4a6e77f6aea945c07dfbf87743835b44cc2c39a1f9aeecfa83135abc92e18e50396f2e6a06c44e0188b0081effbfb4160d28f118d4ff73dd199a102e47cffd8c7ff2bacd83ae72b5820c021a486766dd587b5da61"
const signedEncryptedMessage2Hex = "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"
const signatureEncryptedMessage2Hex = "c24604001102000605024dfd0166000a091033af447ccd759b09bae600a096ec5e63ecf0a403085e10f75cc3bab327663282009f51fad9df457ed8d2b70d8a73c76e0443eac0f377"
const symmetricallyEncryptedCompressedHex = "c32e040903085a357c1a7b5614ed00cc0d1d92f428162058b3f558a0fb0980d221ebac6c97d5eda4e0fe32f6e706e94dd263012d6ca1ef8c4bbd324098225e603a10c85ebf09cbf7b5aeeb5ce46381a52edc51038b76a8454483be74e6dcd1e50d5689a8ae7eceaeefed98a0023d49b22eb1f65c2aa1ef1783bb5e1995713b0457102ec3c3075fe871267ffa4b686ad5d52000d857"
const dsaTestKeyHex = "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"
const dsaTestKeyPrivateHex = "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"
const p256TestKeyHex = "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"
const p256TestKeyPrivateHex = "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"
const armoredPrivateKeyBlock = `-----BEGIN PGP PRIVATE KEY BLOCK-----
Version: GnuPG v1.4.10 (GNU/Linux)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=zNCn
-----END PGP PRIVATE KEY BLOCK-----`
const e2ePublicKey = `-----BEGIN PGP PUBLIC KEY BLOCK-----
Charset: UTF-8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=7MrZ
-----END PGP PUBLIC KEY BLOCK-----`
const dsaKeyWithSHA512 = `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`
const unknownHashFunctionHex = `8a00000040040001990006050253863c24000a09103b4fe6acc0b21f32ffff0101010101010101010101010101010101010101010101010101010101010101010101010101`
const rsaSignatureBadMPIlength = `8a00000040040001030006050253863c24000a09103b4fe6acc0b21f32ffff0101010101010101010101010101010101010101010101010101010101010101010101010101`
const missingHashFunctionHex = `8a00000040040001030006050253863c24000a09103b4fe6acc0b21f32ffff0101010101010101010101010101010101010101010101010101010101010101010101010101`
const campbellQuine = `a0b001000300fcffa0b001000d00f2ff000300fcffa0b001000d00f2ff8270a01c00000500faff8270a01c00000500faff000500faff001400ebff8270a01c00000500faff000500faff001400ebff428821c400001400ebff428821c400001400ebff428821c400001400ebff428821c400001400ebff428821c400000000ffff000000ffff000b00f4ff428821c400000000ffff000000ffff000b00f4ff0233214c40000100feff000233214c40000100feff0000`
const keyV4forVerifyingSignedMessageV3 = `-----BEGIN PGP PUBLIC KEY BLOCK-----
Comment: GPGTools - https://gpgtools.org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=RZia
-----END PGP PUBLIC KEY BLOCK-----
`
const signedMessageV3 = `-----BEGIN PGP MESSAGE-----
Comment: GPGTools - https://gpgtools.org
owGbwMvMwMVYWXlhlrhb9GXG03JJDKF/MtxDMjKLFYAoUaEktbhEITe1uDgxPVWP
q5NhKjMrWAVcC9evD8z/bF/uWNjqtk/X3y5/38XGRQHm/57rrDRYuGnTw597Xqka
uM3137/hH3Os+Jf2dc0fXOITKwJvXJvecPVs0ta+Vg7ZO1MLn8w58Xx+6L58mbka
DGHyU9yTueZE8D+QF/Tz28Y78dqtF56R1VPn9Xw4uJqrWYdd7b3vIZ1V6R4Nh05d
iT57d/OhWwA=
=hG7R
-----END PGP MESSAGE-----
`
// https://mailarchive.ietf.org/arch/msg/openpgp/9SheW_LENE0Kxf7haNllovPyAdY/
const v5PrivKey = `-----BEGIN PGP PRIVATE KEY BLOCK-----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=IiS2
-----END PGP PRIVATE KEY BLOCK-----`
// See OpenPGP crypto refresh Section A.3.
const v6PrivKey = `-----BEGIN PGP PRIVATE KEY BLOCK-----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-----END PGP PRIVATE KEY BLOCK-----`
// See OpenPGP crypto refresh merge request:
// https://gitlab.com/openpgp-wg/rfc4880bis/-/merge_requests/304
const v6PrivKeyMsg = `-----BEGIN PGP MESSAGE-----
wV0GIQYSyD8ecG9jCP4VGkF3Q6HwM3kOk+mXhIjR2zeNqZMIhRmHzxjV8bU/gXzO
WgBM85PMiVi93AZfJfhK9QmxfdNnZBjeo1VDeVZheQHgaVf7yopqR6W1FT6NOrfS
aQIHAgZhZBZTW+CwcW1g4FKlbExAf56zaw76/prQoN+bAzxpohup69LA7JW/Vp0l
yZnuSj3hcFj0DfqLTGgr4/u717J+sPWbtQBfgMfG9AOIwwrUBqsFE9zW+f1zdlYo
bhF30A+IitsxxA==
-----END PGP MESSAGE-----`
// See OpenPGP crypto refresh merge request:
// https://gitlab.com/openpgp-wg/rfc4880bis/-/merge_requests/305
const v6PrivKeyInlineSignMsg = `-----BEGIN PGP MESSAGE-----
wV0GIQYSyD8ecG9jCP4VGkF3Q6HwM3kOk+mXhIjR2zeNqZMIhRmHzxjV8bU/gXzO
WgBM85PMiVi93AZfJfhK9QmxfdNnZBjeo1VDeVZheQHgaVf7yopqR6W1FT6NOrfS
aQIHAgZhZBZTW+CwcW1g4FKlbExAf56zaw76/prQoN+bAzxpohup69LA7JW/Vp0l
yZnuSj3hcFj0DfqLTGgr4/u717J+sPWbtQBfgMfG9AOIwwrUBqsFE9zW+f1zdlYo
bhF30A+IitsxxA==
-----END PGP MESSAGE-----`
// See https://gitlab.com/openpgp-wg/rfc4880bis/-/merge_requests/274
// decryption password: "correct horse battery staple"
const v6ArgonSealedPrivKey = `-----BEGIN PGP PRIVATE KEY BLOCK-----
xYIGY4d/4xsAAAAg+U2nu0jWCmHlZ3BqZYfQMxmZu52JGggkLq2EVD34laP9JgkC
FARdb9ccngltHraRe25uHuyuAQQVtKipJ0+r5jL4dacGWSAheCWPpITYiyfyIOPS
3gIDyg8f7strd1OB4+LZsUhcIjOMpVHgmiY/IutJkulneoBYwrEGHxsKAAAAQgWC
Y4d/4wMLCQcFFQoOCAwCFgACmwMCHgkiIQbLGGxPBgmml+TVLfpscisMHx4nwYpW
cI9lJewnutmsyQUnCQIHAgAAAACtKCAQPi19In7A5tfORHHbNr/JcIMlNpAnFJin
7wV2wH+q4UWFs7kDsBJ+xP2i8CMEWi7Ha8tPlXGpZR4UruETeh1mhELIj5UeM8T/
0z+5oX1RHu11j8bZzFDLX9eTsgOdWATHggZjh3/jGQAAACCGkySDZ/nlAV25Ivj0
gJXdp4SYfy1ZhbEvutFsr15ENf0mCQIUBA5hhGgp2oaavg6mFUXcFMwBBBUuE8qf
9Ock+xwusd+GAglBr5LVyr/lup3xxQvHXFSjjA2haXfoN6xUGRdDEHI6+uevKjVR
v5oAxgu7eJpaXNjCmwYYGwoAAAAsBYJjh3/jApsMIiEGyxhsTwYJppfk1S36bHIr
DB8eJ8GKVnCPZSXsJ7rZrMkAAAAABAEgpukYbZ1ZNfyP5WMUzbUnSGpaUSD5t2Ki
Nacp8DkBClZRa2c3AMQzSDXa9jGhYzxjzVb5scHDzTkjyRZWRdTq8U6L4da+/+Kt
ruh8m7Xo2ehSSFyWRSuTSZe5tm/KXgYG
-----END PGP PRIVATE KEY BLOCK-----`
const v4Key25519 = `-----BEGIN PGP PRIVATE KEY BLOCK-----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=e7jT
-----END PGP PRIVATE KEY BLOCK-----`
const keyWithExpiredCrossSig = `-----BEGIN PGP PUBLIC KEY BLOCK-----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=Nt+N
-----END PGP PUBLIC KEY BLOCK-----
`
const sigFromKeyWithExpiredCrossSig = `-----BEGIN PGP SIGNATURE-----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=aQkm
-----END PGP SIGNATURE-----
`
const signedMessageWithCriticalNotation = `-----BEGIN PGP MESSAGE-----
owGbwMvMwMH4oOW7S46CznTG09xJDDE3Wl1KUotLuDousDAwcjBYiSmyXL+48d6x
U1PSGUxcj8IUszKBVMpMaWAAAgEGZpAeh9SKxNyCnFS95PzcytRiBi5OAZjyXXzM
f8WYLqv7TXP61Sa4rqT12CI3xaN73YS2pt089f96odCKaEPnWJ3iSGmzJaW/ug10
2Zo8Wj2k4s7t8wt4H3HtTu+y5UZfV3VOO+l//sdE/o+Lsub8FZH7/eOq7OnbNp4n
vwjE8mqJXetNMfj8r2SCyvkEnlVRYR+/mnge+ib56FdJ8uKtqSxyvgA=
=fRXs
-----END PGP MESSAGE-----`
const criticalNotationSigner = `-----BEGIN PGP PUBLIC KEY BLOCK-----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=h/aX
-----END PGP PUBLIC KEY BLOCK-----`
const keyv5Test = `-----BEGIN PGP PRIVATE KEY BLOCK-----
Comment: Bob's OpenPGP Transferable Secret Key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bUdZqZeef2xJe8gMQg05DoD1DF3GipZ0Ies65beh+d5hegb7N4pzh0LzrBrVNHar
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IRDMXDOPyzEfjwARAQABAAv9F2CwsjS+Sjh1M1vegJbZjei4gF1HHpEM0K0PSXsp
SfVvpR4AoSJ4He6CXSMWg0ot8XKtDuZoV9jnJaES5UL9pMAD7JwIOqZm/DYVJM5h
OASCh1c356/wSbFbzRHPtUdZO9Q30WFNJM5pHbCJPjtNoRmRGkf71RxtvHBzy7np
Ga+W6U/NVKHw0i0CYwMI0YlKDakYW3Pm+QL+gHZFvngGweTod0f9l2VLLAmeQR/c
+EZs7lNumhuZ8mXcwhUc9JQIhOkpO+wreDysEFkAcsKbkQP3UDUsA1gFx9pbMzT0
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gCnwdpbgdHP6v/hvZY54ZaJjz6L2e8unNEkYLxDt8cmAyGPgH2XgL7giHIp9jrsQ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=miES
-----END PGP PRIVATE KEY BLOCK-----
`
const certv5Test = `-----BEGIN PGP PRIVATE KEY BLOCK-----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=IiS2
-----END PGP PRIVATE KEY BLOCK-----
`
const msgv5Test = `-----BEGIN PGP MESSAGE-----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=liRq
-----END PGP MESSAGE-----
`
// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Package s2k implements the various OpenPGP string-to-key transforms as
// specified in RFC 4800 section 3.7.1, and Argon2 specified in
// draft-ietf-openpgp-crypto-refresh-08 section 3.7.1.4.
package s2k // import "github.com/ProtonMail/go-crypto/openpgp/s2k"
import (
"crypto"
"hash"
"io"
"strconv"
"github.com/ProtonMail/go-crypto/openpgp/errors"
"github.com/ProtonMail/go-crypto/openpgp/internal/algorithm"
"golang.org/x/crypto/argon2"
)
type Mode uint8
// Defines the default S2KMode constants
//
// 0 (simple), 1(salted), 3(iterated), 4(argon2)
const (
SimpleS2K Mode = 0
SaltedS2K Mode = 1
IteratedSaltedS2K Mode = 3
Argon2S2K Mode = 4
GnuS2K Mode = 101
)
const Argon2SaltSize int = 16
// Params contains all the parameters of the s2k packet
type Params struct {
// mode is the mode of s2k function.
// It can be 0 (simple), 1(salted), 3(iterated)
// 2(reserved) 100-110(private/experimental).
mode Mode
// hashId is the ID of the hash function used in any of the modes
hashId byte
// salt is a byte array to use as a salt in hashing process or argon2
saltBytes [Argon2SaltSize]byte
// countByte is used to determine how many rounds of hashing are to
// be performed in s2k mode 3. See RFC 4880 Section 3.7.1.3.
countByte byte
// passes is a parameter in Argon2 to determine the number of iterations
// See RFC the crypto refresh Section 3.7.1.4.
passes byte
// parallelism is a parameter in Argon2 to determine the degree of paralellism
// See RFC the crypto refresh Section 3.7.1.4.
parallelism byte
// memoryExp is a parameter in Argon2 to determine the memory usage
// i.e., 2 ** memoryExp kibibytes
// See RFC the crypto refresh Section 3.7.1.4.
memoryExp byte
}
// encodeCount converts an iterative "count" in the range 1024 to
// 65011712, inclusive, to an encoded count. The return value is the
// octet that is actually stored in the GPG file. encodeCount panics
// if i is not in the above range (encodedCount above takes care to
// pass i in the correct range). See RFC 4880 Section 3.7.7.1.
func encodeCount(i int) uint8 {
if i < 65536 || i > 65011712 {
panic("count arg i outside the required range")
}
for encoded := 96; encoded < 256; encoded++ {
count := decodeCount(uint8(encoded))
if count >= i {
return uint8(encoded)
}
}
return 255
}
// decodeCount returns the s2k mode 3 iterative "count" corresponding to
// the encoded octet c.
func decodeCount(c uint8) int {
return (16 + int(c&15)) << (uint32(c>>4) + 6)
}
// encodeMemory converts the Argon2 "memory" in the range parallelism*8 to
// 2**31, inclusive, to an encoded memory. The return value is the
// octet that is actually stored in the GPG file. encodeMemory panics
// if is not in the above range
// See OpenPGP crypto refresh Section 3.7.1.4.
func encodeMemory(memory uint32, parallelism uint8) uint8 {
if memory < (8*uint32(parallelism)) || memory > uint32(2147483648) {
panic("Memory argument memory is outside the required range")
}
for exp := 3; exp < 31; exp++ {
compare := decodeMemory(uint8(exp))
if compare >= memory {
return uint8(exp)
}
}
return 31
}
// decodeMemory computes the decoded memory in kibibytes as 2**memoryExponent
func decodeMemory(memoryExponent uint8) uint32 {
return uint32(1) << memoryExponent
}
// Simple writes to out the result of computing the Simple S2K function (RFC
// 4880, section 3.7.1.1) using the given hash and input passphrase.
func Simple(out []byte, h hash.Hash, in []byte) {
Salted(out, h, in, nil)
}
var zero [1]byte
// Salted writes to out the result of computing the Salted S2K function (RFC
// 4880, section 3.7.1.2) using the given hash, input passphrase and salt.
func Salted(out []byte, h hash.Hash, in []byte, salt []byte) {
done := 0
var digest []byte
for i := 0; done < len(out); i++ {
h.Reset()
for j := 0; j < i; j++ {
h.Write(zero[:])
}
h.Write(salt)
h.Write(in)
digest = h.Sum(digest[:0])
n := copy(out[done:], digest)
done += n
}
}
// Iterated writes to out the result of computing the Iterated and Salted S2K
// function (RFC 4880, section 3.7.1.3) using the given hash, input passphrase,
// salt and iteration count.
func Iterated(out []byte, h hash.Hash, in []byte, salt []byte, count int) {
combined := make([]byte, len(in)+len(salt))
copy(combined, salt)
copy(combined[len(salt):], in)
if count < len(combined) {
count = len(combined)
}
done := 0
var digest []byte
for i := 0; done < len(out); i++ {
h.Reset()
for j := 0; j < i; j++ {
h.Write(zero[:])
}
written := 0
for written < count {
if written+len(combined) > count {
todo := count - written
h.Write(combined[:todo])
written = count
} else {
h.Write(combined)
written += len(combined)
}
}
digest = h.Sum(digest[:0])
n := copy(out[done:], digest)
done += n
}
}
// Argon2 writes to out the key derived from the password (in) with the Argon2
// function (the crypto refresh, section 3.7.1.4)
func Argon2(out []byte, in []byte, salt []byte, passes uint8, paralellism uint8, memoryExp uint8) {
key := argon2.IDKey(in, salt, uint32(passes), decodeMemory(memoryExp), paralellism, uint32(len(out)))
copy(out[:], key)
}
// Generate generates valid parameters from given configuration.
// It will enforce the Iterated and Salted or Argon2 S2K method.
func Generate(rand io.Reader, c *Config) (*Params, error) {
var params *Params
if c != nil && c.Mode() == Argon2S2K {
// handle Argon2 case
argonConfig := c.Argon2()
params = &Params{
mode: Argon2S2K,
passes: argonConfig.Passes(),
parallelism: argonConfig.Parallelism(),
memoryExp: argonConfig.EncodedMemory(),
}
} else if c != nil && c.PassphraseIsHighEntropy && c.Mode() == SaltedS2K { // Allow SaltedS2K if PassphraseIsHighEntropy
hashId, ok := algorithm.HashToHashId(c.hash())
if !ok {
return nil, errors.UnsupportedError("no such hash")
}
params = &Params{
mode: SaltedS2K,
hashId: hashId,
}
} else { // Enforce IteratedSaltedS2K method otherwise
hashId, ok := algorithm.HashToHashId(c.hash())
if !ok {
return nil, errors.UnsupportedError("no such hash")
}
if c != nil {
c.S2KMode = IteratedSaltedS2K
}
params = &Params{
mode: IteratedSaltedS2K,
hashId: hashId,
countByte: c.EncodedCount(),
}
}
if _, err := io.ReadFull(rand, params.salt()); err != nil {
return nil, err
}
return params, nil
}
// Parse reads a binary specification for a string-to-key transformation from r
// and returns a function which performs that transform. If the S2K is a special
// GNU extension that indicates that the private key is missing, then the error
// returned is errors.ErrDummyPrivateKey.
func Parse(r io.Reader) (f func(out, in []byte), err error) {
params, err := ParseIntoParams(r)
if err != nil {
return nil, err
}
return params.Function()
}
// ParseIntoParams reads a binary specification for a string-to-key
// transformation from r and returns a struct describing the s2k parameters.
func ParseIntoParams(r io.Reader) (params *Params, err error) {
var buf [Argon2SaltSize + 3]byte
_, err = io.ReadFull(r, buf[:1])
if err != nil {
return
}
params = &Params{
mode: Mode(buf[0]),
}
switch params.mode {
case SimpleS2K:
_, err = io.ReadFull(r, buf[:1])
if err != nil {
return nil, err
}
params.hashId = buf[0]
return params, nil
case SaltedS2K:
_, err = io.ReadFull(r, buf[:9])
if err != nil {
return nil, err
}
params.hashId = buf[0]
copy(params.salt(), buf[1:9])
return params, nil
case IteratedSaltedS2K:
_, err = io.ReadFull(r, buf[:10])
if err != nil {
return nil, err
}
params.hashId = buf[0]
copy(params.salt(), buf[1:9])
params.countByte = buf[9]
return params, nil
case Argon2S2K:
_, err = io.ReadFull(r, buf[:Argon2SaltSize+3])
if err != nil {
return nil, err
}
copy(params.salt(), buf[:Argon2SaltSize])
params.passes = buf[Argon2SaltSize]
params.parallelism = buf[Argon2SaltSize+1]
params.memoryExp = buf[Argon2SaltSize+2]
if err := validateArgon2Params(params); err != nil {
return nil, err
}
return params, nil
case GnuS2K:
// This is a GNU extension. See
// https://git.gnupg.org/cgi-bin/gitweb.cgi?p=gnupg.git;a=blob;f=doc/DETAILS;h=fe55ae16ab4e26d8356dc574c9e8bc935e71aef1;hb=23191d7851eae2217ecdac6484349849a24fd94a#l1109
if _, err = io.ReadFull(r, buf[:5]); err != nil {
return nil, err
}
params.hashId = buf[0]
if buf[1] == 'G' && buf[2] == 'N' && buf[3] == 'U' && buf[4] == 1 {
return params, nil
}
return nil, errors.UnsupportedError("GNU S2K extension")
}
return nil, errors.UnsupportedError("S2K function")
}
func (params *Params) Mode() Mode {
return params.mode
}
func (params *Params) Dummy() bool {
return params != nil && params.mode == GnuS2K
}
func (params *Params) salt() []byte {
switch params.mode {
case SaltedS2K, IteratedSaltedS2K:
return params.saltBytes[:8]
case Argon2S2K:
return params.saltBytes[:Argon2SaltSize]
default:
return nil
}
}
func (params *Params) Function() (f func(out, in []byte), err error) {
if params.Dummy() {
return nil, errors.ErrDummyPrivateKey("dummy key found")
}
var hashObj crypto.Hash
if params.mode != Argon2S2K {
var ok bool
hashObj, ok = algorithm.HashIdToHashWithSha1(params.hashId)
if !ok {
return nil, errors.UnsupportedError("hash for S2K function: " + strconv.Itoa(int(params.hashId)))
}
if !hashObj.Available() {
return nil, errors.UnsupportedError("hash not available: " + strconv.Itoa(int(hashObj)))
}
}
switch params.mode {
case SimpleS2K:
f := func(out, in []byte) {
Simple(out, hashObj.New(), in)
}
return f, nil
case SaltedS2K:
f := func(out, in []byte) {
Salted(out, hashObj.New(), in, params.salt())
}
return f, nil
case IteratedSaltedS2K:
f := func(out, in []byte) {
Iterated(out, hashObj.New(), in, params.salt(), decodeCount(params.countByte))
}
return f, nil
case Argon2S2K:
f := func(out, in []byte) {
Argon2(out, in, params.salt(), params.passes, params.parallelism, params.memoryExp)
}
return f, nil
}
return nil, errors.UnsupportedError("S2K function")
}
func (params *Params) Serialize(w io.Writer) (err error) {
if _, err = w.Write([]byte{uint8(params.mode)}); err != nil {
return
}
if params.mode != Argon2S2K {
if _, err = w.Write([]byte{params.hashId}); err != nil {
return
}
}
if params.Dummy() {
_, err = w.Write(append([]byte("GNU"), 1))
return
}
if params.mode > 0 {
if _, err = w.Write(params.salt()); err != nil {
return
}
if params.mode == IteratedSaltedS2K {
_, err = w.Write([]byte{params.countByte})
}
if params.mode == Argon2S2K {
_, err = w.Write([]byte{params.passes, params.parallelism, params.memoryExp})
}
}
return
}
// Serialize salts and stretches the given passphrase and writes the
// resulting key into key. It also serializes an S2K descriptor to
// w. The key stretching can be configured with c, which may be
// nil. In that case, sensible defaults will be used.
func Serialize(w io.Writer, key []byte, rand io.Reader, passphrase []byte, c *Config) error {
params, err := Generate(rand, c)
if err != nil {
return err
}
err = params.Serialize(w)
if err != nil {
return err
}
f, err := params.Function()
if err != nil {
return err
}
f(key, passphrase)
return nil
}
// validateArgon2Params checks that the argon2 parameters are valid according to RFC9580.
func validateArgon2Params(params *Params) error {
// The number of passes t and the degree of parallelism p MUST be non-zero.
if params.parallelism == 0 {
return errors.StructuralError("invalid argon2 params: parallelism is 0")
}
if params.passes == 0 {
return errors.StructuralError("invalid argon2 params: iterations is 0")
}
// The encoded memory size MUST be a value from 3+ceil(log2(p)) to 31,
// such that the decoded memory size m is a value from 8*p to 2^31.
if params.memoryExp > 31 || decodeMemory(params.memoryExp) < 8*uint32(params.parallelism) {
return errors.StructuralError("invalid argon2 params: memory is out of bounds")
}
return nil
}
package s2k
// Cache stores keys derived with s2k functions from one passphrase
// to avoid recomputation if multiple items are encrypted with
// the same parameters.
type Cache map[Params][]byte
// GetOrComputeDerivedKey tries to retrieve the key
// for the given s2k parameters from the cache.
// If there is no hit, it derives the key with the s2k function from the passphrase,
// updates the cache, and returns the key.
func (c *Cache) GetOrComputeDerivedKey(passphrase []byte, params *Params, expectedKeySize int) ([]byte, error) {
key, found := (*c)[*params]
if !found || len(key) != expectedKeySize {
var err error
derivedKey := make([]byte, expectedKeySize)
s2k, err := params.Function()
if err != nil {
return nil, err
}
s2k(derivedKey, passphrase)
(*c)[*params] = key
return derivedKey, nil
}
return key, nil
}
package s2k
import "crypto"
// Config collects configuration parameters for s2k key-stretching
// transformations. A nil *Config is valid and results in all default
// values.
type Config struct {
// S2K (String to Key) mode, used for key derivation in the context of secret key encryption
// and passphrase-encrypted data. Either s2k.Argon2S2K or s2k.IteratedSaltedS2K may be used.
// If the passphrase is a high-entropy key, indicated by setting PassphraseIsHighEntropy to true,
// s2k.SaltedS2K can also be used.
// Note: Argon2 is the strongest option but not all OpenPGP implementations are compatible with it
//(pending standardisation).
// 0 (simple), 1(salted), 3(iterated), 4(argon2)
// 2(reserved) 100-110(private/experimental).
S2KMode Mode
// Only relevant if S2KMode is not set to s2k.Argon2S2K.
// Hash is the default hash function to be used. If
// nil, SHA256 is used.
Hash crypto.Hash
// Argon2 parameters for S2K (String to Key).
// Only relevant if S2KMode is set to s2k.Argon2S2K.
// If nil, default parameters are used.
// For more details on the choice of parameters, see https://tools.ietf.org/html/rfc9106#section-4.
Argon2Config *Argon2Config
// Only relevant if S2KMode is set to s2k.IteratedSaltedS2K.
// Iteration count for Iterated S2K (String to Key). It
// determines the strength of the passphrase stretching when
// the said passphrase is hashed to produce a key. S2KCount
// should be between 65536 and 65011712, inclusive. If Config
// is nil or S2KCount is 0, the value 16777216 used. Not all
// values in the above range can be represented. S2KCount will
// be rounded up to the next representable value if it cannot
// be encoded exactly. When set, it is strongly encrouraged to
// use a value that is at least 65536. See RFC 4880 Section
// 3.7.1.3.
S2KCount int
// Indicates whether the passphrase passed by the application is a
// high-entropy key (e.g. it's randomly generated or derived from
// another passphrase using a strong key derivation function).
// When true, allows the S2KMode to be s2k.SaltedS2K.
// When the passphrase is not a high-entropy key, using SaltedS2K is
// insecure, and not allowed by draft-ietf-openpgp-crypto-refresh-08.
PassphraseIsHighEntropy bool
}
// Argon2Config stores the Argon2 parameters
// A nil *Argon2Config is valid and results in all default
type Argon2Config struct {
NumberOfPasses uint8
DegreeOfParallelism uint8
// Memory specifies the desired Argon2 memory usage in kibibytes.
// For example memory=64*1024 sets the memory cost to ~64 MB.
Memory uint32
}
func (c *Config) Mode() Mode {
if c == nil {
return IteratedSaltedS2K
}
return c.S2KMode
}
func (c *Config) hash() crypto.Hash {
if c == nil || uint(c.Hash) == 0 {
return crypto.SHA256
}
return c.Hash
}
func (c *Config) Argon2() *Argon2Config {
if c == nil || c.Argon2Config == nil {
return nil
}
return c.Argon2Config
}
// EncodedCount get encoded count
func (c *Config) EncodedCount() uint8 {
if c == nil || c.S2KCount == 0 {
return 224 // The common case. Corresponding to 16777216
}
i := c.S2KCount
switch {
case i < 65536:
i = 65536
case i > 65011712:
i = 65011712
}
return encodeCount(i)
}
func (c *Argon2Config) Passes() uint8 {
if c == nil || c.NumberOfPasses == 0 {
return 3
}
return c.NumberOfPasses
}
func (c *Argon2Config) Parallelism() uint8 {
if c == nil || c.DegreeOfParallelism == 0 {
return 4
}
return c.DegreeOfParallelism
}
func (c *Argon2Config) EncodedMemory() uint8 {
if c == nil || c.Memory == 0 {
return 16 // 64 MiB of RAM
}
memory := c.Memory
lowerBound := uint32(c.Parallelism()) * 8
upperBound := uint32(2147483648)
switch {
case memory < lowerBound:
memory = lowerBound
case memory > upperBound:
memory = upperBound
}
return encodeMemory(memory, c.Parallelism())
}
// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package openpgp
import (
"crypto"
"hash"
"io"
"strconv"
"time"
"github.com/ProtonMail/go-crypto/openpgp/armor"
"github.com/ProtonMail/go-crypto/openpgp/errors"
"github.com/ProtonMail/go-crypto/openpgp/internal/algorithm"
"github.com/ProtonMail/go-crypto/openpgp/packet"
)
// DetachSign signs message with the private key from signer (which must
// already have been decrypted) and writes the signature to w.
// If config is nil, sensible defaults will be used.
func DetachSign(w io.Writer, signer *Entity, message io.Reader, config *packet.Config) error {
return detachSign(w, signer, message, packet.SigTypeBinary, config)
}
// ArmoredDetachSign signs message with the private key from signer (which
// must already have been decrypted) and writes an armored signature to w.
// If config is nil, sensible defaults will be used.
func ArmoredDetachSign(w io.Writer, signer *Entity, message io.Reader, config *packet.Config) (err error) {
return armoredDetachSign(w, signer, message, packet.SigTypeBinary, config)
}
// DetachSignText signs message (after canonicalising the line endings) with
// the private key from signer (which must already have been decrypted) and
// writes the signature to w.
// If config is nil, sensible defaults will be used.
func DetachSignText(w io.Writer, signer *Entity, message io.Reader, config *packet.Config) error {
return detachSign(w, signer, message, packet.SigTypeText, config)
}
// ArmoredDetachSignText signs message (after canonicalising the line endings)
// with the private key from signer (which must already have been decrypted)
// and writes an armored signature to w.
// If config is nil, sensible defaults will be used.
func ArmoredDetachSignText(w io.Writer, signer *Entity, message io.Reader, config *packet.Config) error {
return armoredDetachSign(w, signer, message, packet.SigTypeText, config)
}
func armoredDetachSign(w io.Writer, signer *Entity, message io.Reader, sigType packet.SignatureType, config *packet.Config) (err error) {
out, err := armor.Encode(w, SignatureType, nil)
if err != nil {
return
}
err = detachSign(out, signer, message, sigType, config)
if err != nil {
return
}
return out.Close()
}
func detachSign(w io.Writer, signer *Entity, message io.Reader, sigType packet.SignatureType, config *packet.Config) (err error) {
signingKey, ok := signer.SigningKeyById(config.Now(), config.SigningKey())
if !ok {
return errors.InvalidArgumentError("no valid signing keys")
}
if signingKey.PrivateKey == nil {
return errors.InvalidArgumentError("signing key doesn't have a private key")
}
if signingKey.PrivateKey.Encrypted {
return errors.InvalidArgumentError("signing key is encrypted")
}
if _, ok := algorithm.HashToHashId(config.Hash()); !ok {
return errors.InvalidArgumentError("invalid hash function")
}
sig := createSignaturePacket(signingKey.PublicKey, sigType, config)
h, err := sig.PrepareSign(config)
if err != nil {
return
}
wrappedHash, err := wrapHashForSignature(h, sig.SigType)
if err != nil {
return
}
if _, err = io.Copy(wrappedHash, message); err != nil {
return err
}
err = sig.Sign(h, signingKey.PrivateKey, config)
if err != nil {
return
}
return sig.Serialize(w)
}
// FileHints contains metadata about encrypted files. This metadata is, itself,
// encrypted.
type FileHints struct {
// IsBinary can be set to hint that the contents are binary data.
IsBinary bool
// FileName hints at the name of the file that should be written. It's
// truncated to 255 bytes if longer. It may be empty to suggest that the
// file should not be written to disk. It may be equal to "_CONSOLE" to
// suggest the data should not be written to disk.
FileName string
// ModTime contains the modification time of the file, or the zero time if not applicable.
ModTime time.Time
}
// SymmetricallyEncrypt acts like gpg -c: it encrypts a file with a passphrase.
// The resulting WriteCloser must be closed after the contents of the file have
// been written.
// If config is nil, sensible defaults will be used.
func SymmetricallyEncrypt(ciphertext io.Writer, passphrase []byte, hints *FileHints, config *packet.Config) (plaintext io.WriteCloser, err error) {
if hints == nil {
hints = &FileHints{}
}
key, err := packet.SerializeSymmetricKeyEncrypted(ciphertext, passphrase, config)
if err != nil {
return
}
var w io.WriteCloser
cipherSuite := packet.CipherSuite{
Cipher: config.Cipher(),
Mode: config.AEAD().Mode(),
}
w, err = packet.SerializeSymmetricallyEncrypted(ciphertext, config.Cipher(), config.AEAD() != nil, cipherSuite, key, config)
if err != nil {
return
}
literalData := w
if algo := config.Compression(); algo != packet.CompressionNone {
var compConfig *packet.CompressionConfig
if config != nil {
compConfig = config.CompressionConfig
}
literalData, err = packet.SerializeCompressed(w, algo, compConfig)
if err != nil {
return
}
}
var epochSeconds uint32
if !hints.ModTime.IsZero() {
epochSeconds = uint32(hints.ModTime.Unix())
}
return packet.SerializeLiteral(literalData, hints.IsBinary, hints.FileName, epochSeconds)
}
// intersectPreferences mutates and returns a prefix of a that contains only
// the values in the intersection of a and b. The order of a is preserved.
func intersectPreferences(a []uint8, b []uint8) (intersection []uint8) {
var j int
for _, v := range a {
for _, v2 := range b {
if v == v2 {
a[j] = v
j++
break
}
}
}
return a[:j]
}
// intersectPreferences mutates and returns a prefix of a that contains only
// the values in the intersection of a and b. The order of a is preserved.
func intersectCipherSuites(a [][2]uint8, b [][2]uint8) (intersection [][2]uint8) {
var j int
for _, v := range a {
for _, v2 := range b {
if v[0] == v2[0] && v[1] == v2[1] {
a[j] = v
j++
break
}
}
}
return a[:j]
}
func hashToHashId(h crypto.Hash) uint8 {
v, ok := algorithm.HashToHashId(h)
if !ok {
panic("tried to convert unknown hash")
}
return v
}
// EncryptText encrypts a message to a number of recipients and, optionally,
// signs it. Optional information is contained in 'hints', also encrypted, that
// aids the recipients in processing the message. The resulting WriteCloser
// must be closed after the contents of the file have been written. If config
// is nil, sensible defaults will be used. The signing is done in text mode.
func EncryptText(ciphertext io.Writer, to []*Entity, signed *Entity, hints *FileHints, config *packet.Config) (plaintext io.WriteCloser, err error) {
return encrypt(ciphertext, ciphertext, to, signed, hints, packet.SigTypeText, config)
}
// Encrypt encrypts a message to a number of recipients and, optionally, signs
// it. hints contains optional information, that is also encrypted, that aids
// the recipients in processing the message. The resulting WriteCloser must
// be closed after the contents of the file have been written.
// If config is nil, sensible defaults will be used.
func Encrypt(ciphertext io.Writer, to []*Entity, signed *Entity, hints *FileHints, config *packet.Config) (plaintext io.WriteCloser, err error) {
return encrypt(ciphertext, ciphertext, to, signed, hints, packet.SigTypeBinary, config)
}
// EncryptSplit encrypts a message to a number of recipients and, optionally, signs
// it. hints contains optional information, that is also encrypted, that aids
// the recipients in processing the message. The resulting WriteCloser must
// be closed after the contents of the file have been written.
// If config is nil, sensible defaults will be used.
func EncryptSplit(keyWriter io.Writer, dataWriter io.Writer, to []*Entity, signed *Entity, hints *FileHints, config *packet.Config) (plaintext io.WriteCloser, err error) {
return encrypt(keyWriter, dataWriter, to, signed, hints, packet.SigTypeBinary, config)
}
// EncryptTextSplit encrypts a message to a number of recipients and, optionally, signs
// it. hints contains optional information, that is also encrypted, that aids
// the recipients in processing the message. The resulting WriteCloser must
// be closed after the contents of the file have been written.
// If config is nil, sensible defaults will be used.
func EncryptTextSplit(keyWriter io.Writer, dataWriter io.Writer, to []*Entity, signed *Entity, hints *FileHints, config *packet.Config) (plaintext io.WriteCloser, err error) {
return encrypt(keyWriter, dataWriter, to, signed, hints, packet.SigTypeText, config)
}
// writeAndSign writes the data as a payload package and, optionally, signs
// it. hints contains optional information, that is also encrypted,
// that aids the recipients in processing the message. The resulting
// WriteCloser must be closed after the contents of the file have been
// written. If config is nil, sensible defaults will be used.
func writeAndSign(payload io.WriteCloser, candidateHashes []uint8, signed *Entity, hints *FileHints, sigType packet.SignatureType, config *packet.Config) (plaintext io.WriteCloser, err error) {
var signer *packet.PrivateKey
if signed != nil {
signKey, ok := signed.SigningKeyById(config.Now(), config.SigningKey())
if !ok {
return nil, errors.InvalidArgumentError("no valid signing keys")
}
signer = signKey.PrivateKey
if signer == nil {
return nil, errors.InvalidArgumentError("no private key in signing key")
}
if signer.Encrypted {
return nil, errors.InvalidArgumentError("signing key must be decrypted")
}
}
var hash crypto.Hash
for _, hashId := range candidateHashes {
if h, ok := algorithm.HashIdToHash(hashId); ok && h.Available() {
hash = h
break
}
}
// If the hash specified by config is a candidate, we'll use that.
if configuredHash := config.Hash(); configuredHash.Available() {
for _, hashId := range candidateHashes {
if h, ok := algorithm.HashIdToHash(hashId); ok && h == configuredHash {
hash = h
break
}
}
}
if hash == 0 {
hashId := candidateHashes[0]
name, ok := algorithm.HashIdToString(hashId)
if !ok {
name = "#" + strconv.Itoa(int(hashId))
}
return nil, errors.InvalidArgumentError("cannot encrypt because no candidate hash functions are compiled in. (Wanted " + name + " in this case.)")
}
var salt []byte
if signer != nil {
var opsVersion = 3
if signer.Version == 6 {
opsVersion = signer.Version
}
ops := &packet.OnePassSignature{
Version: opsVersion,
SigType: sigType,
Hash: hash,
PubKeyAlgo: signer.PubKeyAlgo,
KeyId: signer.KeyId,
IsLast: true,
}
if opsVersion == 6 {
ops.KeyFingerprint = signer.Fingerprint
salt, err = packet.SignatureSaltForHash(hash, config.Random())
if err != nil {
return nil, err
}
ops.Salt = salt
}
if err := ops.Serialize(payload); err != nil {
return nil, err
}
}
if hints == nil {
hints = &FileHints{}
}
w := payload
if signer != nil {
// If we need to write a signature packet after the literal
// data then we need to stop literalData from closing
// encryptedData.
w = noOpCloser{w}
}
var epochSeconds uint32
if !hints.ModTime.IsZero() {
epochSeconds = uint32(hints.ModTime.Unix())
}
literalData, err := packet.SerializeLiteral(w, hints.IsBinary, hints.FileName, epochSeconds)
if err != nil {
return nil, err
}
if signer != nil {
h, wrappedHash, err := hashForSignature(hash, sigType, salt)
if err != nil {
return nil, err
}
metadata := &packet.LiteralData{
Format: 'u',
FileName: hints.FileName,
Time: epochSeconds,
}
if hints.IsBinary {
metadata.Format = 'b'
}
return signatureWriter{payload, literalData, hash, wrappedHash, h, salt, signer, sigType, config, metadata}, nil
}
return literalData, nil
}
// encrypt encrypts a message to a number of recipients and, optionally, signs
// it. hints contains optional information, that is also encrypted, that aids
// the recipients in processing the message. The resulting WriteCloser must
// be closed after the contents of the file have been written.
// If config is nil, sensible defaults will be used.
func encrypt(keyWriter io.Writer, dataWriter io.Writer, to []*Entity, signed *Entity, hints *FileHints, sigType packet.SignatureType, config *packet.Config) (plaintext io.WriteCloser, err error) {
if len(to) == 0 {
return nil, errors.InvalidArgumentError("no encryption recipient provided")
}
// These are the possible ciphers that we'll use for the message.
candidateCiphers := []uint8{
uint8(packet.CipherAES256),
uint8(packet.CipherAES128),
}
// These are the possible hash functions that we'll use for the signature.
candidateHashes := []uint8{
hashToHashId(crypto.SHA256),
hashToHashId(crypto.SHA384),
hashToHashId(crypto.SHA512),
hashToHashId(crypto.SHA3_256),
hashToHashId(crypto.SHA3_512),
}
// Prefer GCM if everyone supports it
candidateCipherSuites := [][2]uint8{
{uint8(packet.CipherAES256), uint8(packet.AEADModeGCM)},
{uint8(packet.CipherAES256), uint8(packet.AEADModeEAX)},
{uint8(packet.CipherAES256), uint8(packet.AEADModeOCB)},
{uint8(packet.CipherAES128), uint8(packet.AEADModeGCM)},
{uint8(packet.CipherAES128), uint8(packet.AEADModeEAX)},
{uint8(packet.CipherAES128), uint8(packet.AEADModeOCB)},
}
candidateCompression := []uint8{
uint8(packet.CompressionNone),
uint8(packet.CompressionZIP),
uint8(packet.CompressionZLIB),
}
encryptKeys := make([]Key, len(to))
// AEAD is used only if config enables it and every key supports it
aeadSupported := config.AEAD() != nil
for i := range to {
var ok bool
encryptKeys[i], ok = to[i].EncryptionKey(config.Now())
if !ok {
return nil, errors.InvalidArgumentError("cannot encrypt a message to key id " + strconv.FormatUint(to[i].PrimaryKey.KeyId, 16) + " because it has no valid encryption keys")
}
primarySelfSignature, _ := to[i].PrimarySelfSignature()
if primarySelfSignature == nil {
return nil, errors.InvalidArgumentError("entity without a self-signature")
}
if !primarySelfSignature.SEIPDv2 {
aeadSupported = false
}
candidateCiphers = intersectPreferences(candidateCiphers, primarySelfSignature.PreferredSymmetric)
candidateHashes = intersectPreferences(candidateHashes, primarySelfSignature.PreferredHash)
candidateCipherSuites = intersectCipherSuites(candidateCipherSuites, primarySelfSignature.PreferredCipherSuites)
candidateCompression = intersectPreferences(candidateCompression, primarySelfSignature.PreferredCompression)
}
// In the event that the intersection of supported algorithms is empty we use the ones
// labelled as MUST that every implementation supports.
if len(candidateCiphers) == 0 {
// https://www.ietf.org/archive/id/draft-ietf-openpgp-crypto-refresh-07.html#section-9.3
candidateCiphers = []uint8{uint8(packet.CipherAES128)}
}
if len(candidateHashes) == 0 {
// https://www.ietf.org/archive/id/draft-ietf-openpgp-crypto-refresh-07.html#hash-algos
candidateHashes = []uint8{hashToHashId(crypto.SHA256)}
}
if len(candidateCipherSuites) == 0 {
// https://www.ietf.org/archive/id/draft-ietf-openpgp-crypto-refresh-07.html#section-9.6
candidateCipherSuites = [][2]uint8{{uint8(packet.CipherAES128), uint8(packet.AEADModeOCB)}}
}
cipher := packet.CipherFunction(candidateCiphers[0])
aeadCipherSuite := packet.CipherSuite{
Cipher: packet.CipherFunction(candidateCipherSuites[0][0]),
Mode: packet.AEADMode(candidateCipherSuites[0][1]),
}
// If the cipher specified by config is a candidate, we'll use that.
configuredCipher := config.Cipher()
for _, c := range candidateCiphers {
cipherFunc := packet.CipherFunction(c)
if cipherFunc == configuredCipher {
cipher = cipherFunc
break
}
}
var symKey []byte
if aeadSupported {
symKey = make([]byte, aeadCipherSuite.Cipher.KeySize())
} else {
symKey = make([]byte, cipher.KeySize())
}
if _, err := io.ReadFull(config.Random(), symKey); err != nil {
return nil, err
}
for _, key := range encryptKeys {
if err := packet.SerializeEncryptedKeyAEAD(keyWriter, key.PublicKey, cipher, aeadSupported, symKey, config); err != nil {
return nil, err
}
}
var payload io.WriteCloser
payload, err = packet.SerializeSymmetricallyEncrypted(dataWriter, cipher, aeadSupported, aeadCipherSuite, symKey, config)
if err != nil {
return
}
payload, err = handleCompression(payload, candidateCompression, config)
if err != nil {
return nil, err
}
return writeAndSign(payload, candidateHashes, signed, hints, sigType, config)
}
// Sign signs a message. The resulting WriteCloser must be closed after the
// contents of the file have been written. hints contains optional information
// that aids the recipients in processing the message.
// If config is nil, sensible defaults will be used.
func Sign(output io.Writer, signed *Entity, hints *FileHints, config *packet.Config) (input io.WriteCloser, err error) {
if signed == nil {
return nil, errors.InvalidArgumentError("no signer provided")
}
// These are the possible hash functions that we'll use for the signature.
candidateHashes := []uint8{
hashToHashId(crypto.SHA256),
hashToHashId(crypto.SHA384),
hashToHashId(crypto.SHA512),
hashToHashId(crypto.SHA3_256),
hashToHashId(crypto.SHA3_512),
}
defaultHashes := candidateHashes[0:1]
primarySelfSignature, _ := signed.PrimarySelfSignature()
if primarySelfSignature == nil {
return nil, errors.StructuralError("signed entity has no self-signature")
}
preferredHashes := primarySelfSignature.PreferredHash
if len(preferredHashes) == 0 {
preferredHashes = defaultHashes
}
candidateHashes = intersectPreferences(candidateHashes, preferredHashes)
if len(candidateHashes) == 0 {
return nil, errors.StructuralError("cannot sign because signing key shares no common algorithms with candidate hashes")
}
return writeAndSign(noOpCloser{output}, candidateHashes, signed, hints, packet.SigTypeBinary, config)
}
// signatureWriter hashes the contents of a message while passing it along to
// literalData. When closed, it closes literalData, writes a signature packet
// to encryptedData and then also closes encryptedData.
type signatureWriter struct {
encryptedData io.WriteCloser
literalData io.WriteCloser
hashType crypto.Hash
wrappedHash hash.Hash
h hash.Hash
salt []byte // v6 only
signer *packet.PrivateKey
sigType packet.SignatureType
config *packet.Config
metadata *packet.LiteralData // V5 signatures protect document metadata
}
func (s signatureWriter) Write(data []byte) (int, error) {
s.wrappedHash.Write(data)
switch s.sigType {
case packet.SigTypeBinary:
return s.literalData.Write(data)
case packet.SigTypeText:
flag := 0
return writeCanonical(s.literalData, data, &flag)
}
return 0, errors.UnsupportedError("unsupported signature type: " + strconv.Itoa(int(s.sigType)))
}
func (s signatureWriter) Close() error {
sig := createSignaturePacket(&s.signer.PublicKey, s.sigType, s.config)
sig.Hash = s.hashType
sig.Metadata = s.metadata
if err := sig.SetSalt(s.salt); err != nil {
return err
}
if err := sig.Sign(s.h, s.signer, s.config); err != nil {
return err
}
if err := s.literalData.Close(); err != nil {
return err
}
if err := sig.Serialize(s.encryptedData); err != nil {
return err
}
return s.encryptedData.Close()
}
func createSignaturePacket(signer *packet.PublicKey, sigType packet.SignatureType, config *packet.Config) *packet.Signature {
sigLifetimeSecs := config.SigLifetime()
return &packet.Signature{
Version: signer.Version,
SigType: sigType,
PubKeyAlgo: signer.PubKeyAlgo,
Hash: config.Hash(),
CreationTime: config.Now(),
IssuerKeyId: &signer.KeyId,
IssuerFingerprint: signer.Fingerprint,
Notations: config.Notations(),
SigLifetimeSecs: &sigLifetimeSecs,
}
}
// noOpCloser is like an ioutil.NopCloser, but for an io.Writer.
// TODO: we have two of these in OpenPGP packages alone. This probably needs
// to be promoted somewhere more common.
type noOpCloser struct {
w io.Writer
}
func (c noOpCloser) Write(data []byte) (n int, err error) {
return c.w.Write(data)
}
func (c noOpCloser) Close() error {
return nil
}
func handleCompression(compressed io.WriteCloser, candidateCompression []uint8, config *packet.Config) (data io.WriteCloser, err error) {
data = compressed
confAlgo := config.Compression()
if confAlgo == packet.CompressionNone {
return
}
// Set algorithm labelled as MUST as fallback
// https://www.ietf.org/archive/id/draft-ietf-openpgp-crypto-refresh-07.html#section-9.4
finalAlgo := packet.CompressionNone
// if compression specified by config available we will use it
for _, c := range candidateCompression {
if uint8(confAlgo) == c {
finalAlgo = confAlgo
break
}
}
if finalAlgo != packet.CompressionNone {
var compConfig *packet.CompressionConfig
if config != nil {
compConfig = config.CompressionConfig
}
data, err = packet.SerializeCompressed(compressed, finalAlgo, compConfig)
if err != nil {
return
}
}
return data, nil
}
package x25519
import (
"crypto/sha256"
"crypto/subtle"
"io"
"github.com/ProtonMail/go-crypto/openpgp/aes/keywrap"
"github.com/ProtonMail/go-crypto/openpgp/errors"
x25519lib "github.com/cloudflare/circl/dh/x25519"
"golang.org/x/crypto/hkdf"
)
const (
hkdfInfo = "OpenPGP X25519"
aes128KeySize = 16
// The size of a public or private key in bytes.
KeySize = x25519lib.Size
)
type PublicKey struct {
// Point represents the encoded elliptic curve point of the public key.
Point []byte
}
type PrivateKey struct {
PublicKey
// Secret represents the secret of the private key.
Secret []byte
}
// NewPrivateKey creates a new empty private key including the public key.
func NewPrivateKey(key PublicKey) *PrivateKey {
return &PrivateKey{
PublicKey: key,
}
}
// Validate validates that the provided public key matches the private key.
func Validate(pk *PrivateKey) (err error) {
var expectedPublicKey, privateKey x25519lib.Key
subtle.ConstantTimeCopy(1, privateKey[:], pk.Secret)
x25519lib.KeyGen(&expectedPublicKey, &privateKey)
if subtle.ConstantTimeCompare(expectedPublicKey[:], pk.PublicKey.Point) == 0 {
return errors.KeyInvalidError("x25519: invalid key")
}
return nil
}
// GenerateKey generates a new x25519 key pair.
func GenerateKey(rand io.Reader) (*PrivateKey, error) {
var privateKey, publicKey x25519lib.Key
privateKeyOut := new(PrivateKey)
err := generateKey(rand, &privateKey, &publicKey)
if err != nil {
return nil, err
}
privateKeyOut.PublicKey.Point = publicKey[:]
privateKeyOut.Secret = privateKey[:]
return privateKeyOut, nil
}
func generateKey(rand io.Reader, privateKey *x25519lib.Key, publicKey *x25519lib.Key) error {
maxRounds := 10
isZero := true
for round := 0; isZero; round++ {
if round == maxRounds {
return errors.InvalidArgumentError("x25519: zero keys only, randomness source might be corrupt")
}
_, err := io.ReadFull(rand, privateKey[:])
if err != nil {
return err
}
isZero = constantTimeIsZero(privateKey[:])
}
x25519lib.KeyGen(publicKey, privateKey)
return nil
}
// Encrypt encrypts a sessionKey with x25519 according to
// the OpenPGP crypto refresh specification section 5.1.6. The function assumes that the
// sessionKey has the correct format and padding according to the specification.
func Encrypt(rand io.Reader, publicKey *PublicKey, sessionKey []byte) (ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, err error) {
var ephemeralPrivate, ephemeralPublic, staticPublic, shared x25519lib.Key
// Check that the input static public key has 32 bytes
if len(publicKey.Point) != KeySize {
err = errors.KeyInvalidError("x25519: the public key has the wrong size")
return
}
copy(staticPublic[:], publicKey.Point)
// Generate ephemeral keyPair
err = generateKey(rand, &ephemeralPrivate, &ephemeralPublic)
if err != nil {
return
}
// Compute shared key
ok := x25519lib.Shared(&shared, &ephemeralPrivate, &staticPublic)
if !ok {
err = errors.KeyInvalidError("x25519: the public key is a low order point")
return
}
// Derive the encryption key from the shared secret
encryptionKey := applyHKDF(ephemeralPublic[:], publicKey.Point[:], shared[:])
ephemeralPublicKey = &PublicKey{
Point: ephemeralPublic[:],
}
// Encrypt the sessionKey with aes key wrapping
encryptedSessionKey, err = keywrap.Wrap(encryptionKey, sessionKey)
return
}
// Decrypt decrypts a session key stored in ciphertext with the provided x25519
// private key and ephemeral public key.
func Decrypt(privateKey *PrivateKey, ephemeralPublicKey *PublicKey, ciphertext []byte) (encodedSessionKey []byte, err error) {
var ephemeralPublic, staticPrivate, shared x25519lib.Key
// Check that the input ephemeral public key has 32 bytes
if len(ephemeralPublicKey.Point) != KeySize {
err = errors.KeyInvalidError("x25519: the public key has the wrong size")
return
}
copy(ephemeralPublic[:], ephemeralPublicKey.Point)
subtle.ConstantTimeCopy(1, staticPrivate[:], privateKey.Secret)
// Compute shared key
ok := x25519lib.Shared(&shared, &staticPrivate, &ephemeralPublic)
if !ok {
err = errors.KeyInvalidError("x25519: the ephemeral public key is a low order point")
return
}
// Derive the encryption key from the shared secret
encryptionKey := applyHKDF(ephemeralPublicKey.Point[:], privateKey.PublicKey.Point[:], shared[:])
// Decrypt the session key with aes key wrapping
encodedSessionKey, err = keywrap.Unwrap(encryptionKey, ciphertext)
return
}
func applyHKDF(ephemeralPublicKey []byte, publicKey []byte, sharedSecret []byte) []byte {
inputKey := make([]byte, 3*KeySize)
// ephemeral public key | recipient public key | shared secret
subtle.ConstantTimeCopy(1, inputKey[:KeySize], ephemeralPublicKey)
subtle.ConstantTimeCopy(1, inputKey[KeySize:2*KeySize], publicKey)
subtle.ConstantTimeCopy(1, inputKey[2*KeySize:], sharedSecret)
hkdfReader := hkdf.New(sha256.New, inputKey, []byte{}, []byte(hkdfInfo))
encryptionKey := make([]byte, aes128KeySize)
_, _ = io.ReadFull(hkdfReader, encryptionKey)
return encryptionKey
}
func constantTimeIsZero(bytes []byte) bool {
isZero := byte(0)
for _, b := range bytes {
isZero |= b
}
return isZero == 0
}
// ENCODING/DECODING ciphertexts:
// EncodeFieldsLength returns the length of the ciphertext encoding
// given the encrypted session key.
func EncodedFieldsLength(encryptedSessionKey []byte, v6 bool) int {
lenCipherFunction := 0
if !v6 {
lenCipherFunction = 1
}
return KeySize + 1 + len(encryptedSessionKey) + lenCipherFunction
}
// EncodeField encodes x25519 session key encryption fields as
// ephemeral x25519 public key | follow byte length | cipherFunction (v3 only) | encryptedSessionKey
// and writes it to writer.
func EncodeFields(writer io.Writer, ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, cipherFunction byte, v6 bool) (err error) {
lenAlgorithm := 0
if !v6 {
lenAlgorithm = 1
}
if _, err = writer.Write(ephemeralPublicKey.Point); err != nil {
return err
}
if _, err = writer.Write([]byte{byte(len(encryptedSessionKey) + lenAlgorithm)}); err != nil {
return err
}
if !v6 {
if _, err = writer.Write([]byte{cipherFunction}); err != nil {
return err
}
}
_, err = writer.Write(encryptedSessionKey)
return err
}
// DecodeField decodes a x25519 session key encryption as
// ephemeral x25519 public key | follow byte length | cipherFunction (v3 only) | encryptedSessionKey.
func DecodeFields(reader io.Reader, v6 bool) (ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, cipherFunction byte, err error) {
var buf [1]byte
ephemeralPublicKey = &PublicKey{
Point: make([]byte, KeySize),
}
// 32 octets representing an ephemeral x25519 public key.
if _, err = io.ReadFull(reader, ephemeralPublicKey.Point); err != nil {
return nil, nil, 0, err
}
// A one-octet size of the following fields.
if _, err = io.ReadFull(reader, buf[:]); err != nil {
return nil, nil, 0, err
}
followingLen := buf[0]
// The one-octet algorithm identifier, if it was passed (in the case of a v3 PKESK packet).
if !v6 {
if _, err = io.ReadFull(reader, buf[:]); err != nil {
return nil, nil, 0, err
}
cipherFunction = buf[0]
followingLen -= 1
}
// The encrypted session key.
encryptedSessionKey = make([]byte, followingLen)
if _, err = io.ReadFull(reader, encryptedSessionKey); err != nil {
return nil, nil, 0, err
}
return ephemeralPublicKey, encryptedSessionKey, cipherFunction, nil
}
package x448
import (
"crypto/sha512"
"crypto/subtle"
"io"
"github.com/ProtonMail/go-crypto/openpgp/aes/keywrap"
"github.com/ProtonMail/go-crypto/openpgp/errors"
x448lib "github.com/cloudflare/circl/dh/x448"
"golang.org/x/crypto/hkdf"
)
const (
hkdfInfo = "OpenPGP X448"
aes256KeySize = 32
// The size of a public or private key in bytes.
KeySize = x448lib.Size
)
type PublicKey struct {
// Point represents the encoded elliptic curve point of the public key.
Point []byte
}
type PrivateKey struct {
PublicKey
// Secret represents the secret of the private key.
Secret []byte
}
// NewPrivateKey creates a new empty private key including the public key.
func NewPrivateKey(key PublicKey) *PrivateKey {
return &PrivateKey{
PublicKey: key,
}
}
// Validate validates that the provided public key matches
// the private key.
func Validate(pk *PrivateKey) (err error) {
var expectedPublicKey, privateKey x448lib.Key
subtle.ConstantTimeCopy(1, privateKey[:], pk.Secret)
x448lib.KeyGen(&expectedPublicKey, &privateKey)
if subtle.ConstantTimeCompare(expectedPublicKey[:], pk.PublicKey.Point) == 0 {
return errors.KeyInvalidError("x448: invalid key")
}
return nil
}
// GenerateKey generates a new x448 key pair.
func GenerateKey(rand io.Reader) (*PrivateKey, error) {
var privateKey, publicKey x448lib.Key
privateKeyOut := new(PrivateKey)
err := generateKey(rand, &privateKey, &publicKey)
if err != nil {
return nil, err
}
privateKeyOut.PublicKey.Point = publicKey[:]
privateKeyOut.Secret = privateKey[:]
return privateKeyOut, nil
}
func generateKey(rand io.Reader, privateKey *x448lib.Key, publicKey *x448lib.Key) error {
maxRounds := 10
isZero := true
for round := 0; isZero; round++ {
if round == maxRounds {
return errors.InvalidArgumentError("x448: zero keys only, randomness source might be corrupt")
}
_, err := io.ReadFull(rand, privateKey[:])
if err != nil {
return err
}
isZero = constantTimeIsZero(privateKey[:])
}
x448lib.KeyGen(publicKey, privateKey)
return nil
}
// Encrypt encrypts a sessionKey with x448 according to
// the OpenPGP crypto refresh specification section 5.1.7. The function assumes that the
// sessionKey has the correct format and padding according to the specification.
func Encrypt(rand io.Reader, publicKey *PublicKey, sessionKey []byte) (ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, err error) {
var ephemeralPrivate, ephemeralPublic, staticPublic, shared x448lib.Key
// Check that the input static public key has 56 bytes.
if len(publicKey.Point) != KeySize {
err = errors.KeyInvalidError("x448: the public key has the wrong size")
return nil, nil, err
}
copy(staticPublic[:], publicKey.Point)
// Generate ephemeral keyPair.
if err = generateKey(rand, &ephemeralPrivate, &ephemeralPublic); err != nil {
return nil, nil, err
}
// Compute shared key.
ok := x448lib.Shared(&shared, &ephemeralPrivate, &staticPublic)
if !ok {
err = errors.KeyInvalidError("x448: the public key is a low order point")
return nil, nil, err
}
// Derive the encryption key from the shared secret.
encryptionKey := applyHKDF(ephemeralPublic[:], publicKey.Point[:], shared[:])
ephemeralPublicKey = &PublicKey{
Point: ephemeralPublic[:],
}
// Encrypt the sessionKey with aes key wrapping.
encryptedSessionKey, err = keywrap.Wrap(encryptionKey, sessionKey)
if err != nil {
return nil, nil, err
}
return ephemeralPublicKey, encryptedSessionKey, nil
}
// Decrypt decrypts a session key stored in ciphertext with the provided x448
// private key and ephemeral public key.
func Decrypt(privateKey *PrivateKey, ephemeralPublicKey *PublicKey, ciphertext []byte) (encodedSessionKey []byte, err error) {
var ephemeralPublic, staticPrivate, shared x448lib.Key
// Check that the input ephemeral public key has 56 bytes.
if len(ephemeralPublicKey.Point) != KeySize {
err = errors.KeyInvalidError("x448: the public key has the wrong size")
return nil, err
}
copy(ephemeralPublic[:], ephemeralPublicKey.Point)
subtle.ConstantTimeCopy(1, staticPrivate[:], privateKey.Secret)
// Compute shared key.
ok := x448lib.Shared(&shared, &staticPrivate, &ephemeralPublic)
if !ok {
err = errors.KeyInvalidError("x448: the ephemeral public key is a low order point")
return nil, err
}
// Derive the encryption key from the shared secret.
encryptionKey := applyHKDF(ephemeralPublicKey.Point[:], privateKey.PublicKey.Point[:], shared[:])
// Decrypt the session key with aes key wrapping.
encodedSessionKey, err = keywrap.Unwrap(encryptionKey, ciphertext)
if err != nil {
return nil, err
}
return encodedSessionKey, nil
}
func applyHKDF(ephemeralPublicKey []byte, publicKey []byte, sharedSecret []byte) []byte {
inputKey := make([]byte, 3*KeySize)
// ephemeral public key | recipient public key | shared secret.
subtle.ConstantTimeCopy(1, inputKey[:KeySize], ephemeralPublicKey)
subtle.ConstantTimeCopy(1, inputKey[KeySize:2*KeySize], publicKey)
subtle.ConstantTimeCopy(1, inputKey[2*KeySize:], sharedSecret)
hkdfReader := hkdf.New(sha512.New, inputKey, []byte{}, []byte(hkdfInfo))
encryptionKey := make([]byte, aes256KeySize)
_, _ = io.ReadFull(hkdfReader, encryptionKey)
return encryptionKey
}
func constantTimeIsZero(bytes []byte) bool {
isZero := byte(0)
for _, b := range bytes {
isZero |= b
}
return isZero == 0
}
// ENCODING/DECODING ciphertexts:
// EncodeFieldsLength returns the length of the ciphertext encoding
// given the encrypted session key.
func EncodedFieldsLength(encryptedSessionKey []byte, v6 bool) int {
lenCipherFunction := 0
if !v6 {
lenCipherFunction = 1
}
return KeySize + 1 + len(encryptedSessionKey) + lenCipherFunction
}
// EncodeField encodes x448 session key encryption fields as
// ephemeral x448 public key | follow byte length | cipherFunction (v3 only) | encryptedSessionKey
// and writes it to writer.
func EncodeFields(writer io.Writer, ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, cipherFunction byte, v6 bool) (err error) {
lenAlgorithm := 0
if !v6 {
lenAlgorithm = 1
}
if _, err = writer.Write(ephemeralPublicKey.Point); err != nil {
return err
}
if _, err = writer.Write([]byte{byte(len(encryptedSessionKey) + lenAlgorithm)}); err != nil {
return err
}
if !v6 {
if _, err = writer.Write([]byte{cipherFunction}); err != nil {
return err
}
}
if _, err = writer.Write(encryptedSessionKey); err != nil {
return err
}
return nil
}
// DecodeField decodes a x448 session key encryption as
// ephemeral x448 public key | follow byte length | cipherFunction (v3 only) | encryptedSessionKey.
func DecodeFields(reader io.Reader, v6 bool) (ephemeralPublicKey *PublicKey, encryptedSessionKey []byte, cipherFunction byte, err error) {
var buf [1]byte
ephemeralPublicKey = &PublicKey{
Point: make([]byte, KeySize),
}
// 56 octets representing an ephemeral x448 public key.
if _, err = io.ReadFull(reader, ephemeralPublicKey.Point); err != nil {
return nil, nil, 0, err
}
// A one-octet size of the following fields.
if _, err = io.ReadFull(reader, buf[:]); err != nil {
return nil, nil, 0, err
}
followingLen := buf[0]
// The one-octet algorithm identifier, if it was passed (in the case of a v3 PKESK packet).
if !v6 {
if _, err = io.ReadFull(reader, buf[:]); err != nil {
return nil, nil, 0, err
}
cipherFunction = buf[0]
followingLen -= 1
}
// The encrypted session key.
encryptedSessionKey = make([]byte, followingLen)
if _, err = io.ReadFull(reader, encryptedSessionKey); err != nil {
return nil, nil, 0, err
}
return ephemeralPublicKey, encryptedSessionKey, cipherFunction, nil
}
Copyright (c) 2019 Cloudflare. All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the
distribution.
* Neither the name of Cloudflare nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
========================================================================
Copyright (c) 2009 The Go Authors. All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the
distribution.
* Neither the name of Google Inc. nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
package x25519
import (
fp "github.com/cloudflare/circl/math/fp25519"
)
// ladderJoye calculates a fixed-point multiplication with the generator point.
// The algorithm is the right-to-left Joye's ladder as described
// in "How to precompute a ladder" in SAC'2017.
func ladderJoye(k *Key) {
w := [5]fp.Elt{} // [mu,x1,z1,x2,z2] order must be preserved.
fp.SetOne(&w[1]) // x1 = 1
fp.SetOne(&w[2]) // z1 = 1
w[3] = fp.Elt{ // x2 = G-S
0xbd, 0xaa, 0x2f, 0xc8, 0xfe, 0xe1, 0x94, 0x7e,
0xf8, 0xed, 0xb2, 0x14, 0xae, 0x95, 0xf0, 0xbb,
0xe2, 0x48, 0x5d, 0x23, 0xb9, 0xa0, 0xc7, 0xad,
0x34, 0xab, 0x7c, 0xe2, 0xee, 0xcd, 0xae, 0x1e,
}
fp.SetOne(&w[4]) // z2 = 1
const n = 255
const h = 3
swap := uint(1)
for s := 0; s < n-h; s++ {
i := (s + h) / 8
j := (s + h) % 8
bit := uint((k[i] >> uint(j)) & 1)
copy(w[0][:], tableGenerator[s*Size:(s+1)*Size])
diffAdd(&w, swap^bit)
swap = bit
}
for s := 0; s < h; s++ {
double(&w[1], &w[2])
}
toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
}
// ladderMontgomery calculates a generic scalar point multiplication
// The algorithm implemented is the left-to-right Montgomery's ladder.
func ladderMontgomery(k, xP *Key) {
w := [5]fp.Elt{} // [x1, x2, z2, x3, z3] order must be preserved.
w[0] = *(*fp.Elt)(xP) // x1 = xP
fp.SetOne(&w[1]) // x2 = 1
w[3] = *(*fp.Elt)(xP) // x3 = xP
fp.SetOne(&w[4]) // z3 = 1
move := uint(0)
for s := 255 - 1; s >= 0; s-- {
i := s / 8
j := s % 8
bit := uint((k[i] >> uint(j)) & 1)
ladderStep(&w, move^bit)
move = bit
}
toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
}
func toAffine(k *[fp.Size]byte, x, z *fp.Elt) {
fp.Inv(z, z)
fp.Mul(x, x, z)
_ = fp.ToBytes(k[:], x)
}
var lowOrderPoints = [5]fp.Elt{
{ /* (0,_,1) point of order 2 on Curve25519 */
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
},
{ /* (1,_,1) point of order 4 on Curve25519 */
0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
},
{ /* (x,_,1) first point of order 8 on Curve25519 */
0xe0, 0xeb, 0x7a, 0x7c, 0x3b, 0x41, 0xb8, 0xae,
0x16, 0x56, 0xe3, 0xfa, 0xf1, 0x9f, 0xc4, 0x6a,
0xda, 0x09, 0x8d, 0xeb, 0x9c, 0x32, 0xb1, 0xfd,
0x86, 0x62, 0x05, 0x16, 0x5f, 0x49, 0xb8, 0x00,
},
{ /* (x,_,1) second point of order 8 on Curve25519 */
0x5f, 0x9c, 0x95, 0xbc, 0xa3, 0x50, 0x8c, 0x24,
0xb1, 0xd0, 0xb1, 0x55, 0x9c, 0x83, 0xef, 0x5b,
0x04, 0x44, 0x5c, 0xc4, 0x58, 0x1c, 0x8e, 0x86,
0xd8, 0x22, 0x4e, 0xdd, 0xd0, 0x9f, 0x11, 0x57,
},
{ /* (-1,_,1) a point of order 4 on the twist of Curve25519 */
0xec, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f,
},
}
//go:build amd64 && !purego
// +build amd64,!purego
package x25519
import (
fp "github.com/cloudflare/circl/math/fp25519"
"golang.org/x/sys/cpu"
)
var hasBmi2Adx = cpu.X86.HasBMI2 && cpu.X86.HasADX
var _ = hasBmi2Adx
func double(x, z *fp.Elt) { doubleAmd64(x, z) }
func diffAdd(w *[5]fp.Elt, b uint) { diffAddAmd64(w, b) }
func ladderStep(w *[5]fp.Elt, b uint) { ladderStepAmd64(w, b) }
func mulA24(z, x *fp.Elt) { mulA24Amd64(z, x) }
//go:noescape
func ladderStepAmd64(w *[5]fp.Elt, b uint)
//go:noescape
func diffAddAmd64(w *[5]fp.Elt, b uint)
//go:noescape
func doubleAmd64(x, z *fp.Elt)
//go:noescape
func mulA24Amd64(z, x *fp.Elt)
#define ladderStepLeg \
addSub(x2,z2) \
addSub(x3,z3) \
integerMulLeg(b0,x2,z3) \
integerMulLeg(b1,x3,z2) \
reduceFromDoubleLeg(t0,b0) \
reduceFromDoubleLeg(t1,b1) \
addSub(t0,t1) \
cselect(x2,x3,regMove) \
cselect(z2,z3,regMove) \
integerSqrLeg(b0,t0) \
integerSqrLeg(b1,t1) \
reduceFromDoubleLeg(x3,b0) \
reduceFromDoubleLeg(z3,b1) \
integerMulLeg(b0,x1,z3) \
reduceFromDoubleLeg(z3,b0) \
integerSqrLeg(b0,x2) \
integerSqrLeg(b1,z2) \
reduceFromDoubleLeg(x2,b0) \
reduceFromDoubleLeg(z2,b1) \
subtraction(t0,x2,z2) \
multiplyA24Leg(t1,t0) \
additionLeg(t1,t1,z2) \
integerMulLeg(b0,x2,z2) \
integerMulLeg(b1,t0,t1) \
reduceFromDoubleLeg(x2,b0) \
reduceFromDoubleLeg(z2,b1)
#define ladderStepBmi2Adx \
addSub(x2,z2) \
addSub(x3,z3) \
integerMulAdx(b0,x2,z3) \
integerMulAdx(b1,x3,z2) \
reduceFromDoubleAdx(t0,b0) \
reduceFromDoubleAdx(t1,b1) \
addSub(t0,t1) \
cselect(x2,x3,regMove) \
cselect(z2,z3,regMove) \
integerSqrAdx(b0,t0) \
integerSqrAdx(b1,t1) \
reduceFromDoubleAdx(x3,b0) \
reduceFromDoubleAdx(z3,b1) \
integerMulAdx(b0,x1,z3) \
reduceFromDoubleAdx(z3,b0) \
integerSqrAdx(b0,x2) \
integerSqrAdx(b1,z2) \
reduceFromDoubleAdx(x2,b0) \
reduceFromDoubleAdx(z2,b1) \
subtraction(t0,x2,z2) \
multiplyA24Adx(t1,t0) \
additionAdx(t1,t1,z2) \
integerMulAdx(b0,x2,z2) \
integerMulAdx(b1,t0,t1) \
reduceFromDoubleAdx(x2,b0) \
reduceFromDoubleAdx(z2,b1)
#define difAddLeg \
addSub(x1,z1) \
integerMulLeg(b0,z1,ui) \
reduceFromDoubleLeg(z1,b0) \
addSub(x1,z1) \
integerSqrLeg(b0,x1) \
integerSqrLeg(b1,z1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1) \
integerMulLeg(b0,x1,z2) \
integerMulLeg(b1,z1,x2) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1)
#define difAddBmi2Adx \
addSub(x1,z1) \
integerMulAdx(b0,z1,ui) \
reduceFromDoubleAdx(z1,b0) \
addSub(x1,z1) \
integerSqrAdx(b0,x1) \
integerSqrAdx(b1,z1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1) \
integerMulAdx(b0,x1,z2) \
integerMulAdx(b1,z1,x2) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1)
#define doubleLeg \
addSub(x1,z1) \
integerSqrLeg(b0,x1) \
integerSqrLeg(b1,z1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1) \
subtraction(t0,x1,z1) \
multiplyA24Leg(t1,t0) \
additionLeg(t1,t1,z1) \
integerMulLeg(b0,x1,z1) \
integerMulLeg(b1,t0,t1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1)
#define doubleBmi2Adx \
addSub(x1,z1) \
integerSqrAdx(b0,x1) \
integerSqrAdx(b1,z1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1) \
subtraction(t0,x1,z1) \
multiplyA24Adx(t1,t0) \
additionAdx(t1,t1,z1) \
integerMulAdx(b0,x1,z1) \
integerMulAdx(b1,t0,t1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1)
//go:build amd64 && !purego
// +build amd64,!purego
#include "textflag.h"
// Depends on circl/math/fp25519 package
#include "../../math/fp25519/fp_amd64.h"
#include "curve_amd64.h"
// CTE_A24 is (A+2)/4 from Curve25519
#define CTE_A24 121666
#define Size 32
// multiplyA24Leg multiplies x times CTE_A24 and stores in z
// Uses: AX, DX, R8-R13, FLAGS
// Instr: x86_64, cmov
#define multiplyA24Leg(z,x) \
MOVL $CTE_A24, AX; MULQ 0+x; MOVQ AX, R8; MOVQ DX, R9; \
MOVL $CTE_A24, AX; MULQ 8+x; MOVQ AX, R12; MOVQ DX, R10; \
MOVL $CTE_A24, AX; MULQ 16+x; MOVQ AX, R13; MOVQ DX, R11; \
MOVL $CTE_A24, AX; MULQ 24+x; \
ADDQ R12, R9; \
ADCQ R13, R10; \
ADCQ AX, R11; \
ADCQ $0, DX; \
MOVL $38, AX; /* 2*C = 38 = 2^256 MOD 2^255-19*/ \
IMULQ AX, DX; \
ADDQ DX, R8; \
ADCQ $0, R9; MOVQ R9, 8+z; \
ADCQ $0, R10; MOVQ R10, 16+z; \
ADCQ $0, R11; MOVQ R11, 24+z; \
MOVQ $0, DX; \
CMOVQCS AX, DX; \
ADDQ DX, R8; MOVQ R8, 0+z;
// multiplyA24Adx multiplies x times CTE_A24 and stores in z
// Uses: AX, DX, R8-R12, FLAGS
// Instr: x86_64, cmov, bmi2
#define multiplyA24Adx(z,x) \
MOVQ $CTE_A24, DX; \
MULXQ 0+x, R8, R10; \
MULXQ 8+x, R9, R11; ADDQ R10, R9; \
MULXQ 16+x, R10, AX; ADCQ R11, R10; \
MULXQ 24+x, R11, R12; ADCQ AX, R11; \
;;;;;;;;;;;;;;;;;;;;; ADCQ $0, R12; \
MOVL $38, DX; /* 2*C = 38 = 2^256 MOD 2^255-19*/ \
IMULQ DX, R12; \
ADDQ R12, R8; \
ADCQ $0, R9; MOVQ R9, 8+z; \
ADCQ $0, R10; MOVQ R10, 16+z; \
ADCQ $0, R11; MOVQ R11, 24+z; \
MOVQ $0, R12; \
CMOVQCS DX, R12; \
ADDQ R12, R8; MOVQ R8, 0+z;
#define mulA24Legacy \
multiplyA24Leg(0(DI),0(SI))
#define mulA24Bmi2Adx \
multiplyA24Adx(0(DI),0(SI))
// func mulA24Amd64(z, x *fp255.Elt)
TEXT ·mulA24Amd64(SB),NOSPLIT,$0-16
MOVQ z+0(FP), DI
MOVQ x+8(FP), SI
CHECK_BMI2ADX(LMA24, mulA24Legacy, mulA24Bmi2Adx)
// func ladderStepAmd64(w *[5]fp255.Elt, b uint)
// ladderStepAmd64 calculates a point addition and doubling as follows:
// (x2,z2) = 2*(x2,z2) and (x3,z3) = (x2,z2)+(x3,z3) using as a difference (x1,-).
// work = (x1,x2,z2,x3,z3) are five fp255.Elt of 32 bytes.
// stack = (t0,t1) are two fp.Elt of fp.Size bytes, and
// (b0,b1) are two-double precision fp.Elt of 2*fp.Size bytes.
TEXT ·ladderStepAmd64(SB),NOSPLIT,$192-16
// Parameters
#define regWork DI
#define regMove SI
#define x1 0*Size(regWork)
#define x2 1*Size(regWork)
#define z2 2*Size(regWork)
#define x3 3*Size(regWork)
#define z3 4*Size(regWork)
// Local variables
#define t0 0*Size(SP)
#define t1 1*Size(SP)
#define b0 2*Size(SP)
#define b1 4*Size(SP)
MOVQ w+0(FP), regWork
MOVQ b+8(FP), regMove
CHECK_BMI2ADX(LLADSTEP, ladderStepLeg, ladderStepBmi2Adx)
#undef regWork
#undef regMove
#undef x1
#undef x2
#undef z2
#undef x3
#undef z3
#undef t0
#undef t1
#undef b0
#undef b1
// func diffAddAmd64(w *[5]fp255.Elt, b uint)
// diffAddAmd64 calculates a differential point addition using a precomputed point.
// (x1,z1) = (x1,z1)+(mu) using a difference point (x2,z2)
// w = (mu,x1,z1,x2,z2) are five fp.Elt, and
// stack = (b0,b1) are two-double precision fp.Elt of 2*fp.Size bytes.
TEXT ·diffAddAmd64(SB),NOSPLIT,$128-16
// Parameters
#define regWork DI
#define regSwap SI
#define ui 0*Size(regWork)
#define x1 1*Size(regWork)
#define z1 2*Size(regWork)
#define x2 3*Size(regWork)
#define z2 4*Size(regWork)
// Local variables
#define b0 0*Size(SP)
#define b1 2*Size(SP)
MOVQ w+0(FP), regWork
MOVQ b+8(FP), regSwap
cswap(x1,x2,regSwap)
cswap(z1,z2,regSwap)
CHECK_BMI2ADX(LDIFADD, difAddLeg, difAddBmi2Adx)
#undef regWork
#undef regSwap
#undef ui
#undef x1
#undef z1
#undef x2
#undef z2
#undef b0
#undef b1
// func doubleAmd64(x, z *fp255.Elt)
// doubleAmd64 calculates a point doubling (x1,z1) = 2*(x1,z1).
// stack = (t0,t1) are two fp.Elt of fp.Size bytes, and
// (b0,b1) are two-double precision fp.Elt of 2*fp.Size bytes.
TEXT ·doubleAmd64(SB),NOSPLIT,$192-16
// Parameters
#define x1 0(DI)
#define z1 0(SI)
// Local variables
#define t0 0*Size(SP)
#define t1 1*Size(SP)
#define b0 2*Size(SP)
#define b1 4*Size(SP)
MOVQ x+0(FP), DI
MOVQ z+8(FP), SI
CHECK_BMI2ADX(LDOUB,doubleLeg,doubleBmi2Adx)
#undef x1
#undef z1
#undef t0
#undef t1
#undef b0
#undef b1
package x25519
import (
"encoding/binary"
"math/bits"
fp "github.com/cloudflare/circl/math/fp25519"
)
func doubleGeneric(x, z *fp.Elt) {
t0, t1 := &fp.Elt{}, &fp.Elt{}
fp.AddSub(x, z)
fp.Sqr(x, x)
fp.Sqr(z, z)
fp.Sub(t0, x, z)
mulA24Generic(t1, t0)
fp.Add(t1, t1, z)
fp.Mul(x, x, z)
fp.Mul(z, t0, t1)
}
func diffAddGeneric(w *[5]fp.Elt, b uint) {
mu, x1, z1, x2, z2 := &w[0], &w[1], &w[2], &w[3], &w[4]
fp.Cswap(x1, x2, b)
fp.Cswap(z1, z2, b)
fp.AddSub(x1, z1)
fp.Mul(z1, z1, mu)
fp.AddSub(x1, z1)
fp.Sqr(x1, x1)
fp.Sqr(z1, z1)
fp.Mul(x1, x1, z2)
fp.Mul(z1, z1, x2)
}
func ladderStepGeneric(w *[5]fp.Elt, b uint) {
x1, x2, z2, x3, z3 := &w[0], &w[1], &w[2], &w[3], &w[4]
t0 := &fp.Elt{}
t1 := &fp.Elt{}
fp.AddSub(x2, z2)
fp.AddSub(x3, z3)
fp.Mul(t0, x2, z3)
fp.Mul(t1, x3, z2)
fp.AddSub(t0, t1)
fp.Cmov(x2, x3, b)
fp.Cmov(z2, z3, b)
fp.Sqr(x3, t0)
fp.Sqr(z3, t1)
fp.Mul(z3, x1, z3)
fp.Sqr(x2, x2)
fp.Sqr(z2, z2)
fp.Sub(t0, x2, z2)
mulA24Generic(t1, t0)
fp.Add(t1, t1, z2)
fp.Mul(x2, x2, z2)
fp.Mul(z2, t0, t1)
}
func mulA24Generic(z, x *fp.Elt) {
const A24 = 121666
const n = 8
var xx [4]uint64
for i := range xx {
xx[i] = binary.LittleEndian.Uint64(x[i*n : (i+1)*n])
}
h0, l0 := bits.Mul64(xx[0], A24)
h1, l1 := bits.Mul64(xx[1], A24)
h2, l2 := bits.Mul64(xx[2], A24)
h3, l3 := bits.Mul64(xx[3], A24)
var c3 uint64
l1, c0 := bits.Add64(h0, l1, 0)
l2, c1 := bits.Add64(h1, l2, c0)
l3, c2 := bits.Add64(h2, l3, c1)
l4, _ := bits.Add64(h3, 0, c2)
_, l4 = bits.Mul64(l4, 38)
l0, c0 = bits.Add64(l0, l4, 0)
xx[1], c1 = bits.Add64(l1, 0, c0)
xx[2], c2 = bits.Add64(l2, 0, c1)
xx[3], c3 = bits.Add64(l3, 0, c2)
xx[0], _ = bits.Add64(l0, (-c3)&38, 0)
for i := range xx {
binary.LittleEndian.PutUint64(z[i*n:(i+1)*n], xx[i])
}
}
//go:build !amd64 || purego
// +build !amd64 purego
package x25519
import fp "github.com/cloudflare/circl/math/fp25519"
func double(x, z *fp.Elt) { doubleGeneric(x, z) }
func diffAdd(w *[5]fp.Elt, b uint) { diffAddGeneric(w, b) }
func ladderStep(w *[5]fp.Elt, b uint) { ladderStepGeneric(w, b) }
func mulA24(z, x *fp.Elt) { mulA24Generic(z, x) }
/*
Package x25519 provides Diffie-Hellman functions as specified in RFC-7748.
Validation of public keys.
The Diffie-Hellman function, as described in RFC-7748 [1], works for any
public key. However, if a different protocol requires contributory
behaviour [2,3], then the public keys must be validated against low-order
points [3,4]. To do that, the Shared function performs this validation
internally and returns false when the public key is invalid (i.e., it
is a low-order point).
References:
- [1] RFC7748 by Langley, Hamburg, Turner (https://rfc-editor.org/rfc/rfc7748.txt)
- [2] Curve25519 by Bernstein (https://cr.yp.to/ecdh.html)
- [3] Bernstein (https://cr.yp.to/ecdh.html#validate)
- [4] Cremers&Jackson (https://eprint.iacr.org/2019/526)
*/
package x25519
package x25519
import (
"crypto/subtle"
fp "github.com/cloudflare/circl/math/fp25519"
)
// Size is the length in bytes of a X25519 key.
const Size = 32
// Key represents a X25519 key.
type Key [Size]byte
func (k *Key) clamp(in *Key) *Key {
*k = *in
k[0] &= 248
k[31] = (k[31] & 127) | 64
return k
}
// isValidPubKey verifies if the public key is not a low-order point.
func (k *Key) isValidPubKey() bool {
fp.Modp((*fp.Elt)(k))
var isLowOrder int
for _, P := range lowOrderPoints {
isLowOrder |= subtle.ConstantTimeCompare(P[:], k[:])
}
return isLowOrder == 0
}
// KeyGen obtains a public key given a secret key.
func KeyGen(public, secret *Key) {
ladderJoye(public.clamp(secret))
}
// Shared calculates Alice's shared key from Alice's secret key and Bob's
// public key returning true on success. A failure case happens when the public
// key is a low-order point, thus the shared key is all-zeros and the function
// returns false.
func Shared(shared, secret, public *Key) bool {
validPk := *public
validPk[31] &= (1 << (255 % 8)) - 1
ok := validPk.isValidPubKey()
ladderMontgomery(shared.clamp(secret), &validPk)
return ok
}
package x25519
import "github.com/cloudflare/circl/math/fp25519"
// tableGenerator contains the set of points:
//
// t[i] = (xi+1)/(xi-1),
//
// where (xi,yi) = 2^iG and G is the generator point
// Size = (256)*(256/8) = 8192 bytes.
var tableGenerator = [256 * fp25519.Size]byte{
/* (2^ 0)P */ 0xf3, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x5f,
/* (2^ 1)P */ 0x96, 0xfe, 0xaa, 0x16, 0xf4, 0x20, 0x82, 0x6b, 0x34, 0x6a, 0x56, 0x4f, 0x2b, 0xeb, 0xeb, 0x82, 0x0f, 0x95, 0xa5, 0x75, 0xb0, 0xa5, 0xa9, 0xd5, 0xf4, 0x88, 0x24, 0x4b, 0xcf, 0xb2, 0x42, 0x51,
/* (2^ 2)P */ 0x0c, 0x68, 0x69, 0x00, 0x75, 0xbc, 0xae, 0x6a, 0x41, 0x9c, 0xf9, 0xa0, 0x20, 0x78, 0xcf, 0x89, 0xf4, 0xd0, 0x56, 0x3b, 0x18, 0xd9, 0x58, 0x2a, 0xa4, 0x11, 0x60, 0xe3, 0x80, 0xca, 0x5a, 0x4b,
/* (2^ 3)P */ 0x5d, 0x74, 0x29, 0x8c, 0x34, 0x32, 0x91, 0x32, 0xd7, 0x2f, 0x64, 0xe1, 0x16, 0xe6, 0xa2, 0xf4, 0x34, 0xbc, 0x67, 0xff, 0x03, 0xbb, 0x45, 0x1e, 0x4a, 0x9b, 0x2a, 0xf4, 0xd0, 0x12, 0x69, 0x30,
/* (2^ 4)P */ 0x54, 0x71, 0xaf, 0xe6, 0x07, 0x65, 0x88, 0xff, 0x2f, 0xc8, 0xee, 0xdf, 0x13, 0x0e, 0xf5, 0x04, 0xce, 0xb5, 0xba, 0x2a, 0xe8, 0x2f, 0x51, 0xaa, 0x22, 0xf2, 0xd5, 0x68, 0x1a, 0x25, 0x4e, 0x17,
/* (2^ 5)P */ 0x98, 0x88, 0x02, 0x82, 0x0d, 0x70, 0x96, 0xcf, 0xc5, 0x02, 0x2c, 0x0a, 0x37, 0xe3, 0x43, 0x17, 0xaa, 0x6e, 0xe8, 0xb4, 0x98, 0xec, 0x9e, 0x37, 0x2e, 0x48, 0xe0, 0x51, 0x8a, 0x88, 0x59, 0x0c,
/* (2^ 6)P */ 0x89, 0xd1, 0xb5, 0x99, 0xd6, 0xf1, 0xcb, 0xfb, 0x84, 0xdc, 0x9f, 0x8e, 0xd5, 0xf0, 0xae, 0xac, 0x14, 0x76, 0x1f, 0x23, 0x06, 0x0d, 0xc2, 0xc1, 0x72, 0xf9, 0x74, 0xa2, 0x8d, 0x21, 0x38, 0x29,
/* (2^ 7)P */ 0x18, 0x7f, 0x1d, 0xff, 0xbe, 0x49, 0xaf, 0xf6, 0xc2, 0xc9, 0x7a, 0x38, 0x22, 0x1c, 0x54, 0xcc, 0x6b, 0xc5, 0x15, 0x40, 0xef, 0xc9, 0xfc, 0x96, 0xa9, 0x13, 0x09, 0x69, 0x7c, 0x62, 0xc1, 0x69,
/* (2^ 8)P */ 0x0e, 0xdb, 0x33, 0x47, 0x2f, 0xfd, 0x86, 0x7a, 0xe9, 0x7d, 0x08, 0x9e, 0xf2, 0xc4, 0xb8, 0xfd, 0x29, 0xa2, 0xa2, 0x8e, 0x1a, 0x4b, 0x5e, 0x09, 0x79, 0x7a, 0xb3, 0x29, 0xc8, 0xa7, 0xd7, 0x1a,
/* (2^ 9)P */ 0xc0, 0xa0, 0x7e, 0xd1, 0xca, 0x89, 0x2d, 0x34, 0x51, 0x20, 0xed, 0xcc, 0xa6, 0xdd, 0xbe, 0x67, 0x74, 0x2f, 0xb4, 0x2b, 0xbf, 0x31, 0xca, 0x19, 0xbb, 0xac, 0x80, 0x49, 0xc8, 0xb4, 0xf7, 0x3d,
/* (2^ 10)P */ 0x83, 0xd8, 0x0a, 0xc8, 0x4d, 0x44, 0xc6, 0xa8, 0x85, 0xab, 0xe3, 0x66, 0x03, 0x44, 0x1e, 0xb9, 0xd8, 0xf6, 0x64, 0x01, 0xa0, 0xcd, 0x15, 0xc2, 0x68, 0xe6, 0x47, 0xf2, 0x6e, 0x7c, 0x86, 0x3d,
/* (2^ 11)P */ 0x8c, 0x65, 0x3e, 0xcc, 0x2b, 0x58, 0xdd, 0xc7, 0x28, 0x55, 0x0e, 0xee, 0x48, 0x47, 0x2c, 0xfd, 0x71, 0x4f, 0x9f, 0xcc, 0x95, 0x9b, 0xfd, 0xa0, 0xdf, 0x5d, 0x67, 0xb0, 0x71, 0xd8, 0x29, 0x75,
/* (2^ 12)P */ 0x78, 0xbd, 0x3c, 0x2d, 0xb4, 0x68, 0xf5, 0xb8, 0x82, 0xda, 0xf3, 0x91, 0x1b, 0x01, 0x33, 0x12, 0x62, 0x3b, 0x7c, 0x4a, 0xcd, 0x6c, 0xce, 0x2d, 0x03, 0x86, 0x49, 0x9e, 0x8e, 0xfc, 0xe7, 0x75,
/* (2^ 13)P */ 0xec, 0xb6, 0xd0, 0xfc, 0xf1, 0x13, 0x4f, 0x2f, 0x45, 0x7a, 0xff, 0x29, 0x1f, 0xca, 0xa8, 0xf1, 0x9b, 0xe2, 0x81, 0x29, 0xa7, 0xc1, 0x49, 0xc2, 0x6a, 0xb5, 0x83, 0x8c, 0xbb, 0x0d, 0xbe, 0x6e,
/* (2^ 14)P */ 0x22, 0xb2, 0x0b, 0x17, 0x8d, 0xfa, 0x14, 0x71, 0x5f, 0x93, 0x93, 0xbf, 0xd5, 0xdc, 0xa2, 0x65, 0x9a, 0x97, 0x9c, 0xb5, 0x68, 0x1f, 0xc4, 0xbd, 0x89, 0x92, 0xce, 0xa2, 0x79, 0xef, 0x0e, 0x2f,
/* (2^ 15)P */ 0xce, 0x37, 0x3c, 0x08, 0x0c, 0xbf, 0xec, 0x42, 0x22, 0x63, 0x49, 0xec, 0x09, 0xbc, 0x30, 0x29, 0x0d, 0xac, 0xfe, 0x9c, 0xc1, 0xb0, 0x94, 0xf2, 0x80, 0xbb, 0xfa, 0xed, 0x4b, 0xaa, 0x80, 0x37,
/* (2^ 16)P */ 0x29, 0xd9, 0xea, 0x7c, 0x3e, 0x7d, 0xc1, 0x56, 0xc5, 0x22, 0x57, 0x2e, 0xeb, 0x4b, 0xcb, 0xe7, 0x5a, 0xe1, 0xbf, 0x2d, 0x73, 0x31, 0xe9, 0x0c, 0xf8, 0x52, 0x10, 0x62, 0xc7, 0x83, 0xb8, 0x41,
/* (2^ 17)P */ 0x50, 0x53, 0xd2, 0xc3, 0xa0, 0x5c, 0xf7, 0xdb, 0x51, 0xe3, 0xb1, 0x6e, 0x08, 0xbe, 0x36, 0x29, 0x12, 0xb2, 0xa9, 0xb4, 0x3c, 0xe0, 0x36, 0xc9, 0xaa, 0x25, 0x22, 0x32, 0x82, 0xbf, 0x45, 0x1d,
/* (2^ 18)P */ 0xc5, 0x4c, 0x02, 0x6a, 0x03, 0xb1, 0x1a, 0xe8, 0x72, 0x9a, 0x4c, 0x30, 0x1c, 0x20, 0x12, 0xe2, 0xfc, 0xb1, 0x32, 0x68, 0xba, 0x3f, 0xd7, 0xc5, 0x81, 0x95, 0x83, 0x4d, 0x5a, 0xdb, 0xff, 0x20,
/* (2^ 19)P */ 0xad, 0x0f, 0x5d, 0xbe, 0x67, 0xd3, 0x83, 0xa2, 0x75, 0x44, 0x16, 0x8b, 0xca, 0x25, 0x2b, 0x6c, 0x2e, 0xf2, 0xaa, 0x7c, 0x46, 0x35, 0x49, 0x9d, 0x49, 0xff, 0x85, 0xee, 0x8e, 0x40, 0x66, 0x51,
/* (2^ 20)P */ 0x61, 0xe3, 0xb4, 0xfa, 0xa2, 0xba, 0x67, 0x3c, 0xef, 0x5c, 0xf3, 0x7e, 0xc6, 0x33, 0xe4, 0xb3, 0x1c, 0x9b, 0x15, 0x41, 0x92, 0x72, 0x59, 0x52, 0x33, 0xab, 0xb0, 0xd5, 0x92, 0x18, 0x62, 0x6a,
/* (2^ 21)P */ 0xcb, 0xcd, 0x55, 0x75, 0x38, 0x4a, 0xb7, 0x20, 0x3f, 0x92, 0x08, 0x12, 0x0e, 0xa1, 0x2a, 0x53, 0xd1, 0x1d, 0x28, 0x62, 0x77, 0x7b, 0xa1, 0xea, 0xbf, 0x44, 0x5c, 0xf0, 0x43, 0x34, 0xab, 0x61,
/* (2^ 22)P */ 0xf8, 0xde, 0x24, 0x23, 0x42, 0x6c, 0x7a, 0x25, 0x7f, 0xcf, 0xe3, 0x17, 0x10, 0x6c, 0x1c, 0x13, 0x57, 0xa2, 0x30, 0xf6, 0x39, 0x87, 0x75, 0x23, 0x80, 0x85, 0xa7, 0x01, 0x7a, 0x40, 0x5a, 0x29,
/* (2^ 23)P */ 0xd9, 0xa8, 0x5d, 0x6d, 0x24, 0x43, 0xc4, 0xf8, 0x5d, 0xfa, 0x52, 0x0c, 0x45, 0x75, 0xd7, 0x19, 0x3d, 0xf8, 0x1b, 0x73, 0x92, 0xfc, 0xfc, 0x2a, 0x00, 0x47, 0x2b, 0x1b, 0xe8, 0xc8, 0x10, 0x7d,
/* (2^ 24)P */ 0x0b, 0xa2, 0xba, 0x70, 0x1f, 0x27, 0xe0, 0xc8, 0x57, 0x39, 0xa6, 0x7c, 0x86, 0x48, 0x37, 0x99, 0xbb, 0xd4, 0x7e, 0xcb, 0xb3, 0xef, 0x12, 0x54, 0x75, 0x29, 0xe6, 0x73, 0x61, 0xd3, 0x96, 0x31,
/* (2^ 25)P */ 0xfc, 0xdf, 0xc7, 0x41, 0xd1, 0xca, 0x5b, 0xde, 0x48, 0xc8, 0x95, 0xb3, 0xd2, 0x8c, 0xcc, 0x47, 0xcb, 0xf3, 0x1a, 0xe1, 0x42, 0xd9, 0x4c, 0xa3, 0xc2, 0xce, 0x4e, 0xd0, 0xf2, 0xdb, 0x56, 0x02,
/* (2^ 26)P */ 0x7f, 0x66, 0x0e, 0x4b, 0xe9, 0xb7, 0x5a, 0x87, 0x10, 0x0d, 0x85, 0xc0, 0x83, 0xdd, 0xd4, 0xca, 0x9f, 0xc7, 0x72, 0x4e, 0x8f, 0x2e, 0xf1, 0x47, 0x9b, 0xb1, 0x85, 0x8c, 0xbb, 0x87, 0x1a, 0x5f,
/* (2^ 27)P */ 0xb8, 0x51, 0x7f, 0x43, 0xb6, 0xd0, 0xe9, 0x7a, 0x65, 0x90, 0x87, 0x18, 0x55, 0xce, 0xc7, 0x12, 0xee, 0x7a, 0xf7, 0x5c, 0xfe, 0x09, 0xde, 0x2a, 0x27, 0x56, 0x2c, 0x7d, 0x2f, 0x5a, 0xa0, 0x23,
/* (2^ 28)P */ 0x9a, 0x16, 0x7c, 0xf1, 0x28, 0xe1, 0x08, 0x59, 0x2d, 0x85, 0xd0, 0x8a, 0xdd, 0x98, 0x74, 0xf7, 0x64, 0x2f, 0x10, 0xab, 0xce, 0xc4, 0xb4, 0x74, 0x45, 0x98, 0x13, 0x10, 0xdd, 0xba, 0x3a, 0x18,
/* (2^ 29)P */ 0xac, 0xaa, 0x92, 0xaa, 0x8d, 0xba, 0x65, 0xb1, 0x05, 0x67, 0x38, 0x99, 0x95, 0xef, 0xc5, 0xd5, 0xd1, 0x40, 0xfc, 0xf8, 0x0c, 0x8f, 0x2f, 0xbe, 0x14, 0x45, 0x20, 0xee, 0x35, 0xe6, 0x01, 0x27,
/* (2^ 30)P */ 0x14, 0x65, 0x15, 0x20, 0x00, 0xa8, 0x9f, 0x62, 0xce, 0xc1, 0xa8, 0x64, 0x87, 0x86, 0x23, 0xf2, 0x0e, 0x06, 0x3f, 0x0b, 0xff, 0x4f, 0x89, 0x5b, 0xfa, 0xa3, 0x08, 0xf7, 0x4c, 0x94, 0xd9, 0x60,
/* (2^ 31)P */ 0x1f, 0x20, 0x7a, 0x1c, 0x1a, 0x00, 0xea, 0xae, 0x63, 0xce, 0xe2, 0x3e, 0x63, 0x6a, 0xf1, 0xeb, 0xe1, 0x07, 0x7a, 0x4c, 0x59, 0x09, 0x77, 0x6f, 0xcb, 0x08, 0x02, 0x0d, 0x15, 0x58, 0xb9, 0x79,
/* (2^ 32)P */ 0xe7, 0x10, 0xd4, 0x01, 0x53, 0x5e, 0xb5, 0x24, 0x4d, 0xc8, 0xfd, 0xf3, 0xdf, 0x4e, 0xa3, 0xe3, 0xd8, 0x32, 0x40, 0x90, 0xe4, 0x68, 0x87, 0xd8, 0xec, 0xae, 0x3a, 0x7b, 0x42, 0x84, 0x13, 0x13,
/* (2^ 33)P */ 0x14, 0x4f, 0x23, 0x86, 0x12, 0xe5, 0x05, 0x84, 0x29, 0xc5, 0xb4, 0xad, 0x39, 0x47, 0xdc, 0x14, 0xfd, 0x4f, 0x63, 0x50, 0xb2, 0xb5, 0xa2, 0xb8, 0x93, 0xff, 0xa7, 0xd8, 0x4a, 0xa9, 0xe2, 0x2f,
/* (2^ 34)P */ 0xdd, 0xfa, 0x43, 0xe8, 0xef, 0x57, 0x5c, 0xec, 0x18, 0x99, 0xbb, 0xf0, 0x40, 0xce, 0x43, 0x28, 0x05, 0x63, 0x3d, 0xcf, 0xd6, 0x61, 0xb5, 0xa4, 0x7e, 0x77, 0xfb, 0xe8, 0xbd, 0x29, 0x36, 0x74,
/* (2^ 35)P */ 0x8f, 0x73, 0xaf, 0xbb, 0x46, 0xdd, 0x3e, 0x34, 0x51, 0xa6, 0x01, 0xb1, 0x28, 0x18, 0x98, 0xed, 0x7a, 0x79, 0x2c, 0x88, 0x0b, 0x76, 0x01, 0xa4, 0x30, 0x87, 0xc8, 0x8d, 0xe2, 0x23, 0xc2, 0x1f,
/* (2^ 36)P */ 0x0e, 0xba, 0x0f, 0xfc, 0x91, 0x4e, 0x60, 0x48, 0xa4, 0x6f, 0x2c, 0x05, 0x8f, 0xf7, 0x37, 0xb6, 0x9c, 0x23, 0xe9, 0x09, 0x3d, 0xac, 0xcc, 0x91, 0x7c, 0x68, 0x7a, 0x43, 0xd4, 0xee, 0xf7, 0x23,
/* (2^ 37)P */ 0x00, 0xd8, 0x9b, 0x8d, 0x11, 0xb1, 0x73, 0x51, 0xa7, 0xd4, 0x89, 0x31, 0xb6, 0x41, 0xd6, 0x29, 0x86, 0xc5, 0xbb, 0x88, 0x79, 0x17, 0xbf, 0xfd, 0xf5, 0x1d, 0xd8, 0xca, 0x4f, 0x89, 0x59, 0x29,
/* (2^ 38)P */ 0x99, 0xc8, 0xbb, 0xb4, 0xf3, 0x8e, 0xbc, 0xae, 0xb9, 0x92, 0x69, 0xb2, 0x5a, 0x99, 0x48, 0x41, 0xfb, 0x2c, 0xf9, 0x34, 0x01, 0x0b, 0xe2, 0x24, 0xe8, 0xde, 0x05, 0x4a, 0x89, 0x58, 0xd1, 0x40,
/* (2^ 39)P */ 0xf6, 0x76, 0xaf, 0x85, 0x11, 0x0b, 0xb0, 0x46, 0x79, 0x7a, 0x18, 0x73, 0x78, 0xc7, 0xba, 0x26, 0x5f, 0xff, 0x8f, 0xab, 0x95, 0xbf, 0xc0, 0x3d, 0xd7, 0x24, 0x55, 0x94, 0xd8, 0x8b, 0x60, 0x2a,
/* (2^ 40)P */ 0x02, 0x63, 0x44, 0xbd, 0x88, 0x95, 0x44, 0x26, 0x9c, 0x43, 0x88, 0x03, 0x1c, 0xc2, 0x4b, 0x7c, 0xb2, 0x11, 0xbd, 0x83, 0xf3, 0xa4, 0x98, 0x8e, 0xb9, 0x76, 0xd8, 0xc9, 0x7b, 0x8d, 0x21, 0x26,
/* (2^ 41)P */ 0x8a, 0x17, 0x7c, 0x99, 0x42, 0x15, 0x08, 0xe3, 0x6f, 0x60, 0xb6, 0x6f, 0xa8, 0x29, 0x2d, 0x3c, 0x74, 0x93, 0x27, 0xfa, 0x36, 0x77, 0x21, 0x5c, 0xfa, 0xb1, 0xfe, 0x4a, 0x73, 0x05, 0xde, 0x7d,
/* (2^ 42)P */ 0xab, 0x2b, 0xd4, 0x06, 0x39, 0x0e, 0xf1, 0x3b, 0x9c, 0x64, 0x80, 0x19, 0x3e, 0x80, 0xf7, 0xe4, 0x7a, 0xbf, 0x95, 0x95, 0xf8, 0x3b, 0x05, 0xe6, 0x30, 0x55, 0x24, 0xda, 0x38, 0xaf, 0x4f, 0x39,
/* (2^ 43)P */ 0xf4, 0x28, 0x69, 0x89, 0x58, 0xfb, 0x8e, 0x7a, 0x3c, 0x11, 0x6a, 0xcc, 0xe9, 0x78, 0xc7, 0xfb, 0x6f, 0x59, 0xaf, 0x30, 0xe3, 0x0c, 0x67, 0x72, 0xf7, 0x6c, 0x3d, 0x1d, 0xa8, 0x22, 0xf2, 0x48,
/* (2^ 44)P */ 0xa7, 0xca, 0x72, 0x0d, 0x41, 0xce, 0x1f, 0xf0, 0x95, 0x55, 0x3b, 0x21, 0xc7, 0xec, 0x20, 0x5a, 0x83, 0x14, 0xfa, 0xc1, 0x65, 0x11, 0xc2, 0x7b, 0x41, 0xa7, 0xa8, 0x1d, 0xe3, 0x9a, 0xf8, 0x07,
/* (2^ 45)P */ 0xf9, 0x0f, 0x83, 0xc6, 0xb4, 0xc2, 0xd2, 0x05, 0x93, 0x62, 0x31, 0xc6, 0x0f, 0x33, 0x3e, 0xd4, 0x04, 0xa9, 0xd3, 0x96, 0x0a, 0x59, 0xa5, 0xa5, 0xb6, 0x33, 0x53, 0xa6, 0x91, 0xdb, 0x5e, 0x70,
/* (2^ 46)P */ 0xf7, 0xa5, 0xb9, 0x0b, 0x5e, 0xe1, 0x8e, 0x04, 0x5d, 0xaf, 0x0a, 0x9e, 0xca, 0xcf, 0x40, 0x32, 0x0b, 0xa4, 0xc4, 0xed, 0xce, 0x71, 0x4b, 0x8f, 0x6d, 0x4a, 0x54, 0xde, 0xa3, 0x0d, 0x1c, 0x62,
/* (2^ 47)P */ 0x91, 0x40, 0x8c, 0xa0, 0x36, 0x28, 0x87, 0x92, 0x45, 0x14, 0xc9, 0x10, 0xb0, 0x75, 0x83, 0xce, 0x94, 0x63, 0x27, 0x4f, 0x52, 0xeb, 0x72, 0x8a, 0x35, 0x36, 0xc8, 0x7e, 0xfa, 0xfc, 0x67, 0x26,
/* (2^ 48)P */ 0x2a, 0x75, 0xe8, 0x45, 0x33, 0x17, 0x4c, 0x7f, 0xa5, 0x79, 0x70, 0xee, 0xfe, 0x47, 0x1b, 0x06, 0x34, 0xff, 0x86, 0x9f, 0xfa, 0x9a, 0xdd, 0x25, 0x9c, 0xc8, 0x5d, 0x42, 0xf5, 0xce, 0x80, 0x37,
/* (2^ 49)P */ 0xe9, 0xb4, 0x3b, 0x51, 0x5a, 0x03, 0x46, 0x1a, 0xda, 0x5a, 0x57, 0xac, 0x79, 0xf3, 0x1e, 0x3e, 0x50, 0x4b, 0xa2, 0x5f, 0x1c, 0x5f, 0x8c, 0xc7, 0x22, 0x9f, 0xfd, 0x34, 0x76, 0x96, 0x1a, 0x32,
/* (2^ 50)P */ 0xfa, 0x27, 0x6e, 0x82, 0xb8, 0x07, 0x67, 0x94, 0xd0, 0x6f, 0x50, 0x4c, 0xd6, 0x84, 0xca, 0x3d, 0x36, 0x14, 0xe9, 0x75, 0x80, 0x21, 0x89, 0xc1, 0x84, 0x84, 0x3b, 0x9b, 0x16, 0x84, 0x92, 0x6d,
/* (2^ 51)P */ 0xdf, 0x2d, 0x3f, 0x38, 0x40, 0xe8, 0x67, 0x3a, 0x75, 0x9b, 0x4f, 0x0c, 0xa3, 0xc9, 0xee, 0x33, 0x47, 0xef, 0x83, 0xa7, 0x6f, 0xc8, 0xc7, 0x3e, 0xc4, 0xfb, 0xc9, 0xba, 0x9f, 0x44, 0xec, 0x26,
/* (2^ 52)P */ 0x7d, 0x9e, 0x9b, 0xa0, 0xcb, 0x38, 0x0f, 0x5c, 0x8c, 0x47, 0xa3, 0x62, 0xc7, 0x8c, 0x16, 0x81, 0x1c, 0x12, 0xfc, 0x06, 0xd3, 0xb0, 0x23, 0x3e, 0xdd, 0xdc, 0xef, 0xa5, 0xa0, 0x8a, 0x23, 0x5a,
/* (2^ 53)P */ 0xff, 0x43, 0xea, 0xc4, 0x21, 0x61, 0xa2, 0x1b, 0xb5, 0x32, 0x88, 0x7c, 0x7f, 0xc7, 0xf8, 0x36, 0x9a, 0xf9, 0xdc, 0x0a, 0x0b, 0xea, 0xfb, 0x88, 0xf9, 0xeb, 0x5b, 0xc2, 0x8e, 0x93, 0xa9, 0x5c,
/* (2^ 54)P */ 0xa0, 0xcd, 0xfc, 0x51, 0x5e, 0x6a, 0x43, 0xd5, 0x3b, 0x89, 0xcd, 0xc2, 0x97, 0x47, 0xbc, 0x1d, 0x08, 0x4a, 0x22, 0xd3, 0x65, 0x6a, 0x34, 0x19, 0x66, 0xf4, 0x9a, 0x9b, 0xe4, 0x34, 0x50, 0x0f,
/* (2^ 55)P */ 0x6e, 0xb9, 0xe0, 0xa1, 0x67, 0x39, 0x3c, 0xf2, 0x88, 0x4d, 0x7a, 0x86, 0xfa, 0x08, 0x8b, 0xe5, 0x79, 0x16, 0x34, 0xa7, 0xc6, 0xab, 0x2f, 0xfb, 0x46, 0x69, 0x02, 0xb6, 0x1e, 0x38, 0x75, 0x2a,
/* (2^ 56)P */ 0xac, 0x20, 0x94, 0xc1, 0xe4, 0x3b, 0x0a, 0xc8, 0xdc, 0xb6, 0xf2, 0x81, 0xc6, 0xf6, 0xb1, 0x66, 0x88, 0x33, 0xe9, 0x61, 0x67, 0x03, 0xf7, 0x7c, 0xc4, 0xa4, 0x60, 0xa6, 0xd8, 0xbb, 0xab, 0x25,
/* (2^ 57)P */ 0x98, 0x51, 0xfd, 0x14, 0xba, 0x12, 0xea, 0x91, 0xa9, 0xff, 0x3c, 0x4a, 0xfc, 0x50, 0x49, 0x68, 0x28, 0xad, 0xf5, 0x30, 0x21, 0x84, 0x26, 0xf8, 0x41, 0xa4, 0x01, 0x53, 0xf7, 0x88, 0xa9, 0x3e,
/* (2^ 58)P */ 0x6f, 0x8c, 0x5f, 0x69, 0x9a, 0x10, 0x78, 0xc9, 0xf3, 0xc3, 0x30, 0x05, 0x4a, 0xeb, 0x46, 0x17, 0x95, 0x99, 0x45, 0xb4, 0x77, 0x6d, 0x4d, 0x44, 0xc7, 0x5c, 0x4e, 0x05, 0x8c, 0x2b, 0x95, 0x75,
/* (2^ 59)P */ 0xaa, 0xd6, 0xf4, 0x15, 0x79, 0x3f, 0x70, 0xa3, 0xd8, 0x47, 0x26, 0x2f, 0x20, 0x46, 0xc3, 0x66, 0x4b, 0x64, 0x1d, 0x81, 0xdf, 0x69, 0x14, 0xd0, 0x1f, 0xd7, 0xa5, 0x81, 0x7d, 0xa4, 0xfe, 0x77,
/* (2^ 60)P */ 0x81, 0xa3, 0x7c, 0xf5, 0x9e, 0x52, 0xe9, 0xc5, 0x1a, 0x88, 0x2f, 0xce, 0xb9, 0xb4, 0xee, 0x6e, 0xd6, 0x9b, 0x00, 0xe8, 0x28, 0x1a, 0xe9, 0xb6, 0xec, 0x3f, 0xfc, 0x9a, 0x3e, 0xbe, 0x80, 0x4b,
/* (2^ 61)P */ 0xc5, 0xd2, 0xae, 0x26, 0xc5, 0x73, 0x37, 0x7e, 0x9d, 0xa4, 0xc9, 0x53, 0xb4, 0xfc, 0x4a, 0x1b, 0x4d, 0xb2, 0xff, 0xba, 0xd7, 0xbd, 0x20, 0xa9, 0x0e, 0x40, 0x2d, 0x12, 0x9f, 0x69, 0x54, 0x7c,
/* (2^ 62)P */ 0xc8, 0x4b, 0xa9, 0x4f, 0xe1, 0xc8, 0x46, 0xef, 0x5e, 0xed, 0x52, 0x29, 0xce, 0x74, 0xb0, 0xe0, 0xd5, 0x85, 0xd8, 0xdb, 0xe1, 0x50, 0xa4, 0xbe, 0x2c, 0x71, 0x0f, 0x32, 0x49, 0x86, 0xb6, 0x61,
/* (2^ 63)P */ 0xd1, 0xbd, 0xcc, 0x09, 0x73, 0x5f, 0x48, 0x8a, 0x2d, 0x1a, 0x4d, 0x7d, 0x0d, 0x32, 0x06, 0xbd, 0xf4, 0xbe, 0x2d, 0x32, 0x73, 0x29, 0x23, 0x25, 0x70, 0xf7, 0x17, 0x8c, 0x75, 0xc4, 0x5d, 0x44,
/* (2^ 64)P */ 0x3c, 0x93, 0xc8, 0x7c, 0x17, 0x34, 0x04, 0xdb, 0x9f, 0x05, 0xea, 0x75, 0x21, 0xe8, 0x6f, 0xed, 0x34, 0xdb, 0x53, 0xc0, 0xfd, 0xbe, 0xfe, 0x1e, 0x99, 0xaf, 0x5d, 0xc6, 0x67, 0xe8, 0xdb, 0x4a,
/* (2^ 65)P */ 0xdf, 0x09, 0x06, 0xa9, 0xa2, 0x71, 0xcd, 0x3a, 0x50, 0x40, 0xd0, 0x6d, 0x85, 0x91, 0xe9, 0xe5, 0x3c, 0xc2, 0x57, 0x81, 0x68, 0x9b, 0xc6, 0x1e, 0x4d, 0xfe, 0x5c, 0x88, 0xf6, 0x27, 0x74, 0x69,
/* (2^ 66)P */ 0x51, 0xa8, 0xe1, 0x65, 0x9b, 0x7b, 0xbe, 0xd7, 0xdd, 0x36, 0xc5, 0x22, 0xd5, 0x28, 0x3d, 0xa0, 0x45, 0xb6, 0xd2, 0x8f, 0x65, 0x9d, 0x39, 0x28, 0xe1, 0x41, 0x26, 0x7c, 0xe1, 0xb7, 0xe5, 0x49,
/* (2^ 67)P */ 0xa4, 0x57, 0x04, 0x70, 0x98, 0x3a, 0x8c, 0x6f, 0x78, 0x67, 0xbb, 0x5e, 0xa2, 0xf0, 0x78, 0x50, 0x0f, 0x96, 0x82, 0xc3, 0xcb, 0x3c, 0x3c, 0xd1, 0xb1, 0x84, 0xdf, 0xa7, 0x58, 0x32, 0x00, 0x2e,
/* (2^ 68)P */ 0x1c, 0x6a, 0x29, 0xe6, 0x9b, 0xf3, 0xd1, 0x8a, 0xb2, 0xbf, 0x5f, 0x2a, 0x65, 0xaa, 0xee, 0xc1, 0xcb, 0xf3, 0x26, 0xfd, 0x73, 0x06, 0xee, 0x33, 0xcc, 0x2c, 0x9d, 0xa6, 0x73, 0x61, 0x25, 0x59,
/* (2^ 69)P */ 0x41, 0xfc, 0x18, 0x4e, 0xaa, 0x07, 0xea, 0x41, 0x1e, 0xa5, 0x87, 0x7c, 0x52, 0x19, 0xfc, 0xd9, 0x6f, 0xca, 0x31, 0x58, 0x80, 0xcb, 0xaa, 0xbd, 0x4f, 0x69, 0x16, 0xc9, 0x2d, 0x65, 0x5b, 0x44,
/* (2^ 70)P */ 0x15, 0x23, 0x17, 0xf2, 0xa7, 0xa3, 0x92, 0xce, 0x64, 0x99, 0x1b, 0xe1, 0x2d, 0x28, 0xdc, 0x1e, 0x4a, 0x31, 0x4c, 0xe0, 0xaf, 0x3a, 0x82, 0xa1, 0x86, 0xf5, 0x7c, 0x43, 0x94, 0x2d, 0x0a, 0x79,
/* (2^ 71)P */ 0x09, 0xe0, 0xf6, 0x93, 0xfb, 0x47, 0xc4, 0x71, 0x76, 0x52, 0x84, 0x22, 0x67, 0xa5, 0x22, 0x89, 0x69, 0x51, 0x4f, 0x20, 0x3b, 0x90, 0x70, 0xbf, 0xfe, 0x19, 0xa3, 0x1b, 0x89, 0x89, 0x7a, 0x2f,
/* (2^ 72)P */ 0x0c, 0x14, 0xe2, 0x77, 0xb5, 0x8e, 0xa0, 0x02, 0xf4, 0xdc, 0x7b, 0x42, 0xd4, 0x4e, 0x9a, 0xed, 0xd1, 0x3c, 0x32, 0xe4, 0x44, 0xec, 0x53, 0x52, 0x5b, 0x35, 0xe9, 0x14, 0x3c, 0x36, 0x88, 0x3e,
/* (2^ 73)P */ 0x8c, 0x0b, 0x11, 0x77, 0x42, 0xc1, 0x66, 0xaa, 0x90, 0x33, 0xa2, 0x10, 0x16, 0x39, 0xe0, 0x1a, 0xa2, 0xc2, 0x3f, 0xc9, 0x12, 0xbd, 0x30, 0x20, 0xab, 0xc7, 0x55, 0x95, 0x57, 0x41, 0xe1, 0x3e,
/* (2^ 74)P */ 0x41, 0x7d, 0x6e, 0x6d, 0x3a, 0xde, 0x14, 0x92, 0xfe, 0x7e, 0xf1, 0x07, 0x86, 0xd8, 0xcd, 0x3c, 0x17, 0x12, 0xe1, 0xf8, 0x88, 0x12, 0x4f, 0x67, 0xd0, 0x93, 0x9f, 0x32, 0x0f, 0x25, 0x82, 0x56,
/* (2^ 75)P */ 0x6e, 0x39, 0x2e, 0x6d, 0x13, 0x0b, 0xf0, 0x6c, 0xbf, 0xde, 0x14, 0x10, 0x6f, 0xf8, 0x4c, 0x6e, 0x83, 0x4e, 0xcc, 0xbf, 0xb5, 0xb1, 0x30, 0x59, 0xb6, 0x16, 0xba, 0x8a, 0xb4, 0x69, 0x70, 0x04,
/* (2^ 76)P */ 0x93, 0x07, 0xb2, 0x69, 0xab, 0xe4, 0x4c, 0x0d, 0x9e, 0xfb, 0xd0, 0x97, 0x1a, 0xb9, 0x4d, 0xb2, 0x1d, 0xd0, 0x00, 0x4e, 0xf5, 0x50, 0xfa, 0xcd, 0xb5, 0xdd, 0x8b, 0x36, 0x85, 0x10, 0x1b, 0x22,
/* (2^ 77)P */ 0xd2, 0xd8, 0xe3, 0xb1, 0x68, 0x94, 0xe5, 0xe7, 0x93, 0x2f, 0x12, 0xbd, 0x63, 0x65, 0xc5, 0x53, 0x09, 0x3f, 0x66, 0xe0, 0x03, 0xa9, 0xe8, 0xee, 0x42, 0x3d, 0xbe, 0xcb, 0x62, 0xa6, 0xef, 0x61,
/* (2^ 78)P */ 0x2a, 0xab, 0x6e, 0xde, 0xdd, 0xdd, 0xf8, 0x2c, 0x31, 0xf2, 0x35, 0x14, 0xd5, 0x0a, 0xf8, 0x9b, 0x73, 0x49, 0xf0, 0xc9, 0xce, 0xda, 0xea, 0x5d, 0x27, 0x9b, 0xd2, 0x41, 0x5d, 0x5b, 0x27, 0x29,
/* (2^ 79)P */ 0x4f, 0xf1, 0xeb, 0x95, 0x08, 0x0f, 0xde, 0xcf, 0xa7, 0x05, 0x49, 0x05, 0x6b, 0xb9, 0xaa, 0xb9, 0xfd, 0x20, 0xc4, 0xa1, 0xd9, 0x0d, 0xe8, 0xca, 0xc7, 0xbb, 0x73, 0x16, 0x2f, 0xbf, 0x63, 0x0a,
/* (2^ 80)P */ 0x8c, 0xbc, 0x8f, 0x95, 0x11, 0x6e, 0x2f, 0x09, 0xad, 0x2f, 0x82, 0x04, 0xe8, 0x81, 0x2a, 0x67, 0x17, 0x25, 0xd5, 0x60, 0x15, 0x35, 0xc8, 0xca, 0xf8, 0x92, 0xf1, 0xc8, 0x22, 0x77, 0x3f, 0x6f,
/* (2^ 81)P */ 0xb7, 0x94, 0xe8, 0xc2, 0xcc, 0x90, 0xba, 0xf8, 0x0d, 0x9f, 0xff, 0x38, 0xa4, 0x57, 0x75, 0x2c, 0x59, 0x23, 0xe5, 0x5a, 0x85, 0x1d, 0x4d, 0x89, 0x69, 0x3d, 0x74, 0x7b, 0x15, 0x22, 0xe1, 0x68,
/* (2^ 82)P */ 0xf3, 0x19, 0xb9, 0xcf, 0x70, 0x55, 0x7e, 0xd8, 0xb9, 0x8d, 0x79, 0x95, 0xcd, 0xde, 0x2c, 0x3f, 0xce, 0xa2, 0xc0, 0x10, 0x47, 0x15, 0x21, 0x21, 0xb2, 0xc5, 0x6d, 0x24, 0x15, 0xa1, 0x66, 0x3c,
/* (2^ 83)P */ 0x72, 0xcb, 0x4e, 0x29, 0x62, 0xc5, 0xed, 0xcb, 0x16, 0x0b, 0x28, 0x6a, 0xc3, 0x43, 0x71, 0xba, 0x67, 0x8b, 0x07, 0xd4, 0xef, 0xc2, 0x10, 0x96, 0x1e, 0x4b, 0x6a, 0x94, 0x5d, 0x73, 0x44, 0x61,
/* (2^ 84)P */ 0x50, 0x33, 0x5b, 0xd7, 0x1e, 0x11, 0x6f, 0x53, 0x1b, 0xd8, 0x41, 0x20, 0x8c, 0xdb, 0x11, 0x02, 0x3c, 0x41, 0x10, 0x0e, 0x00, 0xb1, 0x3c, 0xf9, 0x76, 0x88, 0x9e, 0x03, 0x3c, 0xfd, 0x9d, 0x14,
/* (2^ 85)P */ 0x5b, 0x15, 0x63, 0x6b, 0xe4, 0xdd, 0x79, 0xd4, 0x76, 0x79, 0x83, 0x3c, 0xe9, 0x15, 0x6e, 0xb6, 0x38, 0xe0, 0x13, 0x1f, 0x3b, 0xe4, 0xfd, 0xda, 0x35, 0x0b, 0x4b, 0x2e, 0x1a, 0xda, 0xaf, 0x5f,
/* (2^ 86)P */ 0x81, 0x75, 0x19, 0x17, 0xdf, 0xbb, 0x00, 0x36, 0xc2, 0xd2, 0x3c, 0xbe, 0x0b, 0x05, 0x72, 0x39, 0x86, 0xbe, 0xd5, 0xbd, 0x6d, 0x90, 0x38, 0x59, 0x0f, 0x86, 0x9b, 0x3f, 0xe4, 0xe5, 0xfc, 0x34,
/* (2^ 87)P */ 0x02, 0x4d, 0xd1, 0x42, 0xcd, 0xa4, 0xa8, 0x75, 0x65, 0xdf, 0x41, 0x34, 0xc5, 0xab, 0x8d, 0x82, 0xd3, 0x31, 0xe1, 0xd2, 0xed, 0xab, 0xdc, 0x33, 0x5f, 0xd2, 0x14, 0xb8, 0x6f, 0xd7, 0xba, 0x3e,
/* (2^ 88)P */ 0x0f, 0xe1, 0x70, 0x6f, 0x56, 0x6f, 0x90, 0xd4, 0x5a, 0x0f, 0x69, 0x51, 0xaa, 0xf7, 0x12, 0x5d, 0xf2, 0xfc, 0xce, 0x76, 0x6e, 0xb1, 0xad, 0x45, 0x99, 0x29, 0x23, 0xad, 0xae, 0x68, 0xf7, 0x01,
/* (2^ 89)P */ 0xbd, 0xfe, 0x48, 0x62, 0x7b, 0xc7, 0x6c, 0x2b, 0xfd, 0xaf, 0x3a, 0xec, 0x28, 0x06, 0xd3, 0x3c, 0x6a, 0x48, 0xef, 0xd4, 0x80, 0x0b, 0x1c, 0xce, 0x23, 0x6c, 0xf6, 0xa6, 0x2e, 0xff, 0x3b, 0x4c,
/* (2^ 90)P */ 0x5f, 0xeb, 0xea, 0x4a, 0x09, 0xc4, 0x2e, 0x3f, 0xa7, 0x2c, 0x37, 0x6e, 0x28, 0x9b, 0xb1, 0x61, 0x1d, 0x70, 0x2a, 0xde, 0x66, 0xa9, 0xef, 0x5e, 0xef, 0xe3, 0x55, 0xde, 0x65, 0x05, 0xb2, 0x23,
/* (2^ 91)P */ 0x57, 0x85, 0xd5, 0x79, 0x52, 0xca, 0x01, 0xe3, 0x4f, 0x87, 0xc2, 0x27, 0xce, 0xd4, 0xb2, 0x07, 0x67, 0x1d, 0xcf, 0x9d, 0x8a, 0xcd, 0x32, 0xa5, 0x56, 0xff, 0x2b, 0x3f, 0xe2, 0xfe, 0x52, 0x2a,
/* (2^ 92)P */ 0x3d, 0x66, 0xd8, 0x7c, 0xb3, 0xef, 0x24, 0x86, 0x94, 0x75, 0xbd, 0xff, 0x20, 0xac, 0xc7, 0xbb, 0x45, 0x74, 0xd3, 0x82, 0x9c, 0x5e, 0xb8, 0x57, 0x66, 0xec, 0xa6, 0x86, 0xcb, 0x52, 0x30, 0x7b,
/* (2^ 93)P */ 0x1e, 0xe9, 0x25, 0x25, 0xad, 0xf0, 0x82, 0x34, 0xa0, 0xdc, 0x8e, 0xd2, 0x43, 0x80, 0xb6, 0x2c, 0x3a, 0x00, 0x1b, 0x2e, 0x05, 0x6d, 0x4f, 0xaf, 0x0a, 0x1b, 0x78, 0x29, 0x25, 0x8c, 0x5f, 0x18,
/* (2^ 94)P */ 0xd6, 0xe0, 0x0c, 0xd8, 0x5b, 0xde, 0x41, 0xaa, 0xd6, 0xe9, 0x53, 0x68, 0x41, 0xb2, 0x07, 0x94, 0x3a, 0x4c, 0x7f, 0x35, 0x6e, 0xc3, 0x3e, 0x56, 0xce, 0x7b, 0x29, 0x0e, 0xdd, 0xb8, 0xc4, 0x4c,
/* (2^ 95)P */ 0x0e, 0x73, 0xb8, 0xff, 0x52, 0x1a, 0xfc, 0xa2, 0x37, 0x8e, 0x05, 0x67, 0x6e, 0xf1, 0x11, 0x18, 0xe1, 0x4e, 0xdf, 0xcd, 0x66, 0xa3, 0xf9, 0x10, 0x99, 0xf0, 0xb9, 0xa0, 0xc4, 0xa0, 0xf4, 0x72,
/* (2^ 96)P */ 0xa7, 0x4e, 0x3f, 0x66, 0x6f, 0xc0, 0x16, 0x8c, 0xba, 0x0f, 0x97, 0x4e, 0xf7, 0x3a, 0x3b, 0x69, 0x45, 0xc3, 0x9e, 0xd6, 0xf1, 0xe7, 0x02, 0x21, 0x89, 0x80, 0x8a, 0x96, 0xbc, 0x3c, 0xa5, 0x0b,
/* (2^ 97)P */ 0x37, 0x55, 0xa1, 0xfe, 0xc7, 0x9d, 0x3d, 0xca, 0x93, 0x64, 0x53, 0x51, 0xbb, 0x24, 0x68, 0x4c, 0xb1, 0x06, 0x40, 0x84, 0x14, 0x63, 0x88, 0xb9, 0x60, 0xcc, 0x54, 0xb4, 0x2a, 0xa7, 0xd2, 0x40,
/* (2^ 98)P */ 0x75, 0x09, 0x57, 0x12, 0xb7, 0xa1, 0x36, 0x59, 0x57, 0xa6, 0xbd, 0xde, 0x48, 0xd6, 0xb9, 0x91, 0xea, 0x30, 0x43, 0xb6, 0x4b, 0x09, 0x44, 0x33, 0xd0, 0x51, 0xee, 0x12, 0x0d, 0xa1, 0x6b, 0x00,
/* (2^ 99)P */ 0x58, 0x5d, 0xde, 0xf5, 0x68, 0x84, 0x22, 0x19, 0xb0, 0x05, 0xcc, 0x38, 0x4c, 0x2f, 0xb1, 0x0e, 0x90, 0x19, 0x60, 0xd5, 0x9d, 0x9f, 0x03, 0xa1, 0x0b, 0x0e, 0xff, 0x4f, 0xce, 0xd4, 0x02, 0x45,
/* (2^100)P */ 0x89, 0xc1, 0x37, 0x68, 0x10, 0x54, 0x20, 0xeb, 0x3c, 0xb9, 0xd3, 0x6d, 0x4c, 0x54, 0xf6, 0xd0, 0x4f, 0xd7, 0x16, 0xc4, 0x64, 0x70, 0x72, 0x40, 0xf0, 0x2e, 0x50, 0x4b, 0x11, 0xc6, 0x15, 0x6e,
/* (2^101)P */ 0x6b, 0xa7, 0xb1, 0xcf, 0x98, 0xa3, 0xf2, 0x4d, 0xb1, 0xf6, 0xf2, 0x19, 0x74, 0x6c, 0x25, 0x11, 0x43, 0x60, 0x6e, 0x06, 0x62, 0x79, 0x49, 0x4a, 0x44, 0x5b, 0x35, 0x41, 0xab, 0x3a, 0x5b, 0x70,
/* (2^102)P */ 0xd8, 0xb1, 0x97, 0xd7, 0x36, 0xf5, 0x5e, 0x36, 0xdb, 0xf0, 0xdd, 0x22, 0xd6, 0x6b, 0x07, 0x00, 0x88, 0x5a, 0x57, 0xe0, 0xb0, 0x33, 0xbf, 0x3b, 0x4d, 0xca, 0xe4, 0xc8, 0x05, 0xaa, 0x77, 0x37,
/* (2^103)P */ 0x5f, 0xdb, 0x78, 0x55, 0xc8, 0x45, 0x27, 0x39, 0xe2, 0x5a, 0xae, 0xdb, 0x49, 0x41, 0xda, 0x6f, 0x67, 0x98, 0xdc, 0x8a, 0x0b, 0xb0, 0xf0, 0xb1, 0xa3, 0x1d, 0x6f, 0xd3, 0x37, 0x34, 0x96, 0x09,
/* (2^104)P */ 0x53, 0x38, 0xdc, 0xa5, 0x90, 0x4e, 0x82, 0x7e, 0xbd, 0x5c, 0x13, 0x1f, 0x64, 0xf6, 0xb5, 0xcc, 0xcc, 0x8f, 0xce, 0x87, 0x6c, 0xd8, 0x36, 0x67, 0x9f, 0x24, 0x04, 0x66, 0xe2, 0x3c, 0x5f, 0x62,
/* (2^105)P */ 0x3f, 0xf6, 0x02, 0x95, 0x05, 0xc8, 0x8a, 0xaf, 0x69, 0x14, 0x35, 0x2e, 0x0a, 0xe7, 0x05, 0x0c, 0x05, 0x63, 0x4b, 0x76, 0x9c, 0x2e, 0x29, 0x35, 0xc3, 0x3a, 0xe2, 0xc7, 0x60, 0x43, 0x39, 0x1a,
/* (2^106)P */ 0x64, 0x32, 0x18, 0x51, 0x32, 0xd5, 0xc6, 0xd5, 0x4f, 0xb7, 0xc2, 0x43, 0xbd, 0x5a, 0x06, 0x62, 0x9b, 0x3f, 0x97, 0x3b, 0xd0, 0xf5, 0xfb, 0xb5, 0x5e, 0x6e, 0x20, 0x61, 0x36, 0xda, 0xa3, 0x13,
/* (2^107)P */ 0xe5, 0x94, 0x5d, 0x72, 0x37, 0x58, 0xbd, 0xc6, 0xc5, 0x16, 0x50, 0x20, 0x12, 0x09, 0xe3, 0x18, 0x68, 0x3c, 0x03, 0x70, 0x15, 0xce, 0x88, 0x20, 0x87, 0x79, 0x83, 0x5c, 0x49, 0x1f, 0xba, 0x7f,
/* (2^108)P */ 0x9d, 0x07, 0xf9, 0xf2, 0x23, 0x74, 0x8c, 0x5a, 0xc5, 0x3f, 0x02, 0x34, 0x7b, 0x15, 0x35, 0x17, 0x51, 0xb3, 0xfa, 0xd2, 0x9a, 0xb4, 0xf9, 0xe4, 0x3c, 0xe3, 0x78, 0xc8, 0x72, 0xff, 0x91, 0x66,
/* (2^109)P */ 0x3e, 0xff, 0x5e, 0xdc, 0xde, 0x2a, 0x2c, 0x12, 0xf4, 0x6c, 0x95, 0xd8, 0xf1, 0x4b, 0xdd, 0xf8, 0xda, 0x5b, 0x9e, 0x9e, 0x5d, 0x20, 0x86, 0xeb, 0x43, 0xc7, 0x75, 0xd9, 0xb9, 0x92, 0x9b, 0x04,
/* (2^110)P */ 0x5a, 0xc0, 0xf6, 0xb0, 0x30, 0x97, 0x37, 0xa5, 0x53, 0xa5, 0xf3, 0xc6, 0xac, 0xff, 0xa0, 0x72, 0x6d, 0xcd, 0x0d, 0xb2, 0x34, 0x2c, 0x03, 0xb0, 0x4a, 0x16, 0xd5, 0x88, 0xbc, 0x9d, 0x0e, 0x47,
/* (2^111)P */ 0x47, 0xc0, 0x37, 0xa2, 0x0c, 0xf1, 0x9c, 0xb1, 0xa2, 0x81, 0x6c, 0x1f, 0x71, 0x66, 0x54, 0xb6, 0x43, 0x0b, 0xd8, 0x6d, 0xd1, 0x1b, 0x32, 0xb3, 0x8e, 0xbe, 0x5f, 0x0c, 0x60, 0x4f, 0xc1, 0x48,
/* (2^112)P */ 0x03, 0xc8, 0xa6, 0x4a, 0x26, 0x1c, 0x45, 0x66, 0xa6, 0x7d, 0xfa, 0xa4, 0x04, 0x39, 0x6e, 0xb6, 0x95, 0x83, 0x12, 0xb3, 0xb0, 0x19, 0x5f, 0xd4, 0x10, 0xbc, 0xc9, 0xc3, 0x27, 0x26, 0x60, 0x31,
/* (2^113)P */ 0x0d, 0xe1, 0xe4, 0x32, 0x48, 0xdc, 0x20, 0x31, 0xf7, 0x17, 0xc7, 0x56, 0x67, 0xc4, 0x20, 0xeb, 0x94, 0x02, 0x28, 0x67, 0x3f, 0x2e, 0xf5, 0x00, 0x09, 0xc5, 0x30, 0x47, 0xc1, 0x4f, 0x6d, 0x56,
/* (2^114)P */ 0x06, 0x72, 0x83, 0xfd, 0x40, 0x5d, 0x3a, 0x7e, 0x7a, 0x54, 0x59, 0x71, 0xdc, 0x26, 0xe9, 0xc1, 0x95, 0x60, 0x8d, 0xa6, 0xfb, 0x30, 0x67, 0x21, 0xa7, 0xce, 0x69, 0x3f, 0x84, 0xc3, 0xe8, 0x22,
/* (2^115)P */ 0x2b, 0x4b, 0x0e, 0x93, 0xe8, 0x74, 0xd0, 0x33, 0x16, 0x58, 0xd1, 0x84, 0x0e, 0x35, 0xe4, 0xb6, 0x65, 0x23, 0xba, 0xd6, 0x6a, 0xc2, 0x34, 0x55, 0xf3, 0xf3, 0xf1, 0x89, 0x2f, 0xc1, 0x73, 0x77,
/* (2^116)P */ 0xaa, 0x62, 0x79, 0xa5, 0x4d, 0x40, 0xba, 0x8c, 0x56, 0xce, 0x99, 0x19, 0xa8, 0x97, 0x98, 0x5b, 0xfc, 0x92, 0x16, 0x12, 0x2f, 0x86, 0x8e, 0x50, 0x91, 0xc2, 0x93, 0xa0, 0x7f, 0x90, 0x81, 0x3a,
/* (2^117)P */ 0x10, 0xa5, 0x25, 0x47, 0xff, 0xd0, 0xde, 0x0d, 0x03, 0xc5, 0x3f, 0x67, 0x10, 0xcc, 0xd8, 0x10, 0x89, 0x4e, 0x1f, 0x9f, 0x1c, 0x15, 0x9d, 0x5b, 0x4c, 0xa4, 0x09, 0xcb, 0xd5, 0xc1, 0xa5, 0x32,
/* (2^118)P */ 0xfb, 0x41, 0x05, 0xb9, 0x42, 0xa4, 0x0a, 0x1e, 0xdb, 0x85, 0xb4, 0xc1, 0x7c, 0xeb, 0x85, 0x5f, 0xe5, 0xf2, 0x9d, 0x8a, 0xce, 0x95, 0xe5, 0xbe, 0x36, 0x22, 0x42, 0x22, 0xc7, 0x96, 0xe4, 0x25,
/* (2^119)P */ 0xb9, 0xe5, 0x0f, 0xcd, 0x46, 0x3c, 0xdf, 0x5e, 0x88, 0x33, 0xa4, 0xd2, 0x7e, 0x5a, 0xe7, 0x34, 0x52, 0xe3, 0x61, 0xd7, 0x11, 0xde, 0x88, 0xe4, 0x5c, 0x54, 0x85, 0xa0, 0x01, 0x8a, 0x87, 0x0e,
/* (2^120)P */ 0x04, 0xbb, 0x21, 0xe0, 0x77, 0x3c, 0x49, 0xba, 0x9a, 0x89, 0xdf, 0xc7, 0x43, 0x18, 0x4d, 0x2b, 0x67, 0x0d, 0xe8, 0x7a, 0x48, 0x7a, 0xa3, 0x9e, 0x94, 0x17, 0xe4, 0x11, 0x80, 0x95, 0xa9, 0x67,
/* (2^121)P */ 0x65, 0xb0, 0x97, 0x66, 0x1a, 0x05, 0x58, 0x4b, 0xd4, 0xa6, 0x6b, 0x8d, 0x7d, 0x3f, 0xe3, 0x47, 0xc1, 0x46, 0xca, 0x83, 0xd4, 0xa8, 0x4d, 0xbb, 0x0d, 0xdb, 0xc2, 0x81, 0xa1, 0xca, 0xbe, 0x68,
/* (2^122)P */ 0xa5, 0x9a, 0x98, 0x0b, 0xe9, 0x80, 0x89, 0x8d, 0x9b, 0xc9, 0x93, 0x2c, 0x4a, 0xb1, 0x5e, 0xf9, 0xa2, 0x73, 0x6e, 0x79, 0xc4, 0xc7, 0xc6, 0x51, 0x69, 0xb5, 0xef, 0xb5, 0x63, 0x83, 0x22, 0x6e,
/* (2^123)P */ 0xc8, 0x24, 0xd6, 0x2d, 0xb0, 0xc0, 0xbb, 0xc6, 0xee, 0x70, 0x81, 0xec, 0x7d, 0xb4, 0x7e, 0x77, 0xa9, 0xaf, 0xcf, 0x04, 0xa0, 0x15, 0xde, 0x3c, 0x9b, 0xbf, 0x60, 0x71, 0x08, 0xbc, 0xc6, 0x1d,
/* (2^124)P */ 0x02, 0x40, 0xc3, 0xee, 0x43, 0xe0, 0x07, 0x2e, 0x7f, 0xdc, 0x68, 0x7a, 0x67, 0xfc, 0xe9, 0x18, 0x9a, 0x5b, 0xd1, 0x8b, 0x18, 0x03, 0xda, 0xd8, 0x53, 0x82, 0x56, 0x00, 0xbb, 0xc3, 0xfb, 0x48,
/* (2^125)P */ 0xe1, 0x4c, 0x65, 0xfb, 0x4c, 0x7d, 0x54, 0x57, 0xad, 0xe2, 0x58, 0xa0, 0x82, 0x5b, 0x56, 0xd3, 0x78, 0x44, 0x15, 0xbf, 0x0b, 0xaf, 0x3e, 0xf6, 0x18, 0xbb, 0xdf, 0x14, 0xf1, 0x1e, 0x53, 0x47,
/* (2^126)P */ 0x87, 0xc5, 0x78, 0x42, 0x0a, 0x63, 0xec, 0xe1, 0xf3, 0x83, 0x8e, 0xca, 0x46, 0xd5, 0x07, 0x55, 0x2b, 0x0c, 0xdc, 0x3a, 0xc6, 0x35, 0xe1, 0x85, 0x4e, 0x84, 0x82, 0x56, 0xa8, 0xef, 0xa7, 0x0a,
/* (2^127)P */ 0x15, 0xf6, 0xe1, 0xb3, 0xa8, 0x1b, 0x69, 0x72, 0xfa, 0x3f, 0xbe, 0x1f, 0x70, 0xe9, 0xb4, 0x32, 0x68, 0x78, 0xbb, 0x39, 0x2e, 0xd9, 0xb6, 0x97, 0xe8, 0x39, 0x2e, 0xa0, 0xde, 0x53, 0xfe, 0x2c,
/* (2^128)P */ 0xb0, 0x52, 0xcd, 0x85, 0xcd, 0x92, 0x73, 0x68, 0x31, 0x98, 0xe2, 0x10, 0xc9, 0x66, 0xff, 0x27, 0x06, 0x2d, 0x83, 0xa9, 0x56, 0x45, 0x13, 0x97, 0xa0, 0xf8, 0x84, 0x0a, 0x36, 0xb0, 0x9b, 0x26,
/* (2^129)P */ 0x5c, 0xf8, 0x43, 0x76, 0x45, 0x55, 0x6e, 0x70, 0x1b, 0x7d, 0x59, 0x9b, 0x8c, 0xa4, 0x34, 0x37, 0x72, 0xa4, 0xef, 0xc6, 0xe8, 0x91, 0xee, 0x7a, 0xe0, 0xd9, 0xa9, 0x98, 0xc1, 0xab, 0xd6, 0x5c,
/* (2^130)P */ 0x1a, 0xe4, 0x3c, 0xcb, 0x06, 0xde, 0x04, 0x0e, 0x38, 0xe1, 0x02, 0x34, 0x89, 0xeb, 0xc6, 0xd8, 0x72, 0x37, 0x6e, 0x68, 0xbb, 0x59, 0x46, 0x90, 0xc8, 0xa8, 0x6b, 0x74, 0x71, 0xc3, 0x15, 0x72,
/* (2^131)P */ 0xd9, 0xa2, 0xe4, 0xea, 0x7e, 0xa9, 0x12, 0xfd, 0xc5, 0xf2, 0x94, 0x63, 0x51, 0xb7, 0x14, 0x95, 0x94, 0xf2, 0x08, 0x92, 0x80, 0xd5, 0x6f, 0x26, 0xb9, 0x26, 0x9a, 0x61, 0x85, 0x70, 0x84, 0x5c,
/* (2^132)P */ 0xea, 0x94, 0xd6, 0xfe, 0x10, 0x54, 0x98, 0x52, 0x54, 0xd2, 0x2e, 0x4a, 0x93, 0x5b, 0x90, 0x3c, 0x67, 0xe4, 0x3b, 0x2d, 0x69, 0x47, 0xbb, 0x10, 0xe1, 0xe9, 0xe5, 0x69, 0x2d, 0x3d, 0x3b, 0x06,
/* (2^133)P */ 0xeb, 0x7d, 0xa5, 0xdd, 0xee, 0x26, 0x27, 0x47, 0x91, 0x18, 0xf4, 0x10, 0xae, 0xc4, 0xb6, 0xef, 0x14, 0x76, 0x30, 0x7b, 0x91, 0x41, 0x16, 0x2b, 0x7c, 0x5b, 0xf4, 0xc4, 0x4f, 0x55, 0x7c, 0x11,
/* (2^134)P */ 0x12, 0x88, 0x9d, 0x8f, 0x11, 0xf3, 0x7c, 0xc0, 0x39, 0x79, 0x01, 0x50, 0x20, 0xd8, 0xdb, 0x01, 0x27, 0x28, 0x1b, 0x17, 0xf4, 0x03, 0xe8, 0xd7, 0xea, 0x25, 0xd2, 0x87, 0x74, 0xe8, 0x15, 0x10,
/* (2^135)P */ 0x4d, 0xcc, 0x3a, 0xd2, 0xfe, 0xe3, 0x8d, 0xc5, 0x2d, 0xbe, 0xa7, 0x94, 0xc2, 0x91, 0xdb, 0x50, 0x57, 0xf4, 0x9c, 0x1c, 0x3d, 0xd4, 0x94, 0x0b, 0x4a, 0x52, 0x37, 0x6e, 0xfa, 0x40, 0x16, 0x6b,
/* (2^136)P */ 0x09, 0x0d, 0xda, 0x5f, 0x6c, 0x34, 0x2f, 0x69, 0x51, 0x31, 0x4d, 0xfa, 0x59, 0x1c, 0x0b, 0x20, 0x96, 0xa2, 0x77, 0x07, 0x76, 0x6f, 0xc4, 0xb8, 0xcf, 0xfb, 0xfd, 0x3f, 0x5f, 0x39, 0x38, 0x4b,
/* (2^137)P */ 0x71, 0xd6, 0x54, 0xbe, 0x00, 0x5e, 0xd2, 0x18, 0xa6, 0xab, 0xc8, 0xbe, 0x82, 0x05, 0xd5, 0x60, 0x82, 0xb9, 0x78, 0x3b, 0x26, 0x8f, 0xad, 0x87, 0x32, 0x04, 0xda, 0x9c, 0x4e, 0xf6, 0xfd, 0x50,
/* (2^138)P */ 0xf0, 0xdc, 0x78, 0xc5, 0xaa, 0x67, 0xf5, 0x90, 0x3b, 0x13, 0xa3, 0xf2, 0x0e, 0x9b, 0x1e, 0xef, 0x71, 0xde, 0xd9, 0x42, 0x92, 0xba, 0xeb, 0x0e, 0xc7, 0x01, 0x31, 0xf0, 0x9b, 0x3c, 0x47, 0x15,
/* (2^139)P */ 0x95, 0x80, 0xb7, 0x56, 0xae, 0xe8, 0x77, 0x7c, 0x8e, 0x07, 0x6f, 0x6e, 0x66, 0xe7, 0x78, 0xb6, 0x1f, 0xba, 0x48, 0x53, 0x61, 0xb9, 0xa0, 0x2d, 0x0b, 0x3f, 0x73, 0xff, 0xc1, 0x31, 0xf9, 0x7c,
/* (2^140)P */ 0x6c, 0x36, 0x0a, 0x0a, 0xf5, 0x57, 0xb3, 0x26, 0x32, 0xd7, 0x87, 0x2b, 0xf4, 0x8c, 0x70, 0xe9, 0xc0, 0xb2, 0x1c, 0xf9, 0xa5, 0xee, 0x3a, 0xc1, 0x4c, 0xbb, 0x43, 0x11, 0x99, 0x0c, 0xd9, 0x35,
/* (2^141)P */ 0xdc, 0xd9, 0xa0, 0xa9, 0x04, 0xc4, 0xc1, 0x47, 0x51, 0xd2, 0x72, 0x19, 0x45, 0x58, 0x9e, 0x65, 0x31, 0x8c, 0xb3, 0x73, 0xc4, 0xa8, 0x75, 0x38, 0x24, 0x1f, 0x56, 0x79, 0xd3, 0x9e, 0xbd, 0x1f,
/* (2^142)P */ 0x8d, 0xc2, 0x1e, 0xd4, 0x6f, 0xbc, 0xfa, 0x11, 0xca, 0x2d, 0x2a, 0xcd, 0xe3, 0xdf, 0xf8, 0x7e, 0x95, 0x45, 0x40, 0x8c, 0x5d, 0x3b, 0xe7, 0x72, 0x27, 0x2f, 0xb7, 0x54, 0x49, 0xfa, 0x35, 0x61,
/* (2^143)P */ 0x9c, 0xb6, 0x24, 0xde, 0xa2, 0x32, 0xfc, 0xcc, 0x88, 0x5d, 0x09, 0x1f, 0x8c, 0x69, 0x55, 0x3f, 0x29, 0xf9, 0xc3, 0x5a, 0xed, 0x50, 0x33, 0xbe, 0xeb, 0x7e, 0x47, 0xca, 0x06, 0xf8, 0x9b, 0x5e,
/* (2^144)P */ 0x68, 0x9f, 0x30, 0x3c, 0xb6, 0x8f, 0xce, 0xe9, 0xf4, 0xf9, 0xe1, 0x65, 0x35, 0xf6, 0x76, 0x53, 0xf1, 0x93, 0x63, 0x5a, 0xb3, 0xcf, 0xaf, 0xd1, 0x06, 0x35, 0x62, 0xe5, 0xed, 0xa1, 0x32, 0x66,
/* (2^145)P */ 0x4c, 0xed, 0x2d, 0x0c, 0x39, 0x6c, 0x7d, 0x0b, 0x1f, 0xcb, 0x04, 0xdf, 0x81, 0x32, 0xcb, 0x56, 0xc7, 0xc3, 0xec, 0x49, 0x12, 0x5a, 0x30, 0x66, 0x2a, 0xa7, 0x8c, 0xa3, 0x60, 0x8b, 0x58, 0x5d,
/* (2^146)P */ 0x2d, 0xf4, 0xe5, 0xe8, 0x78, 0xbf, 0xec, 0xa6, 0xec, 0x3e, 0x8a, 0x3c, 0x4b, 0xb4, 0xee, 0x86, 0x04, 0x16, 0xd2, 0xfb, 0x48, 0x9c, 0x21, 0xec, 0x31, 0x67, 0xc3, 0x17, 0xf5, 0x1a, 0xaf, 0x1a,
/* (2^147)P */ 0xe7, 0xbd, 0x69, 0x67, 0x83, 0xa2, 0x06, 0xc3, 0xdb, 0x2a, 0x1e, 0x2b, 0x62, 0x80, 0x82, 0x20, 0xa6, 0x94, 0xff, 0xfb, 0x1f, 0xf5, 0x27, 0x80, 0x6b, 0xf2, 0x24, 0x11, 0xce, 0xa1, 0xcf, 0x76,
/* (2^148)P */ 0xb6, 0xab, 0x22, 0x24, 0x56, 0x00, 0xeb, 0x18, 0xc3, 0x29, 0x8c, 0x8f, 0xd5, 0xc4, 0x77, 0xf3, 0x1a, 0x56, 0x31, 0xf5, 0x07, 0xc2, 0xbb, 0x4d, 0x27, 0x8a, 0x12, 0x82, 0xf0, 0xb7, 0x53, 0x02,
/* (2^149)P */ 0xe0, 0x17, 0x2c, 0xb6, 0x1c, 0x09, 0x1f, 0x3d, 0xa9, 0x28, 0x46, 0xd6, 0xab, 0xe1, 0x60, 0x48, 0x53, 0x42, 0x9d, 0x30, 0x36, 0x74, 0xd1, 0x52, 0x76, 0xe5, 0xfa, 0x3e, 0xe1, 0x97, 0x6f, 0x35,
/* (2^150)P */ 0x5b, 0x53, 0x50, 0xa1, 0x1a, 0xe1, 0x51, 0xd3, 0xcc, 0x78, 0xd8, 0x1d, 0xbb, 0x45, 0x6b, 0x3e, 0x98, 0x2c, 0xd9, 0xbe, 0x28, 0x61, 0x77, 0x0c, 0xb8, 0x85, 0x28, 0x03, 0x93, 0xae, 0x34, 0x1d,
/* (2^151)P */ 0xc3, 0xa4, 0x5b, 0xa8, 0x8c, 0x48, 0xa0, 0x4b, 0xce, 0xe6, 0x9c, 0x3c, 0xc3, 0x48, 0x53, 0x98, 0x70, 0xa7, 0xbd, 0x97, 0x6f, 0x4c, 0x12, 0x66, 0x4a, 0x12, 0x54, 0x06, 0x29, 0xa0, 0x81, 0x0f,
/* (2^152)P */ 0xfd, 0x86, 0x9b, 0x56, 0xa6, 0x9c, 0xd0, 0x9e, 0x2d, 0x9a, 0xaf, 0x18, 0xfd, 0x09, 0x10, 0x81, 0x0a, 0xc2, 0xd8, 0x93, 0x3f, 0xd0, 0x08, 0xff, 0x6b, 0xf2, 0xae, 0x9f, 0x19, 0x48, 0xa1, 0x52,
/* (2^153)P */ 0x73, 0x1b, 0x8d, 0x2d, 0xdc, 0xf9, 0x03, 0x3e, 0x70, 0x1a, 0x96, 0x73, 0x18, 0x80, 0x05, 0x42, 0x70, 0x59, 0xa3, 0x41, 0xf0, 0x87, 0xd9, 0xc0, 0x49, 0xd5, 0xc0, 0xa1, 0x15, 0x1f, 0xaa, 0x07,
/* (2^154)P */ 0x24, 0x72, 0xd2, 0x8c, 0xe0, 0x6c, 0xd4, 0xdf, 0x39, 0x42, 0x4e, 0x93, 0x4f, 0x02, 0x0a, 0x6d, 0x59, 0x7b, 0x89, 0x99, 0x63, 0x7a, 0x8a, 0x80, 0xa2, 0x95, 0x3d, 0xe1, 0xe9, 0x56, 0x45, 0x0a,
/* (2^155)P */ 0x45, 0x30, 0xc1, 0xe9, 0x1f, 0x99, 0x1a, 0xd2, 0xb8, 0x51, 0x77, 0xfe, 0x48, 0x85, 0x0e, 0x9b, 0x35, 0x00, 0xf3, 0x4b, 0xcb, 0x43, 0xa6, 0x5d, 0x21, 0xf7, 0x40, 0x39, 0xd6, 0x28, 0xdb, 0x77,
/* (2^156)P */ 0x11, 0x90, 0xdc, 0x4a, 0x61, 0xeb, 0x5e, 0xfc, 0xeb, 0x11, 0xc4, 0xe8, 0x9a, 0x41, 0x29, 0x52, 0x74, 0xcf, 0x1d, 0x7d, 0x78, 0xe7, 0xc3, 0x9e, 0xb5, 0x4c, 0x6e, 0x21, 0x3e, 0x05, 0x0d, 0x34,
/* (2^157)P */ 0xb4, 0xf2, 0x8d, 0xb4, 0x39, 0xaf, 0xc7, 0xca, 0x94, 0x0a, 0xa1, 0x71, 0x28, 0xec, 0xfa, 0xc0, 0xed, 0x75, 0xa5, 0x5c, 0x24, 0x69, 0x0a, 0x14, 0x4c, 0x3a, 0x27, 0x34, 0x71, 0xc3, 0xf1, 0x0c,
/* (2^158)P */ 0xa5, 0xb8, 0x24, 0xc2, 0x6a, 0x30, 0xee, 0xc8, 0xb0, 0x30, 0x49, 0xcb, 0x7c, 0xee, 0xea, 0x57, 0x4f, 0xe7, 0xcb, 0xaa, 0xbd, 0x06, 0xe8, 0xa1, 0x7d, 0x65, 0xeb, 0x2e, 0x74, 0x62, 0x9a, 0x7d,
/* (2^159)P */ 0x30, 0x48, 0x6c, 0x54, 0xef, 0xb6, 0xb6, 0x9e, 0x2e, 0x6e, 0xb3, 0xdd, 0x1f, 0xca, 0x5c, 0x88, 0x05, 0x71, 0x0d, 0xef, 0x83, 0xf3, 0xb9, 0xe6, 0x12, 0x04, 0x2e, 0x9d, 0xef, 0x4f, 0x65, 0x58,
/* (2^160)P */ 0x26, 0x8e, 0x0e, 0xbe, 0xff, 0xc4, 0x05, 0xa9, 0x6e, 0x81, 0x31, 0x9b, 0xdf, 0xe5, 0x2d, 0x94, 0xe1, 0x88, 0x2e, 0x80, 0x3f, 0x72, 0x7d, 0x49, 0x8d, 0x40, 0x2f, 0x60, 0xea, 0x4d, 0x68, 0x30,
/* (2^161)P */ 0x34, 0xcb, 0xe6, 0xa3, 0x78, 0xa2, 0xe5, 0x21, 0xc4, 0x1d, 0x15, 0x5b, 0x6f, 0x6e, 0xfb, 0xae, 0x15, 0xca, 0x77, 0x9d, 0x04, 0x8e, 0x0b, 0xb3, 0x81, 0x89, 0xb9, 0x53, 0xcf, 0xc9, 0xc3, 0x28,
/* (2^162)P */ 0x2a, 0xdd, 0x6c, 0x55, 0x21, 0xb7, 0x7f, 0x28, 0x74, 0x22, 0x02, 0x97, 0xa8, 0x7c, 0x31, 0x0d, 0x58, 0x32, 0x54, 0x3a, 0x42, 0xc7, 0x68, 0x74, 0x2f, 0x64, 0xb5, 0x4e, 0x46, 0x11, 0x7f, 0x4a,
/* (2^163)P */ 0xa6, 0x3a, 0x19, 0x4d, 0x77, 0xa4, 0x37, 0xa2, 0xa1, 0x29, 0x21, 0xa9, 0x6e, 0x98, 0x65, 0xd8, 0x88, 0x1a, 0x7c, 0xf8, 0xec, 0x15, 0xc5, 0x24, 0xeb, 0xf5, 0x39, 0x5f, 0x57, 0x03, 0x40, 0x60,
/* (2^164)P */ 0x27, 0x9b, 0x0a, 0x57, 0x89, 0xf1, 0xb9, 0x47, 0x78, 0x4b, 0x5e, 0x46, 0xde, 0xce, 0x98, 0x2b, 0x20, 0x5c, 0xb8, 0xdb, 0x51, 0xf5, 0x6d, 0x02, 0x01, 0x19, 0xe2, 0x47, 0x10, 0xd9, 0xfc, 0x74,
/* (2^165)P */ 0xa3, 0xbf, 0xc1, 0x23, 0x0a, 0xa9, 0xe2, 0x13, 0xf6, 0x19, 0x85, 0x47, 0x4e, 0x07, 0xb0, 0x0c, 0x44, 0xcf, 0xf6, 0x3a, 0xbe, 0xcb, 0xf1, 0x5f, 0xbe, 0x2d, 0x81, 0xbe, 0x38, 0x54, 0xfe, 0x67,
/* (2^166)P */ 0xb0, 0x05, 0x0f, 0xa4, 0x4f, 0xf6, 0x3c, 0xd1, 0x87, 0x37, 0x28, 0x32, 0x2f, 0xfb, 0x4d, 0x05, 0xea, 0x2a, 0x0d, 0x7f, 0x5b, 0x91, 0x73, 0x41, 0x4e, 0x0d, 0x61, 0x1f, 0x4f, 0x14, 0x2f, 0x48,
/* (2^167)P */ 0x34, 0x82, 0x7f, 0xb4, 0x01, 0x02, 0x21, 0xf6, 0x90, 0xb9, 0x70, 0x9e, 0x92, 0xe1, 0x0a, 0x5d, 0x7c, 0x56, 0x49, 0xb0, 0x55, 0xf4, 0xd7, 0xdc, 0x01, 0x6f, 0x91, 0xf0, 0xf1, 0xd0, 0x93, 0x7e,
/* (2^168)P */ 0xfa, 0xb4, 0x7d, 0x8a, 0xf1, 0xcb, 0x79, 0xdd, 0x2f, 0xc6, 0x74, 0x6f, 0xbf, 0x91, 0x83, 0xbe, 0xbd, 0x91, 0x82, 0x4b, 0xd1, 0x45, 0x71, 0x02, 0x05, 0x17, 0xbf, 0x2c, 0xea, 0x73, 0x5a, 0x58,
/* (2^169)P */ 0xb2, 0x0d, 0x8a, 0x92, 0x3e, 0xa0, 0x5c, 0x48, 0xe7, 0x57, 0x28, 0x74, 0xa5, 0x01, 0xfc, 0x10, 0xa7, 0x51, 0xd5, 0xd6, 0xdb, 0x2e, 0x48, 0x2f, 0x8a, 0xdb, 0x8f, 0x04, 0xb5, 0x33, 0x04, 0x0f,
/* (2^170)P */ 0x47, 0x62, 0xdc, 0xd7, 0x8d, 0x2e, 0xda, 0x60, 0x9a, 0x81, 0xd4, 0x8c, 0xd3, 0xc9, 0xb4, 0x88, 0x97, 0x66, 0xf6, 0x01, 0xc0, 0x3a, 0x03, 0x13, 0x75, 0x7d, 0x36, 0x3b, 0xfe, 0x24, 0x3b, 0x27,
/* (2^171)P */ 0xd4, 0xb9, 0xb3, 0x31, 0x6a, 0xf6, 0xe8, 0xc6, 0xd5, 0x49, 0xdf, 0x94, 0xa4, 0x14, 0x15, 0x28, 0xa7, 0x3d, 0xb2, 0xc8, 0xdf, 0x6f, 0x72, 0xd1, 0x48, 0xe5, 0xde, 0x03, 0xd1, 0xe7, 0x3a, 0x4b,
/* (2^172)P */ 0x7e, 0x9d, 0x4b, 0xce, 0x19, 0x6e, 0x25, 0xc6, 0x1c, 0xc6, 0xe3, 0x86, 0xf1, 0x5c, 0x5c, 0xff, 0x45, 0xc1, 0x8e, 0x4b, 0xa3, 0x3c, 0xc6, 0xac, 0x74, 0x65, 0xe6, 0xfe, 0x88, 0x18, 0x62, 0x74,
/* (2^173)P */ 0x1e, 0x0a, 0x29, 0x45, 0x96, 0x40, 0x6f, 0x95, 0x2e, 0x96, 0x3a, 0x26, 0xe3, 0xf8, 0x0b, 0xef, 0x7b, 0x64, 0xc2, 0x5e, 0xeb, 0x50, 0x6a, 0xed, 0x02, 0x75, 0xca, 0x9d, 0x3a, 0x28, 0x94, 0x06,
/* (2^174)P */ 0xd1, 0xdc, 0xa2, 0x43, 0x36, 0x96, 0x9b, 0x76, 0x53, 0x53, 0xfc, 0x09, 0xea, 0xc8, 0xb7, 0x42, 0xab, 0x7e, 0x39, 0x13, 0xee, 0x2a, 0x00, 0x4f, 0x3a, 0xd6, 0xb7, 0x19, 0x2c, 0x5e, 0x00, 0x63,
/* (2^175)P */ 0xea, 0x3b, 0x02, 0x63, 0xda, 0x36, 0x67, 0xca, 0xb7, 0x99, 0x2a, 0xb1, 0x6d, 0x7f, 0x6c, 0x96, 0xe1, 0xc5, 0x37, 0xc5, 0x90, 0x93, 0xe0, 0xac, 0xee, 0x89, 0xaa, 0xa1, 0x63, 0x60, 0x69, 0x0b,
/* (2^176)P */ 0xe5, 0x56, 0x8c, 0x28, 0x97, 0x3e, 0xb0, 0xeb, 0xe8, 0x8b, 0x8c, 0x93, 0x9f, 0x9f, 0x2a, 0x43, 0x71, 0x7f, 0x71, 0x5b, 0x3d, 0xa9, 0xa5, 0xa6, 0x97, 0x9d, 0x8f, 0xe1, 0xc3, 0xb4, 0x5f, 0x1a,
/* (2^177)P */ 0xce, 0xcd, 0x60, 0x1c, 0xad, 0xe7, 0x94, 0x1c, 0xa0, 0xc4, 0x02, 0xfc, 0x43, 0x2a, 0x20, 0xee, 0x20, 0x6a, 0xc4, 0x67, 0xd8, 0xe4, 0xaf, 0x8d, 0x58, 0x7b, 0xc2, 0x8a, 0x3c, 0x26, 0x10, 0x0a,
/* (2^178)P */ 0x4a, 0x2a, 0x43, 0xe4, 0xdf, 0xa9, 0xde, 0xd0, 0xc5, 0x77, 0x92, 0xbe, 0x7b, 0xf8, 0x6a, 0x85, 0x1a, 0xc7, 0x12, 0xc2, 0xac, 0x72, 0x84, 0xce, 0x91, 0x1e, 0xbb, 0x9b, 0x6d, 0x1b, 0x15, 0x6f,
/* (2^179)P */ 0x6a, 0xd5, 0xee, 0x7c, 0x52, 0x6c, 0x77, 0x26, 0xec, 0xfa, 0xf8, 0xfb, 0xb7, 0x1c, 0x21, 0x7d, 0xcc, 0x09, 0x46, 0xfd, 0xa6, 0x66, 0xae, 0x37, 0x42, 0x0c, 0x77, 0xd2, 0x02, 0xb7, 0x81, 0x1f,
/* (2^180)P */ 0x92, 0x83, 0xc5, 0xea, 0x57, 0xb0, 0xb0, 0x2f, 0x9d, 0x4e, 0x74, 0x29, 0xfe, 0x89, 0xdd, 0xe1, 0xf8, 0xb4, 0xbe, 0x17, 0xeb, 0xf8, 0x64, 0xc9, 0x1e, 0xd4, 0xa2, 0xc9, 0x73, 0x10, 0x57, 0x29,
/* (2^181)P */ 0x54, 0xe2, 0xc0, 0x81, 0x89, 0xa1, 0x48, 0xa9, 0x30, 0x28, 0xb2, 0x65, 0x9b, 0x36, 0xf6, 0x2d, 0xc6, 0xd3, 0xcf, 0x5f, 0xd7, 0xb2, 0x3e, 0xa3, 0x1f, 0xa0, 0x99, 0x41, 0xec, 0xd6, 0x8c, 0x07,
/* (2^182)P */ 0x2f, 0x0d, 0x90, 0xad, 0x41, 0x4a, 0x58, 0x4a, 0x52, 0x4c, 0xc7, 0xe2, 0x78, 0x2b, 0x14, 0x32, 0x78, 0xc9, 0x31, 0x84, 0x33, 0xe8, 0xc4, 0x68, 0xc2, 0x9f, 0x68, 0x08, 0x90, 0xea, 0x69, 0x7f,
/* (2^183)P */ 0x65, 0x82, 0xa3, 0x46, 0x1e, 0xc8, 0xf2, 0x52, 0xfd, 0x32, 0xa8, 0x04, 0x2d, 0x07, 0x78, 0xfd, 0x94, 0x9e, 0x35, 0x25, 0xfa, 0xd5, 0xd7, 0x8c, 0xd2, 0x29, 0xcc, 0x54, 0x74, 0x1b, 0xe7, 0x4d,
/* (2^184)P */ 0xc9, 0x6a, 0xda, 0x1e, 0xad, 0x60, 0xeb, 0x42, 0x3a, 0x9c, 0xc0, 0xdb, 0xdf, 0x37, 0xad, 0x0a, 0x91, 0xc1, 0x3c, 0xe3, 0x71, 0x4b, 0x00, 0x81, 0x3c, 0x80, 0x22, 0x51, 0x34, 0xbe, 0xe6, 0x44,
/* (2^185)P */ 0xdb, 0x20, 0x19, 0xba, 0x88, 0x83, 0xfe, 0x03, 0x08, 0xb0, 0x0d, 0x15, 0x32, 0x7c, 0xd5, 0xf5, 0x29, 0x0c, 0xf6, 0x1a, 0x28, 0xc4, 0xc8, 0x49, 0xee, 0x1a, 0x70, 0xde, 0x18, 0xb5, 0xed, 0x21,
/* (2^186)P */ 0x99, 0xdc, 0x06, 0x8f, 0x41, 0x3e, 0xb6, 0x7f, 0xb8, 0xd7, 0x66, 0xc1, 0x99, 0x0d, 0x46, 0xa4, 0x83, 0x0a, 0x52, 0xce, 0x48, 0x52, 0xdd, 0x24, 0x58, 0x83, 0x92, 0x2b, 0x71, 0xad, 0xc3, 0x5e,
/* (2^187)P */ 0x0f, 0x93, 0x17, 0xbd, 0x5f, 0x2a, 0x02, 0x15, 0xe3, 0x70, 0x25, 0xd8, 0x77, 0x4a, 0xf6, 0xa4, 0x12, 0x37, 0x78, 0x15, 0x69, 0x8d, 0xbc, 0x12, 0xbb, 0x0a, 0x62, 0xfc, 0xc0, 0x94, 0x81, 0x49,
/* (2^188)P */ 0x82, 0x6c, 0x68, 0x55, 0xd2, 0xd9, 0xa2, 0x38, 0xf0, 0x21, 0x3e, 0x19, 0xd9, 0x6b, 0x5c, 0x78, 0x84, 0x54, 0x4a, 0xb2, 0x1a, 0xc8, 0xd5, 0xe4, 0x89, 0x09, 0xe2, 0xb2, 0x60, 0x78, 0x30, 0x56,
/* (2^189)P */ 0xc4, 0x74, 0x4d, 0x8b, 0xf7, 0x55, 0x9d, 0x42, 0x31, 0x01, 0x35, 0x43, 0x46, 0x83, 0xf1, 0x22, 0xff, 0x1f, 0xc7, 0x98, 0x45, 0xc2, 0x60, 0x1e, 0xef, 0x83, 0x99, 0x97, 0x14, 0xf0, 0xf2, 0x59,
/* (2^190)P */ 0x44, 0x4a, 0x49, 0xeb, 0x56, 0x7d, 0xa4, 0x46, 0x8e, 0xa1, 0x36, 0xd6, 0x54, 0xa8, 0x22, 0x3e, 0x3b, 0x1c, 0x49, 0x74, 0x52, 0xe1, 0x46, 0xb3, 0xe7, 0xcd, 0x90, 0x53, 0x4e, 0xfd, 0xea, 0x2c,
/* (2^191)P */ 0x75, 0x66, 0x0d, 0xbe, 0x38, 0x85, 0x8a, 0xba, 0x23, 0x8e, 0x81, 0x50, 0xbb, 0x74, 0x90, 0x4b, 0xc3, 0x04, 0xd3, 0x85, 0x90, 0xb8, 0xda, 0xcb, 0xc4, 0x92, 0x61, 0xe5, 0xe0, 0x4f, 0xa2, 0x61,
/* (2^192)P */ 0xcb, 0x5b, 0x52, 0xdb, 0xe6, 0x15, 0x76, 0xcb, 0xca, 0xe4, 0x67, 0xa5, 0x35, 0x8c, 0x7d, 0xdd, 0x69, 0xdd, 0xfc, 0xca, 0x3a, 0x15, 0xb4, 0xe6, 0x66, 0x97, 0x3c, 0x7f, 0x09, 0x8e, 0x66, 0x2d,
/* (2^193)P */ 0xf0, 0x5e, 0xe5, 0x5c, 0x26, 0x7e, 0x7e, 0xa5, 0x67, 0xb9, 0xd4, 0x7c, 0x52, 0x4e, 0x9f, 0x5d, 0xe5, 0xd1, 0x2f, 0x49, 0x06, 0x36, 0xc8, 0xfb, 0xae, 0xf7, 0xc3, 0xb7, 0xbe, 0x52, 0x0d, 0x09,
/* (2^194)P */ 0x7c, 0x4d, 0x7b, 0x1e, 0x5a, 0x51, 0xb9, 0x09, 0xc0, 0x44, 0xda, 0x99, 0x25, 0x6a, 0x26, 0x1f, 0x04, 0x55, 0xc5, 0xe2, 0x48, 0x95, 0xc4, 0xa1, 0xcc, 0x15, 0x6f, 0x12, 0x87, 0x42, 0xf0, 0x7e,
/* (2^195)P */ 0x15, 0xef, 0x30, 0xbd, 0x9d, 0x65, 0xd1, 0xfe, 0x7b, 0x27, 0xe0, 0xc4, 0xee, 0xb9, 0x4a, 0x8b, 0x91, 0x32, 0xdf, 0xa5, 0x36, 0x62, 0x4d, 0x88, 0x88, 0xf7, 0x5c, 0xbf, 0xa6, 0x6e, 0xd9, 0x1f,
/* (2^196)P */ 0x9a, 0x0d, 0x19, 0x1f, 0x98, 0x61, 0xa1, 0x42, 0xc1, 0x52, 0x60, 0x7e, 0x50, 0x49, 0xd8, 0x61, 0xd5, 0x2c, 0x5a, 0x28, 0xbf, 0x13, 0xe1, 0x9f, 0xd8, 0x85, 0xad, 0xdb, 0x76, 0xd6, 0x22, 0x7c,
/* (2^197)P */ 0x7d, 0xd2, 0xfb, 0x2b, 0xed, 0x70, 0xe7, 0x82, 0xa5, 0xf5, 0x96, 0xe9, 0xec, 0xb2, 0x05, 0x4c, 0x50, 0x01, 0x90, 0xb0, 0xc2, 0xa9, 0x40, 0xcd, 0x64, 0xbf, 0xd9, 0x13, 0x92, 0x31, 0x95, 0x58,
/* (2^198)P */ 0x08, 0x2e, 0xea, 0x3f, 0x70, 0x5d, 0xcc, 0xe7, 0x8c, 0x18, 0xe2, 0x58, 0x12, 0x49, 0x0c, 0xb5, 0xf0, 0x5b, 0x20, 0x48, 0xaa, 0x0b, 0xe3, 0xcc, 0x62, 0x2d, 0xa3, 0xcf, 0x9c, 0x65, 0x7c, 0x53,
/* (2^199)P */ 0x88, 0xc0, 0xcf, 0x98, 0x3a, 0x62, 0xb6, 0x37, 0xa4, 0xac, 0xd6, 0xa4, 0x1f, 0xed, 0x9b, 0xfe, 0xb0, 0xd1, 0xa8, 0x56, 0x8e, 0x9b, 0xd2, 0x04, 0x75, 0x95, 0x51, 0x0b, 0xc4, 0x71, 0x5f, 0x72,
/* (2^200)P */ 0xe6, 0x9c, 0x33, 0xd0, 0x9c, 0xf8, 0xc7, 0x28, 0x8b, 0xc1, 0xdd, 0x69, 0x44, 0xb1, 0x67, 0x83, 0x2c, 0x65, 0xa1, 0xa6, 0x83, 0xda, 0x3a, 0x88, 0x17, 0x6c, 0x4d, 0x03, 0x74, 0x19, 0x5f, 0x58,
/* (2^201)P */ 0x88, 0x91, 0xb1, 0xf1, 0x66, 0xb2, 0xcf, 0x89, 0x17, 0x52, 0xc3, 0xe7, 0x63, 0x48, 0x3b, 0xe6, 0x6a, 0x52, 0xc0, 0xb4, 0xa6, 0x9d, 0x8c, 0xd8, 0x35, 0x46, 0x95, 0xf0, 0x9d, 0x5c, 0x03, 0x3e,
/* (2^202)P */ 0x9d, 0xde, 0x45, 0xfb, 0x12, 0x54, 0x9d, 0xdd, 0x0d, 0xf4, 0xcf, 0xe4, 0x32, 0x45, 0x68, 0xdd, 0x1c, 0x67, 0x1d, 0x15, 0x9b, 0x99, 0x5c, 0x4b, 0x90, 0xf6, 0xe7, 0x11, 0xc8, 0x2c, 0x8c, 0x2d,
/* (2^203)P */ 0x40, 0x5d, 0x05, 0x90, 0x1d, 0xbe, 0x54, 0x7f, 0x40, 0xaf, 0x4a, 0x46, 0xdf, 0xc5, 0x64, 0xa4, 0xbe, 0x17, 0xe9, 0xf0, 0x24, 0x96, 0x97, 0x33, 0x30, 0x6b, 0x35, 0x27, 0xc5, 0x8d, 0x01, 0x2c,
/* (2^204)P */ 0xd4, 0xb3, 0x30, 0xe3, 0x24, 0x50, 0x41, 0xa5, 0xd3, 0x52, 0x16, 0x69, 0x96, 0x3d, 0xff, 0x73, 0xf1, 0x59, 0x9b, 0xef, 0xc4, 0x42, 0xec, 0x94, 0x5a, 0x8e, 0xd0, 0x18, 0x16, 0x20, 0x47, 0x07,
/* (2^205)P */ 0x53, 0x1c, 0x41, 0xca, 0x8a, 0xa4, 0x6c, 0x4d, 0x19, 0x61, 0xa6, 0xcf, 0x2f, 0x5f, 0x41, 0x66, 0xff, 0x27, 0xe2, 0x51, 0x00, 0xd4, 0x4d, 0x9c, 0xeb, 0xf7, 0x02, 0x9a, 0xc0, 0x0b, 0x81, 0x59,
/* (2^206)P */ 0x1d, 0x10, 0xdc, 0xb3, 0x71, 0xb1, 0x7e, 0x2a, 0x8e, 0xf6, 0xfe, 0x9f, 0xb9, 0x5a, 0x1c, 0x44, 0xea, 0x59, 0xb3, 0x93, 0x9b, 0x5c, 0x02, 0x32, 0x2f, 0x11, 0x9d, 0x1e, 0xa7, 0xe0, 0x8c, 0x5e,
/* (2^207)P */ 0xfd, 0x03, 0x95, 0x42, 0x92, 0xcb, 0xcc, 0xbf, 0x55, 0x5d, 0x09, 0x2f, 0x75, 0xba, 0x71, 0xd2, 0x1e, 0x09, 0x2d, 0x97, 0x5e, 0xad, 0x5e, 0x34, 0xba, 0x03, 0x31, 0xa8, 0x11, 0xdf, 0xc8, 0x18,
/* (2^208)P */ 0x4c, 0x0f, 0xed, 0x9a, 0x9a, 0x94, 0xcd, 0x90, 0x7e, 0xe3, 0x60, 0x66, 0xcb, 0xf4, 0xd1, 0xc5, 0x0b, 0x2e, 0xc5, 0x56, 0x2d, 0xc5, 0xca, 0xb8, 0x0d, 0x8e, 0x80, 0xc5, 0x00, 0xe4, 0x42, 0x6e,
/* (2^209)P */ 0x23, 0xfd, 0xae, 0xee, 0x66, 0x69, 0xb4, 0xa3, 0xca, 0xcd, 0x9e, 0xe3, 0x0b, 0x1f, 0x4f, 0x0c, 0x1d, 0xa5, 0x83, 0xd6, 0xc9, 0xc8, 0x9d, 0x18, 0x1b, 0x35, 0x09, 0x4c, 0x05, 0x7f, 0xf2, 0x51,
/* (2^210)P */ 0x82, 0x06, 0x32, 0x2a, 0xcd, 0x7c, 0x48, 0x4c, 0x96, 0x1c, 0xdf, 0xb3, 0x5b, 0xa9, 0x7e, 0x58, 0xe8, 0xb8, 0x5c, 0x55, 0x9e, 0xf7, 0xcc, 0xc8, 0x3d, 0xd7, 0x06, 0xa2, 0x29, 0xc8, 0x7d, 0x54,
/* (2^211)P */ 0x06, 0x9b, 0xc3, 0x80, 0xcd, 0xa6, 0x22, 0xb8, 0xc6, 0xd4, 0x00, 0x20, 0x73, 0x54, 0x6d, 0xe9, 0x4d, 0x3b, 0x46, 0x91, 0x6f, 0x5b, 0x53, 0x28, 0x1d, 0x6e, 0x48, 0xe2, 0x60, 0x46, 0x8f, 0x22,
/* (2^212)P */ 0xbf, 0x3a, 0x8d, 0xde, 0x38, 0x95, 0x79, 0x98, 0x6e, 0xca, 0xeb, 0x45, 0x00, 0x33, 0xd8, 0x8c, 0x38, 0xe7, 0x21, 0x82, 0x00, 0x2a, 0x95, 0x79, 0xbb, 0xd2, 0x5c, 0x53, 0xa7, 0xe1, 0x22, 0x43,
/* (2^213)P */ 0x1c, 0x80, 0xd1, 0x19, 0x18, 0xc1, 0x14, 0xb1, 0xc7, 0x5e, 0x3f, 0x4f, 0xd8, 0xe4, 0x16, 0x20, 0x4c, 0x0f, 0x26, 0x09, 0xf4, 0x2d, 0x0e, 0xdd, 0x66, 0x72, 0x5f, 0xae, 0xc0, 0x62, 0xc3, 0x5e,
/* (2^214)P */ 0xee, 0xb4, 0xb2, 0xb8, 0x18, 0x2b, 0x46, 0xc0, 0xfb, 0x1a, 0x4d, 0x27, 0x50, 0xd9, 0xc8, 0x7c, 0xd2, 0x02, 0x6b, 0x43, 0x05, 0x71, 0x5f, 0xf2, 0xd3, 0xcc, 0xf9, 0xbf, 0xdc, 0xf8, 0xbb, 0x43,
/* (2^215)P */ 0xdf, 0xe9, 0x39, 0xa0, 0x67, 0x17, 0xad, 0xb6, 0x83, 0x35, 0x9d, 0xf6, 0xa8, 0x4d, 0x71, 0xb0, 0xf5, 0x31, 0x29, 0xb4, 0x18, 0xfa, 0x55, 0x5e, 0x61, 0x09, 0xc6, 0x33, 0x8f, 0x55, 0xd5, 0x4e,
/* (2^216)P */ 0xdd, 0xa5, 0x47, 0xc6, 0x01, 0x79, 0xe3, 0x1f, 0x57, 0xd3, 0x81, 0x80, 0x1f, 0xdf, 0x3d, 0x59, 0xa6, 0xd7, 0x3f, 0x81, 0xfd, 0xa4, 0x49, 0x02, 0x61, 0xaf, 0x9c, 0x4e, 0x27, 0xca, 0xac, 0x69,
/* (2^217)P */ 0xc9, 0x21, 0x07, 0x33, 0xea, 0xa3, 0x7b, 0x04, 0xa0, 0x1e, 0x7e, 0x0e, 0xc2, 0x3f, 0x42, 0x83, 0x60, 0x4a, 0x31, 0x01, 0xaf, 0xc0, 0xf4, 0x1d, 0x27, 0x95, 0x28, 0x89, 0xab, 0x2d, 0xa6, 0x09,
/* (2^218)P */ 0x00, 0xcb, 0xc6, 0x9c, 0xa4, 0x25, 0xb3, 0xa5, 0xb6, 0x6c, 0xb5, 0x54, 0xc6, 0x5d, 0x4b, 0xe9, 0xa0, 0x94, 0xc9, 0xad, 0x79, 0x87, 0xe2, 0x3b, 0xad, 0x4a, 0x3a, 0xba, 0xf8, 0xe8, 0x96, 0x42,
/* (2^219)P */ 0xab, 0x1e, 0x45, 0x1e, 0x76, 0x89, 0x86, 0x32, 0x4a, 0x59, 0x59, 0xff, 0x8b, 0x59, 0x4d, 0x2e, 0x4a, 0x08, 0xa7, 0xd7, 0x53, 0x68, 0xb9, 0x49, 0xa8, 0x20, 0x14, 0x60, 0x19, 0xa3, 0x80, 0x49,
/* (2^220)P */ 0x42, 0x2c, 0x55, 0x2f, 0xe1, 0xb9, 0x65, 0x95, 0x96, 0xfe, 0x00, 0x71, 0xdb, 0x18, 0x53, 0x8a, 0xd7, 0xd0, 0xad, 0x43, 0x4d, 0x0b, 0xc9, 0x05, 0xda, 0x4e, 0x5d, 0x6a, 0xd6, 0x4c, 0x8b, 0x53,
/* (2^221)P */ 0x9f, 0x03, 0x9f, 0xe8, 0xc3, 0x4f, 0xe9, 0xf4, 0x45, 0x80, 0x61, 0x6f, 0xf2, 0x9a, 0x2c, 0x59, 0x50, 0x95, 0x4b, 0xfd, 0xb5, 0x6e, 0xa3, 0x08, 0x19, 0x14, 0xed, 0xc2, 0xf6, 0xfa, 0xff, 0x25,
/* (2^222)P */ 0x54, 0xd3, 0x79, 0xcc, 0x59, 0x44, 0x43, 0x34, 0x6b, 0x47, 0xd5, 0xb1, 0xb4, 0xbf, 0xec, 0xee, 0x99, 0x5d, 0x61, 0x61, 0xa0, 0x34, 0xeb, 0xdd, 0x73, 0xb7, 0x64, 0xeb, 0xcc, 0xce, 0x29, 0x51,
/* (2^223)P */ 0x20, 0x35, 0x99, 0x94, 0x58, 0x21, 0x43, 0xee, 0x3b, 0x0b, 0x4c, 0xf1, 0x7c, 0x9c, 0x2f, 0x77, 0xd5, 0xda, 0xbe, 0x06, 0xe3, 0xfc, 0xe2, 0xd2, 0x97, 0x6a, 0xf0, 0x46, 0xb5, 0x42, 0x5f, 0x71,
/* (2^224)P */ 0x1a, 0x5f, 0x5b, 0xda, 0xce, 0xcd, 0x4e, 0x43, 0xa9, 0x41, 0x97, 0xa4, 0x15, 0x71, 0xa1, 0x0d, 0x2e, 0xad, 0xed, 0x73, 0x7c, 0xd7, 0x0b, 0x68, 0x41, 0x90, 0xdd, 0x4e, 0x35, 0x02, 0x7c, 0x48,
/* (2^225)P */ 0xc4, 0xd9, 0x0e, 0xa7, 0xf3, 0xef, 0xef, 0xb8, 0x02, 0xe3, 0x57, 0xe8, 0xa3, 0x2a, 0xa3, 0x56, 0xa0, 0xa5, 0xa2, 0x48, 0xbd, 0x68, 0x3a, 0xdf, 0x44, 0xc4, 0x76, 0x31, 0xb7, 0x50, 0xf6, 0x07,
/* (2^226)P */ 0xb1, 0xcc, 0xe0, 0x26, 0x16, 0x9b, 0x8b, 0xe3, 0x36, 0xfb, 0x09, 0x8b, 0xc1, 0x53, 0xe0, 0x79, 0x64, 0x49, 0xf9, 0xc9, 0x19, 0x03, 0xd9, 0x56, 0xc4, 0xf5, 0x9f, 0xac, 0xe7, 0x41, 0xa9, 0x1c,
/* (2^227)P */ 0xbb, 0xa0, 0x2f, 0x16, 0x29, 0xdf, 0xc4, 0x49, 0x05, 0x33, 0xb3, 0x82, 0x32, 0xcf, 0x88, 0x84, 0x7d, 0x43, 0xbb, 0xca, 0x14, 0xda, 0xdf, 0x95, 0x86, 0xad, 0xd5, 0x64, 0x82, 0xf7, 0x91, 0x33,
/* (2^228)P */ 0x5d, 0x09, 0xb5, 0xe2, 0x6a, 0xe0, 0x9a, 0x72, 0x46, 0xa9, 0x59, 0x32, 0xd7, 0x58, 0x8a, 0xd5, 0xed, 0x21, 0x39, 0xd1, 0x62, 0x42, 0x83, 0xe9, 0x92, 0xb5, 0x4b, 0xa5, 0xfa, 0xda, 0xfe, 0x27,
/* (2^229)P */ 0xbb, 0x48, 0xad, 0x29, 0xb8, 0xc5, 0x9d, 0xa9, 0x60, 0xe2, 0x9e, 0x49, 0x42, 0x57, 0x02, 0x5f, 0xfd, 0x13, 0x75, 0x5d, 0xcd, 0x8e, 0x2c, 0x80, 0x38, 0xd9, 0x6d, 0x3f, 0xef, 0xb3, 0xce, 0x78,
/* (2^230)P */ 0x94, 0x5d, 0x13, 0x8a, 0x4f, 0xf4, 0x42, 0xc3, 0xa3, 0xdd, 0x8c, 0x82, 0x44, 0xdb, 0x9e, 0x7b, 0xe7, 0xcf, 0x37, 0x05, 0x1a, 0xd1, 0x36, 0x94, 0xc8, 0xb4, 0x1a, 0xec, 0x64, 0xb1, 0x64, 0x50,
/* (2^231)P */ 0xfc, 0xb2, 0x7e, 0xd3, 0xcf, 0xec, 0x20, 0x70, 0xfc, 0x25, 0x0d, 0xd9, 0x3e, 0xea, 0x31, 0x1f, 0x34, 0xbb, 0xa1, 0xdf, 0x7b, 0x0d, 0x93, 0x1b, 0x44, 0x30, 0x11, 0x48, 0x7a, 0x46, 0x44, 0x53,
/* (2^232)P */ 0xfb, 0x6d, 0x5e, 0xf2, 0x70, 0x31, 0x07, 0x70, 0xc8, 0x4c, 0x11, 0x50, 0x1a, 0xdc, 0x85, 0xe3, 0x00, 0x4f, 0xfc, 0xc8, 0x8a, 0x69, 0x48, 0x23, 0xd8, 0x40, 0xdd, 0x84, 0x52, 0xa5, 0x77, 0x2a,
/* (2^233)P */ 0xe4, 0x6c, 0x8c, 0xc9, 0xe0, 0xaf, 0x06, 0xfe, 0xe4, 0xd6, 0xdf, 0xdd, 0x96, 0xdf, 0x35, 0xc2, 0xd3, 0x1e, 0xbf, 0x33, 0x1e, 0xd0, 0x28, 0x14, 0xaf, 0xbd, 0x00, 0x93, 0xec, 0x68, 0x57, 0x78,
/* (2^234)P */ 0x3b, 0xb6, 0xde, 0x91, 0x7a, 0xe5, 0x02, 0x97, 0x80, 0x8b, 0xce, 0xe5, 0xbf, 0xb8, 0xbd, 0x61, 0xac, 0x58, 0x1d, 0x3d, 0x6f, 0x42, 0x5b, 0x64, 0xbc, 0x57, 0xa5, 0x27, 0x22, 0xa8, 0x04, 0x48,
/* (2^235)P */ 0x01, 0x26, 0x4d, 0xb4, 0x8a, 0x04, 0x57, 0x8e, 0x35, 0x69, 0x3a, 0x4b, 0x1a, 0x50, 0xd6, 0x68, 0x93, 0xc2, 0xe1, 0xf9, 0xc3, 0x9e, 0x9c, 0xc3, 0xe2, 0x63, 0xde, 0xd4, 0x57, 0xf2, 0x72, 0x41,
/* (2^236)P */ 0x01, 0x64, 0x0c, 0x33, 0x50, 0xb4, 0x68, 0xd3, 0x91, 0x23, 0x8f, 0x41, 0x17, 0x30, 0x0d, 0x04, 0x0d, 0xd9, 0xb7, 0x90, 0x60, 0xbb, 0x34, 0x2c, 0x1f, 0xd5, 0xdf, 0x8f, 0x22, 0x49, 0xf6, 0x16,
/* (2^237)P */ 0xf5, 0x8e, 0x92, 0x2b, 0x8e, 0x81, 0xa6, 0xbe, 0x72, 0x1e, 0xc1, 0xcd, 0x91, 0xcf, 0x8c, 0xe2, 0xcd, 0x36, 0x7a, 0xe7, 0x68, 0xaa, 0x4a, 0x59, 0x0f, 0xfd, 0x7f, 0x6c, 0x80, 0x34, 0x30, 0x31,
/* (2^238)P */ 0x65, 0xbd, 0x49, 0x22, 0xac, 0x27, 0x9d, 0x8a, 0x12, 0x95, 0x8e, 0x01, 0x64, 0xb4, 0xa3, 0x19, 0xc7, 0x7e, 0xb3, 0x52, 0xf3, 0xcf, 0x6c, 0xc2, 0x21, 0x7b, 0x79, 0x1d, 0x34, 0x68, 0x6f, 0x05,
/* (2^239)P */ 0x27, 0x23, 0xfd, 0x7e, 0x75, 0xd6, 0x79, 0x5e, 0x15, 0xfe, 0x3a, 0x55, 0xb6, 0xbc, 0xbd, 0xfa, 0x60, 0x5a, 0xaf, 0x6e, 0x2c, 0x22, 0xe7, 0xd3, 0x3b, 0x74, 0xae, 0x4d, 0x6d, 0xc7, 0x46, 0x70,
/* (2^240)P */ 0x55, 0x4a, 0x8d, 0xb1, 0x72, 0xe8, 0x0b, 0x66, 0x96, 0x14, 0x4e, 0x57, 0x18, 0x25, 0x99, 0x19, 0xbb, 0xdc, 0x2b, 0x30, 0x3a, 0x05, 0x03, 0xc1, 0x8e, 0x8e, 0x21, 0x0b, 0x80, 0xe9, 0xd8, 0x3e,
/* (2^241)P */ 0x3e, 0xe0, 0x75, 0xfa, 0x39, 0x92, 0x0b, 0x7b, 0x83, 0xc0, 0x33, 0x46, 0x68, 0xfb, 0xe9, 0xef, 0x93, 0x77, 0x1a, 0x39, 0xbe, 0x5f, 0xa3, 0x98, 0x34, 0xfe, 0xd0, 0xe2, 0x0f, 0x51, 0x65, 0x60,
/* (2^242)P */ 0x0c, 0xad, 0xab, 0x48, 0x85, 0x66, 0xcb, 0x55, 0x27, 0xe5, 0x87, 0xda, 0x48, 0x45, 0x58, 0xb4, 0xdd, 0xc1, 0x07, 0x01, 0xea, 0xec, 0x43, 0x2c, 0x35, 0xde, 0x72, 0x93, 0x80, 0x28, 0x60, 0x52,
/* (2^243)P */ 0x1f, 0x3b, 0x21, 0xf9, 0x6a, 0xc5, 0x15, 0x34, 0xdb, 0x98, 0x7e, 0x01, 0x4d, 0x1a, 0xee, 0x5b, 0x9b, 0x70, 0xcf, 0xb5, 0x05, 0xb1, 0xf6, 0x13, 0xb6, 0x9a, 0xb2, 0x82, 0x34, 0x0e, 0xf2, 0x5f,
/* (2^244)P */ 0x90, 0x6c, 0x2e, 0xcc, 0x75, 0x9c, 0xa2, 0x0a, 0x06, 0xe2, 0x70, 0x3a, 0xca, 0x73, 0x7d, 0xfc, 0x15, 0xc5, 0xb5, 0xc4, 0x8f, 0xc3, 0x9f, 0x89, 0x07, 0xc2, 0xff, 0x24, 0xb1, 0x86, 0x03, 0x25,
/* (2^245)P */ 0x56, 0x2b, 0x3d, 0xae, 0xd5, 0x28, 0xea, 0x54, 0xce, 0x60, 0xde, 0xd6, 0x9d, 0x14, 0x13, 0x99, 0xc1, 0xd6, 0x06, 0x8f, 0xc5, 0x4f, 0x69, 0x16, 0xc7, 0x8f, 0x01, 0xeb, 0x75, 0x39, 0xb2, 0x46,
/* (2^246)P */ 0xe2, 0xb4, 0xb7, 0xb4, 0x0f, 0x6a, 0x0a, 0x47, 0xde, 0x53, 0x72, 0x8f, 0x5a, 0x47, 0x92, 0x5d, 0xdb, 0x3a, 0xbd, 0x2f, 0xb5, 0xe5, 0xee, 0xab, 0x68, 0x69, 0x80, 0xa0, 0x01, 0x08, 0xa2, 0x7f,
/* (2^247)P */ 0xd2, 0x14, 0x77, 0x9f, 0xf1, 0xfa, 0xf3, 0x76, 0xc3, 0x60, 0x46, 0x2f, 0xc1, 0x40, 0xe8, 0xb3, 0x4e, 0x74, 0x12, 0xf2, 0x8d, 0xcd, 0xb4, 0x0f, 0xd2, 0x2d, 0x3a, 0x1d, 0x25, 0x5a, 0x06, 0x4b,
/* (2^248)P */ 0x4a, 0xcd, 0x77, 0x3d, 0x38, 0xde, 0xeb, 0x5c, 0xb1, 0x9c, 0x2c, 0x88, 0xdf, 0x39, 0xdf, 0x6a, 0x59, 0xf7, 0x9a, 0xb0, 0x2e, 0x24, 0xdd, 0xa2, 0x22, 0x64, 0x5f, 0x0e, 0xe5, 0xc0, 0x47, 0x31,
/* (2^249)P */ 0xdb, 0x50, 0x13, 0x1d, 0x10, 0xa5, 0x4c, 0x16, 0x62, 0xc9, 0x3f, 0xc3, 0x79, 0x34, 0xd1, 0xf8, 0x08, 0xda, 0xe5, 0x13, 0x4d, 0xce, 0x40, 0xe6, 0xba, 0xf8, 0x61, 0x50, 0xc4, 0xe0, 0xde, 0x4b,
/* (2^250)P */ 0xc9, 0xb1, 0xed, 0xa4, 0xc1, 0x6d, 0xc4, 0xd7, 0x8a, 0xd9, 0x7f, 0x43, 0xb6, 0xd7, 0x14, 0x55, 0x0b, 0xc0, 0xa1, 0xb2, 0x6b, 0x2f, 0x94, 0x58, 0x0e, 0x71, 0x70, 0x1d, 0xab, 0xb2, 0xff, 0x2d,
/* (2^251)P */ 0x68, 0x6d, 0x8b, 0xc1, 0x2f, 0xcf, 0xdf, 0xcc, 0x67, 0x61, 0x80, 0xb7, 0xa8, 0xcb, 0xeb, 0xa8, 0xe3, 0x37, 0x29, 0x5e, 0xf9, 0x97, 0x06, 0x98, 0x8c, 0x6e, 0x12, 0xd0, 0x1c, 0xba, 0xfb, 0x02,
/* (2^252)P */ 0x65, 0x45, 0xff, 0xad, 0x60, 0xc3, 0x98, 0xcb, 0x19, 0x15, 0xdb, 0x4b, 0xd2, 0x01, 0x71, 0x44, 0xd5, 0x15, 0xfb, 0x75, 0x74, 0xc8, 0xc4, 0x98, 0x7d, 0xa2, 0x22, 0x6e, 0x6d, 0xc7, 0xf8, 0x05,
/* (2^253)P */ 0x94, 0xf4, 0xb9, 0xfe, 0xdf, 0xe5, 0x69, 0xab, 0x75, 0x6b, 0x40, 0x18, 0x9d, 0xc7, 0x09, 0xae, 0x1d, 0x2d, 0xa4, 0x94, 0xfb, 0x45, 0x9b, 0x19, 0x84, 0xfa, 0x2a, 0xae, 0xeb, 0x0a, 0x71, 0x79,
/* (2^254)P */ 0xdf, 0xd2, 0x34, 0xf3, 0xa7, 0xed, 0xad, 0xa6, 0xb4, 0x57, 0x2a, 0xaf, 0x51, 0x9c, 0xde, 0x7b, 0xa8, 0xea, 0xdc, 0x86, 0x4f, 0xc6, 0x8f, 0xa9, 0x7b, 0xd0, 0x0e, 0xc2, 0x35, 0x03, 0xbe, 0x6b,
/* (2^255)P */ 0x44, 0x43, 0x98, 0x53, 0xbe, 0xdc, 0x7f, 0x66, 0xa8, 0x49, 0x59, 0x00, 0x1c, 0xbc, 0x72, 0x07, 0x8e, 0xd6, 0xbe, 0x4e, 0x9f, 0xa4, 0x07, 0xba, 0xbf, 0x30, 0xdf, 0xba, 0x85, 0xb0, 0xa7, 0x1f,
}
package x448
import (
fp "github.com/cloudflare/circl/math/fp448"
)
// ladderJoye calculates a fixed-point multiplication with the generator point.
// The algorithm is the right-to-left Joye's ladder as described
// in "How to precompute a ladder" in SAC'2017.
func ladderJoye(k *Key) {
w := [5]fp.Elt{} // [mu,x1,z1,x2,z2] order must be preserved.
w[1] = fp.Elt{ // x1 = S
0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
}
fp.SetOne(&w[2]) // z1 = 1
w[3] = fp.Elt{ // x2 = G-S
0x20, 0x27, 0x9d, 0xc9, 0x7d, 0x19, 0xb1, 0xac,
0xf8, 0xba, 0x69, 0x1c, 0xff, 0x33, 0xac, 0x23,
0x51, 0x1b, 0xce, 0x3a, 0x64, 0x65, 0xbd, 0xf1,
0x23, 0xf8, 0xc1, 0x84, 0x9d, 0x45, 0x54, 0x29,
0x67, 0xb9, 0x81, 0x1c, 0x03, 0xd1, 0xcd, 0xda,
0x7b, 0xeb, 0xff, 0x1a, 0x88, 0x03, 0xcf, 0x3a,
0x42, 0x44, 0x32, 0x01, 0x25, 0xb7, 0xfa, 0xf0,
}
fp.SetOne(&w[4]) // z2 = 1
const n = 448
const h = 2
swap := uint(1)
for s := 0; s < n-h; s++ {
i := (s + h) / 8
j := (s + h) % 8
bit := uint((k[i] >> uint(j)) & 1)
copy(w[0][:], tableGenerator[s*Size:(s+1)*Size])
diffAdd(&w, swap^bit)
swap = bit
}
for s := 0; s < h; s++ {
double(&w[1], &w[2])
}
toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
}
// ladderMontgomery calculates a generic scalar point multiplication
// The algorithm implemented is the left-to-right Montgomery's ladder.
func ladderMontgomery(k, xP *Key) {
w := [5]fp.Elt{} // [x1, x2, z2, x3, z3] order must be preserved.
w[0] = *(*fp.Elt)(xP) // x1 = xP
fp.SetOne(&w[1]) // x2 = 1
w[3] = *(*fp.Elt)(xP) // x3 = xP
fp.SetOne(&w[4]) // z3 = 1
move := uint(0)
for s := 448 - 1; s >= 0; s-- {
i := s / 8
j := s % 8
bit := uint((k[i] >> uint(j)) & 1)
ladderStep(&w, move^bit)
move = bit
}
toAffine((*[fp.Size]byte)(k), &w[1], &w[2])
}
func toAffine(k *[fp.Size]byte, x, z *fp.Elt) {
fp.Inv(z, z)
fp.Mul(x, x, z)
_ = fp.ToBytes(k[:], x)
}
var lowOrderPoints = [3]fp.Elt{
{ /* (0,_,1) point of order 2 on Curve448 */
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
},
{ /* (1,_,1) a point of order 4 on the twist of Curve448 */
0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
},
{ /* (-1,_,1) point of order 4 on Curve448 */
0xfe, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
},
}
//go:build amd64 && !purego
// +build amd64,!purego
package x448
import (
fp "github.com/cloudflare/circl/math/fp448"
"golang.org/x/sys/cpu"
)
var hasBmi2Adx = cpu.X86.HasBMI2 && cpu.X86.HasADX
var _ = hasBmi2Adx
func double(x, z *fp.Elt) { doubleAmd64(x, z) }
func diffAdd(w *[5]fp.Elt, b uint) { diffAddAmd64(w, b) }
func ladderStep(w *[5]fp.Elt, b uint) { ladderStepAmd64(w, b) }
func mulA24(z, x *fp.Elt) { mulA24Amd64(z, x) }
//go:noescape
func doubleAmd64(x, z *fp.Elt)
//go:noescape
func diffAddAmd64(w *[5]fp.Elt, b uint)
//go:noescape
func ladderStepAmd64(w *[5]fp.Elt, b uint)
//go:noescape
func mulA24Amd64(z, x *fp.Elt)
#define ladderStepLeg \
addSub(x2,z2) \
addSub(x3,z3) \
integerMulLeg(b0,x2,z3) \
integerMulLeg(b1,x3,z2) \
reduceFromDoubleLeg(t0,b0) \
reduceFromDoubleLeg(t1,b1) \
addSub(t0,t1) \
cselect(x2,x3,regMove) \
cselect(z2,z3,regMove) \
integerSqrLeg(b0,t0) \
integerSqrLeg(b1,t1) \
reduceFromDoubleLeg(x3,b0) \
reduceFromDoubleLeg(z3,b1) \
integerMulLeg(b0,x1,z3) \
reduceFromDoubleLeg(z3,b0) \
integerSqrLeg(b0,x2) \
integerSqrLeg(b1,z2) \
reduceFromDoubleLeg(x2,b0) \
reduceFromDoubleLeg(z2,b1) \
subtraction(t0,x2,z2) \
multiplyA24Leg(t1,t0) \
additionLeg(t1,t1,z2) \
integerMulLeg(b0,x2,z2) \
integerMulLeg(b1,t0,t1) \
reduceFromDoubleLeg(x2,b0) \
reduceFromDoubleLeg(z2,b1)
#define ladderStepBmi2Adx \
addSub(x2,z2) \
addSub(x3,z3) \
integerMulAdx(b0,x2,z3) \
integerMulAdx(b1,x3,z2) \
reduceFromDoubleAdx(t0,b0) \
reduceFromDoubleAdx(t1,b1) \
addSub(t0,t1) \
cselect(x2,x3,regMove) \
cselect(z2,z3,regMove) \
integerSqrAdx(b0,t0) \
integerSqrAdx(b1,t1) \
reduceFromDoubleAdx(x3,b0) \
reduceFromDoubleAdx(z3,b1) \
integerMulAdx(b0,x1,z3) \
reduceFromDoubleAdx(z3,b0) \
integerSqrAdx(b0,x2) \
integerSqrAdx(b1,z2) \
reduceFromDoubleAdx(x2,b0) \
reduceFromDoubleAdx(z2,b1) \
subtraction(t0,x2,z2) \
multiplyA24Adx(t1,t0) \
additionAdx(t1,t1,z2) \
integerMulAdx(b0,x2,z2) \
integerMulAdx(b1,t0,t1) \
reduceFromDoubleAdx(x2,b0) \
reduceFromDoubleAdx(z2,b1)
#define difAddLeg \
addSub(x1,z1) \
integerMulLeg(b0,z1,ui) \
reduceFromDoubleLeg(z1,b0) \
addSub(x1,z1) \
integerSqrLeg(b0,x1) \
integerSqrLeg(b1,z1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1) \
integerMulLeg(b0,x1,z2) \
integerMulLeg(b1,z1,x2) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1)
#define difAddBmi2Adx \
addSub(x1,z1) \
integerMulAdx(b0,z1,ui) \
reduceFromDoubleAdx(z1,b0) \
addSub(x1,z1) \
integerSqrAdx(b0,x1) \
integerSqrAdx(b1,z1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1) \
integerMulAdx(b0,x1,z2) \
integerMulAdx(b1,z1,x2) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1)
#define doubleLeg \
addSub(x1,z1) \
integerSqrLeg(b0,x1) \
integerSqrLeg(b1,z1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1) \
subtraction(t0,x1,z1) \
multiplyA24Leg(t1,t0) \
additionLeg(t1,t1,z1) \
integerMulLeg(b0,x1,z1) \
integerMulLeg(b1,t0,t1) \
reduceFromDoubleLeg(x1,b0) \
reduceFromDoubleLeg(z1,b1)
#define doubleBmi2Adx \
addSub(x1,z1) \
integerSqrAdx(b0,x1) \
integerSqrAdx(b1,z1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1) \
subtraction(t0,x1,z1) \
multiplyA24Adx(t1,t0) \
additionAdx(t1,t1,z1) \
integerMulAdx(b0,x1,z1) \
integerMulAdx(b1,t0,t1) \
reduceFromDoubleAdx(x1,b0) \
reduceFromDoubleAdx(z1,b1)