| Crypto Forum Research Group | A. Cope |
| Internet-Draft | |
| Intended status: Informational | October 26, 2016 |
| Expires: April 29, 2017 |
Hash-Encrypt-Hash, a block cipher mode of operation
draft-cope-heh-00
This memo describes a block cipher mode of operation known as Hash-Encrypt-Hash (HEH).
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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 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 common prefixes of the underlying block size. HEH has neither of those problems.
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].
All implementations MUST support a key size of 48 bytes. For a 48-byte key, the first 16 bytes correspond to tau_key. The second 16 bytes correspond to prf_key. The final 16 bytes correspond to blk_key. Implementations MAY also support key sizes of 64 and 80 bytes, in which case tau_key corresponds to the first 16-byte chunk. The remainder of the key is split in half, with the first half corresponding to prf_key and the second half corresponding to blk_key.
HEH MUST use a block cipher with a block size of 128 bits.
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
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.
Input
Two 128 bit elements X, Y
Output
128 bit element X * Y
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.
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.
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.
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 0's 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:
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 m_{N-1} and m_N are flipped. This is done to simplify the implementation of HEH_hash_inv.
Input
msg, tau_key
Output
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+] padded up to 16 bytes with a 0x01 byte
followed by 0x00 bytes. When msg_length is a multiple of
16, m_N is composed entirely of padding, i.e. 0x0100...00.
pseudo-code:
p = 0^16
For i = 0 to N - 2
p *= tau_key
p += msg[i]
p *= tau_key
p += m_N // as defined above
p *= tau_key
p += msg[N-1]
return p
The Hash step in Hash-Encrypt-Hash. HEH_hash is an invertible hash function used to ensure any change to the msg will result in every full block being modified with high probability.
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
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 block 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
The encryption step of Hash-Encrypt-Hash. Uses a modification of CTS-ECB. Because HEH_hash is the identity function on partial blocks, we instead xor the partial block with the final encrypted full block then re-encrypt the final full block. This technique is discussed in [TET].
Input
msg, blk_key
Output
out_msg
pseudo-code
For i = 0 to N-1
out_msg[i] = block_cipher_encrypt(blk_key, msg[i])
if msg_length % 16 != 0
// XOR the partial block with the first k bytes of out_msg[N-1]
// where k is the number of bytes in the partial block
out_msg[N+] = msg[N+] XOR out_msg[N-1]
out_msg[N-1] = block_cipher_encrypt(blk_key, out_msg[N-1])
return out_msg
Inverse of CTS_2ECB_encrypt.
Input
msg, blk_key
Output
out_msg
pseudo-code
For i = 0 to N-1
out_msg[i] = block_cipher_decrypt(blk_key, msg[i])
if msg_length % 16 != 0
// XOR the partial block with the first k bytes of out_msg[N-1]
// where k is the number of bytes in the partial block
out_msg[N+] = msg[N+] XOR out_msg[N-1]
out_msg[N-1] = block_cipher_decrypt(blk_key, out_msg[N-1])
return out_msg
Core encryption function of HEH.
Input
prf_key, blk_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, blk_key)
out_msg = HEH_hash_inv(out_msg, beta2_key, tau_key)
return out_msg
Core decryption function of HEH.
Input
prf_key, blk_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, blk_key)
out_msg = HEH_hash_inv(out_msg, beta1_key, tau_key)
return out_msg
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 verify authenticity of the message by asserting that the final 16 bytes of the plaintext are the expected zero bytes.
Authenticated encryption function of HEH. Returns ciphertext which is 16 bytes longer than plaintext msg.
Input
prf_key, blk_key, tau_key, nonce, AAD, msg
Output
padded_out_msg
pseudo-code
// append a full block of zeros
padded_msg = msg || 0^16
return HEH_encrypt(prf_key, blk_key, tau_key, nonce, AAD,
padded_msg)
Authenticated decryption function of HEH. Returns either plaintext which is 16 bytes shorter than msg or indication of inauthenticity FAIL.
Input
prf_key, blk_key, tau_key, nonce, AAD, msg,
Output
unpadded_out_msg or FAIL
pseudo-code
out_msg = HEH_DECRYPT(prf_key, blk_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
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.
If no nonce is used (or, equivalently, if a 'nonce' is re-used for multiple messages) then HEH is a strong pseudorandom permutation. In this case the consumer should be aware that if the same plaintext, nonce, and key combination is used more than once it will result in a ciphertext collision.
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.
As HEH is a strong pseudorandom permutation, [AUTH] shows that authentication can be provided by appending a known authentication code to the plaintext, then encrypting the resulting string.
| [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. |
| [AES-GCM-SIV] | Gueronr, 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. |
| [TET] | Halevi, S., "Invertible Universal Hashing and the TET Encryption Mode", 2007. |
AES-128 was used as the block cipher for all of the test vectors
aes_key = 00000000000000000000000000000000 tau_key = 00000000000000000000000000000000 prf_key = 00000000000000000000000000000000 nonce = EMPTY AAD = EMPTY plaintext = 00000000000000000000000000000000 ciphertext = 310f55672a44bf35b3320895e90d3f30
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = EMPTY
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
000000000000000000000000000000
ciphertext = 6e20347c7a0609d04cda4fd26ff3b7d0
3a2e48b13369671c763c24a010d34bd9
2e2707fce73d89a92ad6f191d9cc38cc
c9d8e526885730b4835d6d18c3c55d
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = EMPTY
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000001
000000000000000000000000000000
ciphertext = 77a09f9af01bf2341c8550734e771abc
a41398130c7658d83c075492ece8981d
d5ee21816802cbff60e87fb9ab2cb771
d44fabfbf59dacdf46931e49d632c1
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = 00000000000000000000000000000000
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
000000000000000000000000000000
ciphertext = fb309047c54eccfdc490a29f7c0363c3
cbaf2eee6218eb206297e49bf28bf33f
763baaabf01954dbb4af2ed9a7e09204
5ae481fc58f2dabf5dc9b147d508b1
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = 00000000000000000000000000000000
AAD = EMPTY
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000001
000000000000000000000000000000
ciphertext = 9cdfa55083e0a3b50d3583346e6e40d6
0f81c81a9c4081fbb36eb4bffccac950
cd33fdb34311e632023d3ec6496ecf58
3e14156d392a589983afdd223e7f6c
aes_key = a8da249b5efa13c2c194bf32ba38a377
tau_key = 68f82787dc3033fd655b8e512e02ff9d
prf_key = 21281e64cd9c3388f62c438ff56ff58f
nonce = 4d4761372b4786f0d647b5c2e8cf8527
AAD = EMPTY
plaintext = b8ee29e4a5d1e755d0fde722637636e2
f80cf8fe6576e7cac142f5ca5aa8ac2a
d6a67479105440abdc90b166416ce3cb
6119FA19AA99F0265850BD29C49E2436
4d47
ciphertext = 9726afc277e930f3912c976c779927e0
a9b80ee83db1881300c3752a54f07c1e
66f89d556bda0d2dc318536e1a34e6b7
ab7576469349ea9927cd15429e25d050
9f9a
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = 000102030405060708090A0B0C0D0E0F
AAD = 000102030405060708090A0B0C0D0E0F
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
ciphertext = 7f5eac36f1fee71cc79e4046c1d11f94
cd9219968157de2b3c23c139ff671914
aes_key = 000102030405060708090A0B0C0D0E0F
tau_key = 000102030405060708090A0B0C0D0E0F
prf_key = 000102030405060708090A0B0C0D0E0F
nonce = 000102030405060708090A0B0C0D0E0F
000102030405
AAD = 0102030405060708090A0B0C0D0E0F00
010203
plaintext = 00000000000000000000000000000000
00000000000000000000000000000000
ciphertext = a4f3f950f6f07b892248655a9bc88262
87f7f81312a2a6408d0ad2bed078202a
aes_key = 36DAF975AAE45061AF88079422E5E6A9
tau_key = D0A8C8E6B3FDC335C4E98C9BBB1310E4
prf_key = AA2610D3A619A8F8A222D3DBFB082D17
nonce = 4164A1FFAEEF4B23324C47279AFB02E8
AAD = 948F6D03EA0BDE71A0233AC87753F10E
plaintext = 6A2EDA8E07C10918507F0B5E4F32053C
335D179A8F476ED1D08A458C00726F63
6365BF26A7003F43C0270BBB44EC780E
6119FA19AA99F0265850BD29C49E2436
A9
ciphertext = a962d37c10b43303a522aac165230d67
2cabebfa385d2c7b21468d0af9cab3a7
5bb5c1c332e1afd77b1b98697672c36b
bd05ab6b0f47c759f464689831d3ce9e
93
aes_key = 880D8B115BA55842FF4505C5E45F78F6
tau_key = F83B77EE7445C4190B326489ECA17CF8
prf_key = 9F8BF70E528CC1344300AE428506A937
nonce = 131D6E569B5CCB6E563D2CED8616E6AC
AAD = 01BD52F7065A35A07EE70D9A881EDDB4
plaintext = 00000000000000000000000000000000
B1E0CC8A07264432823C68B2EF59E592
D271271029F6364CEEE577D9FDA8E5C4
131D6E569B5CCB6E563D2CED8616E6AC
C6
ciphertext = a8da249b5efa13c2c194bf32ba38a377
21281e64cd9c3388f62c438ff56ff58f
68f82787dc3033fd655b8e512e02ff9d
c4fb5c2937d3c85c5cb1196c3b0e99af
42
aes_key = 880D8B115BA55842FF4505C5E45F78F6
tau_key = F83B77EE7445C4190B326489ECA17CF8
prf_key = 9F8BF70E528CC1344300AE428506A937
nonce = 131D6E569B5CCB6E563D2CED8616E6AC
AAD = 01BD52F7065A35A07EE70D9A881EDDB4
plaintext = 01000000000000000000000000000000
B1E0CC8A07264432823C68B2EF59E592
D271271029F6364CEEE577D9FDA8E5C4
131D6E569B5CCB6E563D2CED8616E6AC
C6
ciphertext = b8ee29e4a5d1e755d0fde722637636e2
f80cf8fe6576e7cac142f5ca5aa8ac2a
d6a67479105440abdc90b166416ce3cb
4d4761372b4786f0d647b5c2e8cf8527
4b