Crypto Forum Research Group A. Cope
Internet-Draft Google
Intended status: Informational December 21, 2016
Expires: June 24, 2017
Hash-Encrypt-Hash, a block cipher mode of operation
draft-cope-heh-01
Abstract
This memo describes a block cipher mode of operation known as Hash-
Encrypt-Hash (HEH).
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3
2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1. Key size . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2. Block cipher . . . . . . . . . . . . . . . . . . . . . . 4
3.3. Nonce and AAD . . . . . . . . . . . . . . . . . . . . . . 4
4. GF(2^128) math . . . . . . . . . . . . . . . . . . . . . . . 4
4.1. GF(2^128) . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2. Multiplication in GF(2^128) . . . . . . . . . . . . . . . 4
4.3. Addition in GF(2^128) . . . . . . . . . . . . . . . . . . 5
5. Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5.1. generate_keys . . . . . . . . . . . . . . . . . . . . . . 5
5.2. generate_betas . . . . . . . . . . . . . . . . . . . . . 6
5.3. poly_hash . . . . . . . . . . . . . . . . . . . . . . . . 6
5.4. HEH_hash . . . . . . . . . . . . . . . . . . . . . . . . 7
5.5. HEH_hash_inv . . . . . . . . . . . . . . . . . . . . . . 8
5.6. CTS_2ECB_encrypt . . . . . . . . . . . . . . . . . . . . 9
5.7. CTS_2ECB_decrypt . . . . . . . . . . . . . . . . . . . . 9
5.8. HEH_encrypt . . . . . . . . . . . . . . . . . . . . . . . 10
5.9. HEH_decrypt . . . . . . . . . . . . . . . . . . . . . . . 10
6. HEH as an AEAD . . . . . . . . . . . . . . . . . . . . . . . 10
6.1. HEH_AEAD_encrypt . . . . . . . . . . . . . . . . . . . . 10
6.2. HEH_AEAD_decrypt . . . . . . . . . . . . . . . . . . . . 11
7. Security considerations . . . . . . . . . . . . . . . . . . . 11
7.1. Security implementations of nonce use . . . . . . . . . . 11
7.2. Authentication . . . . . . . . . . . . . . . . . . . . . 12
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 12
8.1. Normative References . . . . . . . . . . . . . . . . . . 12
8.2. Informative References . . . . . . . . . . . . . . . . . 12
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 12
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Introduction
This memo describes the implementation of the Hash Encrypt Hash (HEH)
block cipher mode of operation as both an encryption algorithm and an
AEAD. The primary benefit of HEH is that it extends the strong
pseudorandom permutation property of block ciphers to arbitrary-
length messages. This means that if any bit of the plaintext is
flipped, each bit in the ciphertext will flip with 50% probability.
No block cipher mode of operation that is currently in widespread use
has this property. Additionally, HEH is more resistant to misuse
than commonly-used block cipher modes of operation. For example, if
nonces are reused, CTR fails catastrophically, and CBC will leak
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common prefixes of the underlying block size. HEH has neither of
those problems.
1.1. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [RFC2119].
2. Notation
ecb_key - key for the underlying ecb block cipher calls, generated
by generate_keys.
block - 16 bytes.
buffer[i] - block i of buffer. Defined for 0 <= i < N.
buffer[N+] - bytes 16 * N until the end of buffer. The unpadded
partial block.
EMPTY - buffer of length 0.
GF(2^128) - The Galois field of 2^128 elements, as defined in
section 4.1.
msg - shorthand for message, a buffer that is an input to a
function.
N - FLOOR(msg_length / 16), number of full blocks of msg.
out_msg - buffer that is a transformation of msg. out_msg_length =
msg_length unless otherwise explicitly specified.
prf_key - pseudo-random function key. The key passed as input to
HEH.
tau_key - 16 byte key used to compute the hash, generated by
generate_keys.
XOR - bitwise exclusive-or.
XXXX_length - length of XXXX in bytes.
* - Multiplication in GF(2^128) as defined in section 4.2.
+ - Addition in GF(2^128) as defined in section 4.3.
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0^i - buffer of i zero bytes.
|| - concatenation.
3. Overview
3.1. Key size
The HEH key is a single key of the same length as the underlying
block cipher key. HEH uses CMAC to derive subkeys from the HEH key.
3.2. Block cipher
HEH MUST use a block cipher with a block size of 128-bits.
3.3. Nonce and AAD
HEH SHOULD support a 16-byte nonce. Support for other nonce lengths
between 0 and 2^32-1 (inclusive) bytes is OPTIONAL. Support for
additional authenticated data (AAD) and support for varying AAD
lengths between 0 and 2^32-1 (inclusive) bytes is OPTIONAL. Security
implications are discussed in section 7.1
4. GF(2^128) math
4.1. GF(2^128)
GF(2^128) is the Galois field of 2^128 elements defined by the
irreducible polynomial x^128 + x^7 + x^2 + x + 1.
Elements in the field are converted to and from 128-bit strings by
taking the least-significant bit of the first byte to be the
coefficient of x^0, the most-significant bit of the first byte to the
the coefficient of x^7, and so on, until the most-significant bit of
the last byte is the coefficient of x^127 [AES-GCM-SIV].
Examples:
10000111 || 0^15 = x^7 + x^2 + x + 1
0^15 || 00000001 = x^120.
0^15 || 10000000 = x^127.
4.2. Multiplication in GF(2^128)
Input
Two 128-bit elements X, Y
Output
128-bit element X * Y
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Multiplication is defined on 128-bit blocks by converting them to
polynomials as described above, and then computing the resulting
product modulo x^128 + x^7 + x^2 + x + 1.
4.3. Addition in GF(2^128)
Input
Two 128-bit elements X, Y
Output
128-bit element X + Y
For any two 128-bit elements X, Y in the Galois field, X + Y is
defined as X XOR Y.
The operations + and XOR are interchangeable within this document.
For consistency we use + on 128-bit strings and XOR if the arguments
are not 128-bits long.
5. Algorithm
When appropriate, we will explain the output as both a mathematical
formula and in pseudo-code. This information is redundant, and it
exists to provide additional clarity. Implementations need not
implement the exact algorithm specified by the pseudocode, so long as
the output matches what the pseudocode would produce.
5.1. generate_keys
ecb_key and tau_key are generated from prf_key by taking the CMAC as
defined in [CMAC] of fixed one-block messages. The input to the CMAC
used to generate ecb and tau key will never collide with the input
used to generate any beta_key, because when generating a beta_key,
the last 4 bytes of the input are always zero.
Input
msg, prf_key
Output
tau_key = CMAC(key = prf_key, message = 0x00...01)
ecb_key = CMAC(key = prf_key, message = 0x00...02) ||
CMAC(key = prf_key, message = 0x00...03)
truncated to perf_key_length bytes
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5.2. generate_betas
To generate the beta_keys needed by HEH_hash, we take the CMAC as
defined in [CMAC] of the nonce, AAD, nonce_length, AAD_length and
plaintext_length. We use CMAC because it is a pseudorandom function
on variable length inputs.
Input
prf_key, nonce, AAD, plaintext_length
Output
beta1_key = CMAC(key = prf_key, message = pad_16(nonce) ||
pad_16(AAD) || pad_16(nonce_length |
| AAD_length || plaintext_length))
beta2_key = x * beta1_key
return beta1_key, beta2_key
Where pad_16(X) = X right-padded with 0s up to a multiple of 16
bytes. If X is already a multiple of 16 bytes (including if X is 0
bytes), this is a no-op.
The following MUST be true in order to generate conformant
ciphertext:
o nonce_length, AAD_length, and plaintext_length MUST be 4 bytes
long.
o nonce_length, AAD_length, and plaintext_length MUST be stored in
little-endian format.
o The input to CMAC MUST be right-padded with 0x00 bytes up to a
multiple of 16 bytes.
o CMAC MUST use the same block cipher that is used in
CTS_2ECB_encrypt.
o CMAC MUST be implemented as described in [CMAC]. In particular,
if CMAC is being reimplemented for HEH, be advised that there is a
multiply-by-x substep of CMAC that uses a different finite field
representation than the one described in section 4.
5.3. poly_hash
Poly_hash treats each block of msg as a coefficient to a polynomial
in GF(2^128), and evaluates that polynomial at tau_key to create a
hash. Poly_hash is called as a subroutine of HEH_hash so that any
minor change to msg will result in every block being changed in
HEH_hash with high probability. Note that the coefficients of
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m_{N-1} and m_N are flipped if there is a partial block. This is
done to simplify the implementation of HEH_hash_inv.
Input
msg, tau_key
Output
if (no partial block)
k^{N-1} * m_0 + ... + k * m_{N-2} + m_{N-1}
else if (partial block)
k^N * m_0 + ... + k^2 * m_{N-2} + k * m_N + m_{N-1}
Where k = tau_key,
m_i = msg[i], for i = 0 to N-1,
m_N = msg[N+] right padded up to 16 bytes with a 0x00 bytes.
Undefined when msg_length is a multiple of 16
pseudo-code:
p = 0^16
For i = 0 to N - 2
p *= tau_key
p += msg[i]
if msg_length % 16 != 0
p *= tau_key
p += m_N // as defined above
p *= tau_key
p += msg[N-1]
return p
5.4. HEH_hash
HEH_hash is the hash step in Hash-Encrypt-Hash. It is an invertible
hash function used to ensure any change to msg will result in every
full block being modified with high probability.
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Input
msg, beta_key, tau_key
Output
out_msg = (m_0 + R, ..., m_{N-2} + R, R, m_N) +
(xb, x^2b, ..., x^{N-1}b, b, 0)
where m_i = msg[i] for i = 0 to N-1,
m_N = msg[N+],
R = out_msg of poly_hash,
b = beta_key,
x is the element x in GF(2^128).
pseudo-code:
R = poly_hash(msg, tau_key)
e = beta_key * x
For i = 0 to N-2
out_msg[i] = msg[i] + R + e
e = e * x
out_msg[N-1] = R + beta_key
out_msg[N+] = msg[N+]
return out_msg
5.5. HEH_hash_inv
Inverse of HEH_hash
Input
msg, beta_key, tau_key
Output
out_msg
pseudo-code
R = msg[N-1] + beta_key
e = beta_key * x
For i = 0 to N-2
out_msg[i] = msg[i] + R + e
e = e * x
out_msg[N+] = msg[N+]
out_msg[N-1] = 0^16
// now all blocks in out_msg are correct except for
// out_msg[N-1], which is all zeroes
R_without_constant_term = poly_hash(out_msg, tau_key)
out_msg[N-1] = R + R_without_constant_term
return out_msg
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5.6. CTS_2ECB_encrypt
The encryption step of Hash-Encrypt-Hash. CTS_2ECB_encrypt uses a
modification of CTS-ECB. Because HEH_hash is the identity function
on partial blocks, we encrypt the partial block by xoring it with a
pad created by encrypting the last full block of plaintext XOR the
last full block ciphertext
Input
msg, ecb_key
Output
out_msg
pseudo-code
For i = 0 to N-1
out_msg[i] = block_cipher_encrypt(ecb_key, msg[i])
if msg_length % 16 != 0
partial_block_pad =
block_cipher_encrypt(ecb_key, out_msg[N-1] XOR msg[N-1])
// XOR the partial block with the first k bytes of
// partial_block_pad, where k is the number of bytes
// in the partial block
out_msg[N+] = msg[N+] XOR partial_block_pad
return out_msg
5.7. CTS_2ECB_decrypt
Inverse of CTS_2ECB_encrypt. CTS_2ECB_decrypt is identical to
CTS_2ECB_encrypt except the initial block_cipher_encrypt calls are
now block_cipher_decrypt calls
Input
msg, ecb_key
Output
out_msg
pseudo-code
For i = 0 to N-1
out_msg[i] = block_cipher_decrypt(ecb_key, msg[i])
if msg_length % 16 != 0
partial_block_pad =
block_cipher_encrypt(ecb_key, out_msg[N-1] XOR msg[N-1])
// XOR the partial block with the first k bytes of
// partial_block_pad, where k is the number of bytes
// in the partial block
out_msg[N+] = msg[N+] XOR partial_block_pad
return out_msg
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5.8. HEH_encrypt
Core encryption function of HEH.
Input
prf_key, ecb_key, tau_key, nonce, AAD, msg
Output
out_msg
pseudo-code
beta1_key, beta2_key = generate_betas(prf_key, nonce, AAD,
msg_length)
out_msg = HEH_hash(msg, beta1_key, tau_key)
out_msg = CTS_2ECB_encrypt(out_msg, ecb_key)
out_msg = HEH_hash_inv(out_msg, beta2_key, tau_key)
return out_msg
5.9. HEH_decrypt
Core decryption function of HEH.
Input
prf_key, ecb_key, tau_key, nonce, AAD, msg
Output
out_msg
pseudo-code
beta1_key, beta2_key = generate_betas(prf_key, nonce, AAD,
msg_length)
out_msg = HEH_hash(msg, beta2_key, tau_key)
out_msg = CTS_2ECB_decrypt(out_msg, ecb_key)
out_msg = HEH_hash_inv(out_msg, beta1_key, tau_key)
return out_msg
6. HEH as an AEAD
Because HEH is a strong pseudorandom permutation, it can also provide
authentication with minimal modification. Support for authentication
is OPTIONAL. To provide authentication, append 16 zero bytes to the
end of the plaintext, then encrypt. When decrypting, we can
determine authenticity of the message by verifying that the final 16
bytes of the plaintext are the expected zero bytes.
6.1. HEH_AEAD_encrypt
The authenticated encryption function of HEH. HEH_AEAD_encrypt
returns ciphertext which is 16 bytes longer than plaintext msg.
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Input
prf_key, ecb_key, tau_key, nonce, AAD, msg
Output
padded_out_msg
pseudo-code
// append a full block of zeroes
padded_msg = msg || 0^16
return HEH_encrypt(prf_key, ecb_key, tau_key, nonce, AAD,
padded_msg)
6.2. HEH_AEAD_decrypt
The authenticated decryption function of HEH. HEH_AEAD_decrypt
returns either plaintext which is 16 bytes shorter than msg or
indication of inauthenticity FAIL.
Input
prf_key, ecb_key, tau_key, nonce, AAD, msg,
Output
unpadded_out_msg or FAIL
pseudo-code
out_msg = HEH_DECRYPT(prf_key, ecb_key, tau_key, nonce, AAD,
msg)
// If final block is not all zeros, FAIL
if out_msg[(out_msg_length - 16):out_msg_length] != 0^16
return FAIL
// Drop the zero-block that was added in HEH_AEAD_encrypt
unpadded_out_msg = out_msg[0:(out_msg_length - 16)]
return unpadded_out_msg
7. Security considerations
The minimum length of the plaintext for HEH is 16 bytes. The maximum
length is 2^32 - 1 bytes. When using HEH as an AEAD, this minimum
and maximum apply to padded_msg.
7.1. Security implementations of nonce use
If no nonce is used (or, equivalently, if a 'nonce' is re-used for
multiple messages) then HEH is a strong pseudorandom permutation. Of
course, if the same plaintext, nonce, and key are used together more
than once, the ciphertext will collide.
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If a unique nonce is used for each plaintext and key combination,
then HEH is semantically secure. We make no claim that using
randomly-generated nonces or using longer nonces generates additional
security.
7.2. Authentication
As HEH is a strong pseudorandom permutation, [AUTH] shows that
authentication can be provided by appending a known authentication
code to the plaintext and then encrypting.
8. References
8.1. Normative References
[CMAC] National Institute of Standards and Technology, "NIST
Special Publication 800-38B", 2005.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
8.2. Informative References
[AES-GCM-SIV]
Gueron, S., Langley, A., and Y. Lindell, "AES-GCM-SIV:
Nonce Misuse-Resistant Authenticated Encryption. draft-
gueron-gcmsiv-03", 2016.
[AUTH] Bellare, M. and P. Rogaway, "Encode-then-encipher
encryption: How to exploit nonces or redundancy in
plaintexts for efficient cryptography", 2000.
[HEH] Sarkar, P., "Efficient Tweakable Enciphering Schemes from
(Block-Wise) Universal Hash Functions", 2008.
[NIST.500-20.1977]
National Institute of Standards and Technology,
"Validating the Correctness of Hardware Implementations of
the NBS Data Encryption Standard", NIST 500-20, November
1977.
Appendix A. Test Vectors
AES-128 was used as the block cipher for all of the test vectors.
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key = 00000000000000000000000000000000
nonce = EMPTY
AAD = EMPTY
plaintext = 00000000000000000000000000000000
ciphertext = a1726260d1450ae4aba906e79e584e07
key = 000102030405060708090A0B0C0D0E0F
nonce = EMPTY
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
000000000000000000000000000000
ciphertext = f5eec4375cefa15fc3eff7a271779231
ce03afa9eeacf9ad060a62602e6dc202
3598944729eb848d13f9b7d362e6d5f6
4e648e38553415f44ff37752954da7
key = 000102030405060708090A0B0C0D0E0F
nonce = EMPTY
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000001
000000000000000000000000000000
ciphertext = 4efc731cacbbaa2713a051e663ddaeb0
4c1b25a3ee7aa4c125c9547154c74923
90b5f7fac503f527b8cdffe21bb96201
b0dc409fed9e9123379e5a6f101bd2
key = 000102030405060708090A0B0C0D0E0F
nonce = 00000000000000000000000000000000
AAD = EMPTY
plaintext = 000102030405060708090A0B0C0D0E0F
ciphertext = d8bd40bfcae5ee810f3d1f1fae890755
key = a8da249b5efa13c2c194bf32ba38a377
nonce = 4d4761372b4786f0d647b5c2e8cf8527
AAD = EMPTY
plaintext = b8ee29e4a5d1e755d0fde722637636e2
f80cf8fe6576e7cac142f5ca5aa8ac2a
ciphertext = 59f2784e1094f95c2223782a30481197
b1fe70c4efdf04ef163904cfc0959a98
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key = 000102030405060708090A0B0C0D0E0F
nonce = 00000000000000000000000000000000
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
000000000000000000000000000000
ciphertext = e040ebe952be6560e46868a37375b852
ef386a872525f604e58ebe148b02141f
a973b7ad15be9ca0d28a2cdcd4e30555
0af5f851eee562a571a77c155d7a9e
key = 000102030405060708090A0B0C0D0E0F
nonce = 00000000000000000000000000000000
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000001
000000000000000000000000000000
ciphertext = 4b1a15a0af086d70f0a797b5314b8cc3
4df27a9dddd4159957adc6b13569f56a
2d70e49749b29f71de22b5708c6924d3
ad80584890e4edba763d717c572587
key = 000102030405060708090A0B0C0D0E0F
nonce = 000102030405060708090A0B0C0D0E0F
AAD = 000102030405060708090A0B0C0D0E0F
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
ciphertext = 16c3f198c970dd0db9b25beedbacb615
b2ee9c51ece5d2426b9f5420be8b1b19
key = 000102030405060708090A0B0C0D0E0F
nonce = 000102030405060708090A0B0C0D0E0F
000102030405
AAD = 0102030405060708090A0B0C0D0E0F00
010203
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
ciphertext = 2aa635491098bc45b711a5d950cc4988
1d110f20056c9d220d125fabfd7ab941
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key = 36DAF975AAE45061AF88079422E5E6A9
nonce = 4164A1FFAEEF4B23324C47279AFB02E8
AAD = 948F6D03EA0BDE71A0233AC87753F10E
plaintext = 6A2EDA8E07C10918507F0B5E4F32053C
335D179A8F476ED1D08A458C00726F63
6365BF26A7003F43C0270BBB44EC780E
6119FA19AA99F0265850BD29C49E2436
A9
ciphertext = d3312031380deb46e7f56220c934a759
55a95fc750f3e535ada7d371ad60b3a7
c6406389a62b1e66be371baa8adba267
225d522936c3829c035ab109526d296f
12
key = 880D8B115BA55842FF4505C5E45F78F6
nonce = 131D6E569B5CCB6E563D2CED8616E6AC
AAD = 01BD52F7065A35A07EE70D9A881EDDB4
plaintext = 00000000000000000000000000000000
B1E0CC8A07264432823C68B2EF59E592
D271271029F6364CEEE577D9FDA8E5C4
131D6E569B5CCB6E563D2CED8616E6AC
C6
ciphertext = 395f5d80788231bd0cc055ef1fd83941
7501a2b4d9c42952ffcbd98be6393305
41968ef42d589e5eb807054af2557905
12bb5dda5be418335ae9b7e9d18c8ce8
f7
key = 880D8B115BA55842FF4505C5E45F78F6
nonce = 131D6E569B5CCB6E563D2CED8616E6AC
AAD = 01BD52F7065A35A07EE70D9A881EDDB4
plaintext = 01000000000000000000000000000000
B1E0CC8A07264432823C68B2EF59E592
D271271029F6364CEEE577D9FDA8E5C4
131D6E569B5CCB6E563D2CED8616E6AC
C6
ciphertext = e6807e6ba3f4fba237c996f91bf6beba
bf243915da3e8ab4c73c5ed9c0a8136b
00dc6d7b4994b7c8551b5bf1a042c2f2
2da1129fcbe45e310f880552a27b3f5b
84
Author's Address
Cope Expires June 24, 2017 [Page 15]
Internet-Draft HEH December 2016
Alex Cope
Google
747 6th St S
Kirkland, WA 98033
USA
Email: alexcope@google.com
Cope Expires June 24, 2017 [Page 16]