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RFC 8032

 
 
 

Edwards-Curve Digital Signature Algorithm (EdDSA)

Part 2 of 3, p. 20 to 39
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6.  Ed25519 Python Illustration

   The rest of this section describes how Ed25519 can be implemented in
   Python (version 3.2 or later) for illustration.  See Appendix A for
   the complete implementation and Appendix B for a test-driver to run
   it through some test vectors.

   Note that this code is not intended for production as it is not
   proven to be correct for all inputs, nor does it protect against
   side-channel attacks.  The purpose is to illustrate the algorithm to
   help implementers with their own implementation.

## First, some preliminaries that will be needed.

import hashlib

def sha512(s):
    return hashlib.sha512(s).digest()

# Base field Z_p
p = 2**255 - 19

def modp_inv(x):
    return pow(x, p-2, p)

# Curve constant
d = -121665 * modp_inv(121666) % p

# Group order
q = 2**252 + 27742317777372353535851937790883648493

def sha512_modq(s):
    return int.from_bytes(sha512(s), "little") % q

## Then follows functions to perform point operations.

# Points are represented as tuples (X, Y, Z, T) of extended
# coordinates, with x = X/Z, y = Y/Z, x*y = T/Z

def point_add(P, Q):
    A, B = (P[1]-P[0]) * (Q[1]-Q[0]) % p, (P[1]+P[0]) * (Q[1]+Q[0]) % p;
    C, D = 2 * P[3] * Q[3] * d % p, 2 * P[2] * Q[2] % p;
    E, F, G, H = B-A, D-C, D+C, B+A;
    return (E*F, G*H, F*G, E*H);

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# Computes Q = s * Q
def point_mul(s, P):
    Q = (0, 1, 1, 0)  # Neutral element
    while s > 0:
        if s & 1:
            Q = point_add(Q, P)
        P = point_add(P, P)
        s >>= 1
    return Q

def point_equal(P, Q):
    # x1 / z1 == x2 / z2  <==>  x1 * z2 == x2 * z1
    if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
        return False
    if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
        return False
    return True

## Now follows functions for point compression.

# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)

# Compute corresponding x-coordinate, with low bit corresponding to
# sign, or return None on failure
def recover_x(y, sign):
    if y >= p:
        return None
    x2 = (y*y-1) * modp_inv(d*y*y+1)
    if x2 == 0:
        if sign:
            return None
        else:
            return 0

    # Compute square root of x2
    x = pow(x2, (p+3) // 8, p)
    if (x*x - x2) % p != 0:
        x = x * modp_sqrt_m1 % p
    if (x*x - x2) % p != 0:
        return None

    if (x & 1) != sign:
        x = p - x
    return x

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# Base point
g_y = 4 * modp_inv(5) % p
g_x = recover_x(g_y, 0)
G = (g_x, g_y, 1, g_x * g_y % p)

def point_compress(P):
    zinv = modp_inv(P[2])
    x = P[0] * zinv % p
    y = P[1] * zinv % p
    return int.to_bytes(y | ((x & 1) << 255), 32, "little")

def point_decompress(s):
    if len(s) != 32:
        raise Exception("Invalid input length for decompression")
    y = int.from_bytes(s, "little")
    sign = y >> 255
    y &= (1 << 255) - 1

    x = recover_x(y, sign)
    if x is None:
        return None
    else:
        return (x, y, 1, x*y % p)

## These are functions for manipulating the private key.

def secret_expand(secret):
    if len(secret) != 32:
        raise Exception("Bad size of private key")
    h = sha512(secret)
    a = int.from_bytes(h[:32], "little")
    a &= (1 << 254) - 8
    a |= (1 << 254)
    return (a, h[32:])

def secret_to_public(secret):
    (a, dummy) = secret_expand(secret)
    return point_compress(point_mul(a, G))

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## The signature function works as below.

def sign(secret, msg):
    a, prefix = secret_expand(secret)
    A = point_compress(point_mul(a, G))
    r = sha512_modq(prefix + msg)
    R = point_mul(r, G)
    Rs = point_compress(R)
    h = sha512_modq(Rs + A + msg)
    s = (r + h * a) % q
    return Rs + int.to_bytes(s, 32, "little")

## And finally the verification function.

def verify(public, msg, signature):
    if len(public) != 32:
        raise Exception("Bad public key length")
    if len(signature) != 64:
        Exception("Bad signature length")
    A = point_decompress(public)
    if not A:
        return False
    Rs = signature[:32]
    R = point_decompress(Rs)
    if not R:
        return False
    s = int.from_bytes(signature[32:], "little")
    if s >= q: return False
    h = sha512_modq(Rs + public + msg)
    sB = point_mul(s, G)
    hA = point_mul(h, A)
    return point_equal(sB, point_add(R, hA))

7.  Test Vectors

   This section contains test vectors for Ed25519ph, Ed25519ctx,
   Ed448ph, Ed25519, and Ed448.

   Each section contains a sequence of test vectors.  The octets are hex
   encoded, and whitespace is inserted for readability.  Ed25519,
   Ed25519ctx, and Ed25519ph private and public keys are 32 octets;
   signatures are 64 octets.  Ed448 and Ed448ph private and public keys
   are 57 octets; signatures are 114 octets.  Messages are of arbitrary
   length.  If the context is non-empty, it is given as 1-255 octets.

Top      Up      ToC       Page 24 
7.1.  Test Vectors for Ed25519

   These test vectors are taken from [ED25519-TEST-VECTORS] (but we
   removed the public key as a suffix of the private key and removed the
   message from the signature) and [ED25519-LIBGCRYPT-TEST-VECTORS].

   -----TEST 1

   ALGORITHM:
   Ed25519

   SECRET KEY:
   9d61b19deffd5a60ba844af492ec2cc4
   4449c5697b326919703bac031cae7f60

   PUBLIC KEY:
   d75a980182b10ab7d54bfed3c964073a
   0ee172f3daa62325af021a68f707511a

   MESSAGE (length 0 bytes):

   SIGNATURE:
   e5564300c360ac729086e2cc806e828a
   84877f1eb8e5d974d873e06522490155
   5fb8821590a33bacc61e39701cf9b46b
   d25bf5f0595bbe24655141438e7a100b

   -----TEST 2

   ALGORITHM:
   Ed25519

   SECRET KEY:
   4ccd089b28ff96da9db6c346ec114e0f
   5b8a319f35aba624da8cf6ed4fb8a6fb

   PUBLIC KEY:
   3d4017c3e843895a92b70aa74d1b7ebc
   9c982ccf2ec4968cc0cd55f12af4660c

   MESSAGE (length 1 byte):
   72

   SIGNATURE:
   92a009a9f0d4cab8720e820b5f642540
   a2b27b5416503f8fb3762223ebdb69da
   085ac1e43e15996e458f3613d0f11d8c
   387b2eaeb4302aeeb00d291612bb0c00

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   -----TEST 3

   ALGORITHM:
   Ed25519

   SECRET KEY:
   c5aa8df43f9f837bedb7442f31dcb7b1
   66d38535076f094b85ce3a2e0b4458f7

   PUBLIC KEY:
   fc51cd8e6218a1a38da47ed00230f058
   0816ed13ba3303ac5deb911548908025

   MESSAGE (length 2 bytes):
   af82

   SIGNATURE:
   6291d657deec24024827e69c3abe01a3
   0ce548a284743a445e3680d7db5ac3ac
   18ff9b538d16f290ae67f760984dc659
   4a7c15e9716ed28dc027beceea1ec40a

   -----TEST 1024

   ALGORITHM:
   Ed25519

   SECRET KEY:
   f5e5767cf153319517630f226876b86c
   8160cc583bc013744c6bf255f5cc0ee5

   PUBLIC KEY:
   278117fc144c72340f67d0f2316e8386
   ceffbf2b2428c9c51fef7c597f1d426e

   MESSAGE (length 1023 bytes):
   08b8b2b733424243760fe426a4b54908
   632110a66c2f6591eabd3345e3e4eb98
   fa6e264bf09efe12ee50f8f54e9f77b1
   e355f6c50544e23fb1433ddf73be84d8
   79de7c0046dc4996d9e773f4bc9efe57
   38829adb26c81b37c93a1b270b20329d
   658675fc6ea534e0810a4432826bf58c
   941efb65d57a338bbd2e26640f89ffbc
   1a858efcb8550ee3a5e1998bd177e93a
   7363c344fe6b199ee5d02e82d522c4fe
   ba15452f80288a821a579116ec6dad2b
   3b310da903401aa62100ab5d1a36553e

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   06203b33890cc9b832f79ef80560ccb9
   a39ce767967ed628c6ad573cb116dbef
   efd75499da96bd68a8a97b928a8bbc10
   3b6621fcde2beca1231d206be6cd9ec7
   aff6f6c94fcd7204ed3455c68c83f4a4
   1da4af2b74ef5c53f1d8ac70bdcb7ed1
   85ce81bd84359d44254d95629e9855a9
   4a7c1958d1f8ada5d0532ed8a5aa3fb2
   d17ba70eb6248e594e1a2297acbbb39d
   502f1a8c6eb6f1ce22b3de1a1f40cc24
   554119a831a9aad6079cad88425de6bd
   e1a9187ebb6092cf67bf2b13fd65f270
   88d78b7e883c8759d2c4f5c65adb7553
   878ad575f9fad878e80a0c9ba63bcbcc
   2732e69485bbc9c90bfbd62481d9089b
   eccf80cfe2df16a2cf65bd92dd597b07
   07e0917af48bbb75fed413d238f5555a
   7a569d80c3414a8d0859dc65a46128ba
   b27af87a71314f318c782b23ebfe808b
   82b0ce26401d2e22f04d83d1255dc51a
   ddd3b75a2b1ae0784504df543af8969b
   e3ea7082ff7fc9888c144da2af58429e
   c96031dbcad3dad9af0dcbaaaf268cb8
   fcffead94f3c7ca495e056a9b47acdb7
   51fb73e666c6c655ade8297297d07ad1
   ba5e43f1bca32301651339e22904cc8c
   42f58c30c04aafdb038dda0847dd988d
   cda6f3bfd15c4b4c4525004aa06eeff8
   ca61783aacec57fb3d1f92b0fe2fd1a8
   5f6724517b65e614ad6808d6f6ee34df
   f7310fdc82aebfd904b01e1dc54b2927
   094b2db68d6f903b68401adebf5a7e08
   d78ff4ef5d63653a65040cf9bfd4aca7
   984a74d37145986780fc0b16ac451649
   de6188a7dbdf191f64b5fc5e2ab47b57
   f7f7276cd419c17a3ca8e1b939ae49e4
   88acba6b965610b5480109c8b17b80e1
   b7b750dfc7598d5d5011fd2dcc5600a3
   2ef5b52a1ecc820e308aa342721aac09
   43bf6686b64b2579376504ccc493d97e
   6aed3fb0f9cd71a43dd497f01f17c0e2
   cb3797aa2a2f256656168e6c496afc5f
   b93246f6b1116398a346f1a641f3b041
   e989f7914f90cc2c7fff357876e506b5
   0d334ba77c225bc307ba537152f3f161
   0e4eafe595f6d9d90d11faa933a15ef1
   369546868a7f3a45a96768d40fd9d034
   12c091c6315cf4fde7cb68606937380d

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   b2eaaa707b4c4185c32eddcdd306705e
   4dc1ffc872eeee475a64dfac86aba41c
   0618983f8741c5ef68d3a101e8a3b8ca
   c60c905c15fc910840b94c00a0b9d0

   SIGNATURE:
   0aab4c900501b3e24d7cdf4663326a3a
   87df5e4843b2cbdb67cbf6e460fec350
   aa5371b1508f9f4528ecea23c436d94b
   5e8fcd4f681e30a6ac00a9704a188a03

   -----TEST SHA(abc)

   ALGORITHM:
   Ed25519

   SECRET KEY:
   833fe62409237b9d62ec77587520911e
   9a759cec1d19755b7da901b96dca3d42

   PUBLIC KEY:
   ec172b93ad5e563bf4932c70e1245034
   c35467ef2efd4d64ebf819683467e2bf

   MESSAGE (length 64 bytes):
   ddaf35a193617abacc417349ae204131
   12e6fa4e89a97ea20a9eeee64b55d39a
   2192992a274fc1a836ba3c23a3feebbd
   454d4423643ce80e2a9ac94fa54ca49f

   SIGNATURE:
   dc2a4459e7369633a52b1bf277839a00
   201009a3efbf3ecb69bea2186c26b589
   09351fc9ac90b3ecfdfbc7c66431e030
   3dca179c138ac17ad9bef1177331a704
   -----

7.2.  Test Vectors for Ed25519ctx

   -----foo

   ALGORITHM:
   Ed25519ctx

   SECRET KEY:
   0305334e381af78f141cb666f6199f57
   bc3495335a256a95bd2a55bf546663f6

Top      Up      ToC       Page 28 
   PUBLIC KEY:
   dfc9425e4f968f7f0c29f0259cf5f9ae
   d6851c2bb4ad8bfb860cfee0ab248292

   MESSAGE (length 16 bytes):
   f726936d19c800494e3fdaff20b276a8

   CONTEXT:
   666f6f

   SIGNATURE:
   55a4cc2f70a54e04288c5f4cd1e45a7b
   b520b36292911876cada7323198dd87a
   8b36950b95130022907a7fb7c4e9b2d5
   f6cca685a587b4b21f4b888e4e7edb0d

   -----bar

   ALGORITHM:
   Ed25519ctx

   SECRET KEY:
   0305334e381af78f141cb666f6199f57
   bc3495335a256a95bd2a55bf546663f6

   PUBLIC KEY:
   dfc9425e4f968f7f0c29f0259cf5f9ae
   d6851c2bb4ad8bfb860cfee0ab248292

   MESSAGE (length 16 bytes):
   f726936d19c800494e3fdaff20b276a8

   CONTEXT:
   626172

   SIGNATURE:
   fc60d5872fc46b3aa69f8b5b4351d580
   8f92bcc044606db097abab6dbcb1aee3
   216c48e8b3b66431b5b186d1d28f8ee1
   5a5ca2df6668346291c2043d4eb3e90d

   -----foo2

   ALGORITHM:
   Ed25519ctx

Top      Up      ToC       Page 29 
   SECRET KEY:
   0305334e381af78f141cb666f6199f57
   bc3495335a256a95bd2a55bf546663f6

   PUBLIC KEY:
   dfc9425e4f968f7f0c29f0259cf5f9ae
   d6851c2bb4ad8bfb860cfee0ab248292

   MESSAGE (length 16 bytes):
   508e9e6882b979fea900f62adceaca35

   CONTEXT:
   666f6f

   SIGNATURE:
   8b70c1cc8310e1de20ac53ce28ae6e72
   07f33c3295e03bb5c0732a1d20dc6490
   8922a8b052cf99b7c4fe107a5abb5b2c
   4085ae75890d02df26269d8945f84b0b

   -----foo3

   ALGORITHM:
   Ed25519ctx

   SECRET KEY:
   ab9c2853ce297ddab85c993b3ae14bca
   d39b2c682beabc27d6d4eb20711d6560

   PUBLIC KEY:
   0f1d1274943b91415889152e893d80e9
   3275a1fc0b65fd71b4b0dda10ad7d772

   MESSAGE (length 16 bytes):
   f726936d19c800494e3fdaff20b276a8

   CONTEXT:
   666f6f

   SIGNATURE:
   21655b5f1aa965996b3f97b3c849eafb
   a922a0a62992f73b3d1b73106a84ad85
   e9b86a7b6005ea868337ff2d20a7f5fb
   d4cd10b0be49a68da2b2e0dc0ad8960f
   -----

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7.3.  Test Vectors for Ed25519ph

   -----TEST abc

   ALGORITHM:
   Ed25519ph

   SECRET KEY:
   833fe62409237b9d62ec77587520911e
   9a759cec1d19755b7da901b96dca3d42

   PUBLIC KEY:
   ec172b93ad5e563bf4932c70e1245034
   c35467ef2efd4d64ebf819683467e2bf

   MESSAGE (length 3 bytes):
   616263

   SIGNATURE:
   98a70222f0b8121aa9d30f813d683f80
   9e462b469c7ff87639499bb94e6dae41
   31f85042463c2a355a2003d062adf5aa
   a10b8c61e636062aaad11c2a26083406
   -----

7.4.  Test Vectors for Ed448

   -----Blank

   ALGORITHM:
   Ed448

   SECRET KEY:
   6c82a562cb808d10d632be89c8513ebf
   6c929f34ddfa8c9f63c9960ef6e348a3
   528c8a3fcc2f044e39a3fc5b94492f8f
   032e7549a20098f95b

   PUBLIC KEY:
   5fd7449b59b461fd2ce787ec616ad46a
   1da1342485a70e1f8a0ea75d80e96778
   edf124769b46c7061bd6783df1e50f6c
   d1fa1abeafe8256180

   MESSAGE (length 0 bytes):

Top      Up      ToC       Page 31 
   SIGNATURE:
   533a37f6bbe457251f023c0d88f976ae
   2dfb504a843e34d2074fd823d41a591f
   2b233f034f628281f2fd7a22ddd47d78
   28c59bd0a21bfd3980ff0d2028d4b18a
   9df63e006c5d1c2d345b925d8dc00b41
   04852db99ac5c7cdda8530a113a0f4db
   b61149f05a7363268c71d95808ff2e65
   2600

   -----1 octet

   ALGORITHM:
   Ed448

   SECRET KEY:
   c4eab05d357007c632f3dbb48489924d
   552b08fe0c353a0d4a1f00acda2c463a
   fbea67c5e8d2877c5e3bc397a659949e
   f8021e954e0a12274e

   PUBLIC KEY:
   43ba28f430cdff456ae531545f7ecd0a
   c834a55d9358c0372bfa0c6c6798c086
   6aea01eb00742802b8438ea4cb82169c
   235160627b4c3a9480

   MESSAGE (length 1 byte):
   03

   SIGNATURE:
   26b8f91727bd62897af15e41eb43c377
   efb9c610d48f2335cb0bd0087810f435
   2541b143c4b981b7e18f62de8ccdf633
   fc1bf037ab7cd779805e0dbcc0aae1cb
   cee1afb2e027df36bc04dcecbf154336
   c19f0af7e0a6472905e799f1953d2a0f
   f3348ab21aa4adafd1d234441cf807c0
   3a00

   -----1 octet (with context)

   ALGORITHM:
   Ed448

Top      Up      ToC       Page 32 
   SECRET KEY:
   c4eab05d357007c632f3dbb48489924d
   552b08fe0c353a0d4a1f00acda2c463a
   fbea67c5e8d2877c5e3bc397a659949e
   f8021e954e0a12274e

   PUBLIC KEY:
   43ba28f430cdff456ae531545f7ecd0a
   c834a55d9358c0372bfa0c6c6798c086
   6aea01eb00742802b8438ea4cb82169c
   235160627b4c3a9480

   MESSAGE (length 1 byte):
   03

   CONTEXT:
   666f6f

   SIGNATURE:
   d4f8f6131770dd46f40867d6fd5d5055
   de43541f8c5e35abbcd001b32a89f7d2
   151f7647f11d8ca2ae279fb842d60721
   7fce6e042f6815ea000c85741de5c8da
   1144a6a1aba7f96de42505d7a7298524
   fda538fccbbb754f578c1cad10d54d0d
   5428407e85dcbc98a49155c13764e66c
   3c00

   -----11 octets

   ALGORITHM:
   Ed448

   SECRET KEY:
   cd23d24f714274e744343237b93290f5
   11f6425f98e64459ff203e8985083ffd
   f60500553abc0e05cd02184bdb89c4cc
   d67e187951267eb328

   PUBLIC KEY:
   dcea9e78f35a1bf3499a831b10b86c90
   aac01cd84b67a0109b55a36e9328b1e3
   65fce161d71ce7131a543ea4cb5f7e9f
   1d8b00696447001400

   MESSAGE (length 11 bytes):
   0c3e544074ec63b0265e0c

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   SIGNATURE:
   1f0a8888ce25e8d458a21130879b840a
   9089d999aaba039eaf3e3afa090a09d3
   89dba82c4ff2ae8ac5cdfb7c55e94d5d
   961a29fe0109941e00b8dbdeea6d3b05
   1068df7254c0cdc129cbe62db2dc957d
   bb47b51fd3f213fb8698f064774250a5
   028961c9bf8ffd973fe5d5c206492b14
   0e00

   -----12 octets

   ALGORITHM:
   Ed448

   SECRET KEY:
   258cdd4ada32ed9c9ff54e63756ae582
   fb8fab2ac721f2c8e676a72768513d93
   9f63dddb55609133f29adf86ec9929dc
   cb52c1c5fd2ff7e21b

   PUBLIC KEY:
   3ba16da0c6f2cc1f30187740756f5e79
   8d6bc5fc015d7c63cc9510ee3fd44adc
   24d8e968b6e46e6f94d19b945361726b
   d75e149ef09817f580

   MESSAGE (length 12 bytes):
   64a65f3cdedcdd66811e2915

   SIGNATURE:
   7eeeab7c4e50fb799b418ee5e3197ff6
   bf15d43a14c34389b59dd1a7b1b85b4a
   e90438aca634bea45e3a2695f1270f07
   fdcdf7c62b8efeaf00b45c2c96ba457e
   b1a8bf075a3db28e5c24f6b923ed4ad7
   47c3c9e03c7079efb87cb110d3a99861
   e72003cbae6d6b8b827e4e6c143064ff
   3c00

   -----13 octets

   ALGORITHM:
   Ed448

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   SECRET KEY:
   7ef4e84544236752fbb56b8f31a23a10
   e42814f5f55ca037cdcc11c64c9a3b29
   49c1bb60700314611732a6c2fea98eeb
   c0266a11a93970100e

   PUBLIC KEY:
   b3da079b0aa493a5772029f0467baebe
   e5a8112d9d3a22532361da294f7bb381
   5c5dc59e176b4d9f381ca0938e13c6c0
   7b174be65dfa578e80

   MESSAGE (length 13 bytes):
   64a65f3cdedcdd66811e2915e7

   SIGNATURE:
   6a12066f55331b6c22acd5d5bfc5d712
   28fbda80ae8dec26bdd306743c5027cb
   4890810c162c027468675ecf645a8317
   6c0d7323a2ccde2d80efe5a1268e8aca
   1d6fbc194d3f77c44986eb4ab4177919
   ad8bec33eb47bbb5fc6e28196fd1caf5
   6b4e7e0ba5519234d047155ac727a105
   3100

   -----64 octets

   ALGORITHM:
   Ed448

   SECRET KEY:
   d65df341ad13e008567688baedda8e9d
   cdc17dc024974ea5b4227b6530e339bf
   f21f99e68ca6968f3cca6dfe0fb9f4fa
   b4fa135d5542ea3f01

   PUBLIC KEY:
   df9705f58edbab802c7f8363cfe5560a
   b1c6132c20a9f1dd163483a26f8ac53a
   39d6808bf4a1dfbd261b099bb03b3fb5
   0906cb28bd8a081f00

   MESSAGE (length 64 bytes):
   bd0f6a3747cd561bdddf4640a332461a
   4a30a12a434cd0bf40d766d9c6d458e5
   512204a30c17d1f50b5079631f64eb31
   12182da3005835461113718d1a5ef944

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   SIGNATURE:
   554bc2480860b49eab8532d2a533b7d5
   78ef473eeb58c98bb2d0e1ce488a98b1
   8dfde9b9b90775e67f47d4a1c3482058
   efc9f40d2ca033a0801b63d45b3b722e
   f552bad3b4ccb667da350192b61c508c
   f7b6b5adadc2c8d9a446ef003fb05cba
   5f30e88e36ec2703b349ca229c267083
   3900

   -----256 octets

   ALGORITHM:
   Ed448

   SECRET KEY:
   2ec5fe3c17045abdb136a5e6a913e32a
   b75ae68b53d2fc149b77e504132d3756
   9b7e766ba74a19bd6162343a21c8590a
   a9cebca9014c636df5

   PUBLIC KEY:
   79756f014dcfe2079f5dd9e718be4171
   e2ef2486a08f25186f6bff43a9936b9b
   fe12402b08ae65798a3d81e22e9ec80e
   7690862ef3d4ed3a00

   MESSAGE (length 256 bytes):
   15777532b0bdd0d1389f636c5f6b9ba7
   34c90af572877e2d272dd078aa1e567c
   fa80e12928bb542330e8409f31745041
   07ecd5efac61ae7504dabe2a602ede89
   e5cca6257a7c77e27a702b3ae39fc769
   fc54f2395ae6a1178cab4738e543072f
   c1c177fe71e92e25bf03e4ecb72f47b6
   4d0465aaea4c7fad372536c8ba516a60
   39c3c2a39f0e4d832be432dfa9a706a6
   e5c7e19f397964ca4258002f7c0541b5
   90316dbc5622b6b2a6fe7a4abffd9610
   5eca76ea7b98816af0748c10df048ce0
   12d901015a51f189f3888145c03650aa
   23ce894c3bd889e030d565071c59f409
   a9981b51878fd6fc110624dcbcde0bf7
   a69ccce38fabdf86f3bef6044819de11

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   SIGNATURE:
   c650ddbb0601c19ca11439e1640dd931
   f43c518ea5bea70d3dcde5f4191fe53f
   00cf966546b72bcc7d58be2b9badef28
   743954e3a44a23f880e8d4f1cfce2d7a
   61452d26da05896f0a50da66a239a8a1
   88b6d825b3305ad77b73fbac0836ecc6
   0987fd08527c1a8e80d5823e65cafe2a
   3d00

   -----1023 octets

   ALGORITHM:
   Ed448

   SECRET KEY:
   872d093780f5d3730df7c212664b37b8
   a0f24f56810daa8382cd4fa3f77634ec
   44dc54f1c2ed9bea86fafb7632d8be19
   9ea165f5ad55dd9ce8

   PUBLIC KEY:
   a81b2e8a70a5ac94ffdbcc9badfc3feb
   0801f258578bb114ad44ece1ec0e799d
   a08effb81c5d685c0c56f64eecaef8cd
   f11cc38737838cf400

   MESSAGE (length 1023 bytes):
   6ddf802e1aae4986935f7f981ba3f035
   1d6273c0a0c22c9c0e8339168e675412
   a3debfaf435ed651558007db4384b650
   fcc07e3b586a27a4f7a00ac8a6fec2cd
   86ae4bf1570c41e6a40c931db27b2faa
   15a8cedd52cff7362c4e6e23daec0fbc
   3a79b6806e316efcc7b68119bf46bc76
   a26067a53f296dafdbdc11c77f7777e9
   72660cf4b6a9b369a6665f02e0cc9b6e
   dfad136b4fabe723d2813db3136cfde9
   b6d044322fee2947952e031b73ab5c60
   3349b307bdc27bc6cb8b8bbd7bd32321
   9b8033a581b59eadebb09b3c4f3d2277
   d4f0343624acc817804728b25ab79717
   2b4c5c21a22f9c7839d64300232eb66e
   53f31c723fa37fe387c7d3e50bdf9813
   a30e5bb12cf4cd930c40cfb4e1fc6225
   92a49588794494d56d24ea4b40c89fc0
   596cc9ebb961c8cb10adde976a5d602b
   1c3f85b9b9a001ed3c6a4d3b1437f520

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   96cd1956d042a597d561a596ecd3d173
   5a8d570ea0ec27225a2c4aaff26306d1
   526c1af3ca6d9cf5a2c98f47e1c46db9
   a33234cfd4d81f2c98538a09ebe76998
   d0d8fd25997c7d255c6d66ece6fa56f1
   1144950f027795e653008f4bd7ca2dee
   85d8e90f3dc315130ce2a00375a318c7
   c3d97be2c8ce5b6db41a6254ff264fa6
   155baee3b0773c0f497c573f19bb4f42
   40281f0b1f4f7be857a4e59d416c06b4
   c50fa09e1810ddc6b1467baeac5a3668
   d11b6ecaa901440016f389f80acc4db9
   77025e7f5924388c7e340a732e554440
   e76570f8dd71b7d640b3450d1fd5f041
   0a18f9a3494f707c717b79b4bf75c984
   00b096b21653b5d217cf3565c9597456
   f70703497a078763829bc01bb1cbc8fa
   04eadc9a6e3f6699587a9e75c94e5bab
   0036e0b2e711392cff0047d0d6b05bd2
   a588bc109718954259f1d86678a579a3
   120f19cfb2963f177aeb70f2d4844826
   262e51b80271272068ef5b3856fa8535
   aa2a88b2d41f2a0e2fda7624c2850272
   ac4a2f561f8f2f7a318bfd5caf969614
   9e4ac824ad3460538fdc25421beec2cc
   6818162d06bbed0c40a387192349db67
   a118bada6cd5ab0140ee273204f628aa
   d1c135f770279a651e24d8c14d75a605
   9d76b96a6fd857def5e0b354b27ab937
   a5815d16b5fae407ff18222c6d1ed263
   be68c95f32d908bd895cd76207ae7264
   87567f9a67dad79abec316f683b17f2d
   02bf07e0ac8b5bc6162cf94697b3c27c
   d1fea49b27f23ba2901871962506520c
   392da8b6ad0d99f7013fbc06c2c17a56
   9500c8a7696481c1cd33e9b14e40b82e
   79a5f5db82571ba97bae3ad3e0479515
   bb0e2b0f3bfcd1fd33034efc6245eddd
   7ee2086ddae2600d8ca73e214e8c2b0b
   db2b047c6a464a562ed77b73d2d841c4
   b34973551257713b753632efba348169
   abc90a68f42611a40126d7cb21b58695
   568186f7e569d2ff0f9e745d0487dd2e
   b997cafc5abf9dd102e62ff66cba87

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   SIGNATURE:
   e301345a41a39a4d72fff8df69c98075
   a0cc082b802fc9b2b6bc503f926b65bd
   df7f4c8f1cb49f6396afc8a70abe6d8a
   ef0db478d4c6b2970076c6a0484fe76d
   76b3a97625d79f1ce240e7c576750d29
   5528286f719b413de9ada3e8eb78ed57
   3603ce30d8bb761785dc30dbc320869e
   1a00
   -----

7.5.  Test Vectors for Ed448ph

   -----TEST abc

   ALGORITHM:
   Ed448ph

   SECRET KEY:
   833fe62409237b9d62ec77587520911e
   9a759cec1d19755b7da901b96dca3d42
   ef7822e0d5104127dc05d6dbefde69e3
   ab2cec7c867c6e2c49

   PUBLIC KEY:
   259b71c19f83ef77a7abd26524cbdb31
   61b590a48f7d17de3ee0ba9c52beb743
   c09428a131d6b1b57303d90d8132c276
   d5ed3d5d01c0f53880

   MESSAGE (length 3 bytes):
   616263

   SIGNATURE:
   822f6901f7480f3d5f562c592994d969
   3602875614483256505600bbc281ae38
   1f54d6bce2ea911574932f52a4e6cadd
   78769375ec3ffd1b801a0d9b3f4030cd
   433964b6457ea39476511214f97469b5
   7dd32dbc560a9a94d00bff07620464a3
   ad203df7dc7ce360c3cd3696d9d9fab9
   0f00

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   -----TEST abc (with context)

   ALGORITHM:
   Ed448ph

   SECRET KEY:
   833fe62409237b9d62ec77587520911e
   9a759cec1d19755b7da901b96dca3d42
   ef7822e0d5104127dc05d6dbefde69e3
   ab2cec7c867c6e2c49

   PUBLIC KEY:
   259b71c19f83ef77a7abd26524cbdb31
   61b590a48f7d17de3ee0ba9c52beb743
   c09428a131d6b1b57303d90d8132c276
   d5ed3d5d01c0f53880

   MESSAGE (length 3 bytes):
   616263

   CONTEXT:
   666f6f

   SIGNATURE:
   c32299d46ec8ff02b54540982814dce9
   a05812f81962b649d528095916a2aa48
   1065b1580423ef927ecf0af5888f90da
   0f6a9a85ad5dc3f280d91224ba9911a3
   653d00e484e2ce232521481c8658df30
   4bb7745a73514cdb9bf3e15784ab7128
   4f8d0704a608c54a6b62d97beb511d13
   2100
   -----


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