SHA-3

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SHA-3
(Keccak)
General
Designers Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche.
Certification SHA-3 winner
Detail
Digest sizes arbitrary
Speed 12.5 cpb on Core 2 [r=1024,c=576].

SHA-3, a subset of the cryptographic primitive family Keccak (/ˈkætʃæk/, or /kɛtʃɑːk/),[3][4] is a cryptographic hash function designed by Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche, building upon RadioGatún.

On October 2, 2012, Keccak was selected as the winner of the NIST hash function competition.[3] SHA-3 is not meant to replace SHA-2, as no significant attack on SHA-2 has been demonstrated. Because of the successful attacks on MD5 and SHA-0 and theoretical attacks on SHA-1 and SHA-2,[5] NIST perceived a need for an alternative, dissimilar cryptographic hash, which became SHA-3.

SHA-3 uses the sponge construction[6][7] in which message blocks are XORed into the initial bits of the state, which is then invertibly permuted. In the version used in SHA-3, the state consists of a 5×5 array of 64-bit words, 1600 bits total. The authors claim 12.5 cycles per byte[8] on an Intel Core 2 CPU. However, in hardware implementations it is notably faster than all other finalists.[9]

Keccak's authors have proposed additional, not-yet-standardized uses for the function, including an authenticated encryption system and a "tree" hash for faster hashing on certain architectures.[10] Keccak is also defined for smaller power-of-2 word sizes w down to 1 bit (25 bits total state). Small state sizes can be used to test cryptanalytic attacks, and intermediate state sizes (e.g., from w=4, 100 bits, to w=32, 800 bits) could potentially provide practical, lightweight alternatives.

The block permutation[edit]

This is defined for any power-of-two word size, w = 2 bits. The main SHA-3 submission uses 64-bit words, ℓ = 6.

The state can be considered to be a 5×5×w array of bits. Let a[i][j][k] be bit (i×5 + jw + k of the input, using a little-endian convention. Index arithmetic is performed modulo 5 for the first two dimensions and modulo w for the third.

The basic block permutation function consists of 12+2ℓ iterations of five sub-rounds, each individually very simple:

θ
Compute the parity of each of the 5×w (320, when w = 64) 5-bit columns, and exclusive-or that into two nearby columns in a regular pattern. To be precise, a[i][j][k] ⊕= parity(a[i][j−1][k]) ⊕ parity(a[i][j+1][k−1])
ρ
Bitwise rotate each of the 25 words by a different triangular number 0, 1, 3, 6, 10, 15, .... To be precise, a[0][0] is not rotated, and for all 0≤t<24, a[i][j][k] = a[i][j][k−(t+1)(t+2)/2], where \begin{pmatrix} i \\ j \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix}^t \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
π
Permute the 25 words in a fixed pattern. a[j][2i+3j] = a[i][j]
χ
Bitwise combine along rows, using a = a ⊕ (¬b & c). To be precise, a[i][j][k] ⊕= ¬a[i][j+1][k] & a[i][j+2][k]. This is the only non-linear operation in SHA-3.
ι
Exclusive-or a round constant into one word of the state. To be precise, in round n, for 0≤m≤ℓ, a[0][0][2m−1] is exclusive-ORed with bit m+7n of a degree-8 LFSR sequence. This breaks the symmetry that is preserved by the other sub-rounds.

Hashing variable-length messages[edit]

Illustration of the sponge construction
The sponge construction for hash functions. pi are input, zi are hashed output. The unused "capacity" c should be twice the desired resistance to collision or preimage attacks.

SHA-3 uses the "sponge construction", where input is "absorbed" into the hash state at a given rate, then an output hash is "squeezed" from it at the same rate.

To absorb r bits of data, the data is XORed into the leading bits of the state, and the block permutation is applied. To squeeze, the first r bits of the state are produced as output, and the block permutation is applied if additional output is desired.

Central to this is the "capacity" of the hash function, which is the c=25wr state bits that are not touched by input or output. This can be adjusted based on security requirements, but the SHA-3 proposal sets a conservative c=2n, where n is the size of the output hash. Thus r, the number of message bits processed per block permutation, depends on the output hash size. The NIST submission sets the rate r as 1152, 1088, 832, or 576 (144, 136, 104 and 72 bytes) for 224, 256, 384 and 512-bit hash sizes, respectively. At RSA Conference 2013, and then at CHES 2013, John Kelsey of NIST announced[11][12] that the capacity is likely to be lowered to 256 bit for the 224 and 256 bit variants, and 512 bit for the 384 and 512 bit variants. Thus, the preimage and collision resistances would be set to the same. The 224/384 bit variants would be truncated versions of the 256/512 variants, similarly to the SHA2 family. NIST also considers standardizing other usage modes of Keccak.

To ensure the message can be evenly divided into r-bit blocks, padding is required. The submission proposes the bit pattern 10*1: a 1 bit, zero or more 0 bits (maximum r−1), and a final 1 bit. The final 1 bit is required because the sponge construction security proof requires that the rate is encoded in the final block ("multi rate padding"). This padding might be changed in the final SHA-3 standard to match the padding of Sakura, a tree hashing scheme proposed by the Keccak authors.

To compute a hash, initialize the state to 0, pad the input, and break it into r-bit pieces. Absorb the input into the state; that is, for each piece, XOR it into the state and then apply the block permutation.

After the final block permutation, the leading n bits of the state are the desired hash. Because r is always greater than n, there is actually never a need for additional block permutations in the squeezing phase. However, arbitrary output length may be useful in applications such as optimal asymmetric encryption padding. In this case, n is a security parameter rather than the output size.

Although not part of the SHA-3 competition requirements, smaller variants of the block permutation can be used, for hash output sizes up to half their state size, if the rate r is limited appropriately. For example, a 256-bit hash can be computed using 25 32-bit words if r = 800−2×256 = 288 (36 bytes per iteration).

Comparison of SHA functions[edit]

In the table below, internal state means the number of bits that are carried over to the next block.

Algorithm and
variant
Output size (bits) Internal state size (bits) Block size (bits) Max message size (bits) Word size (bits) Rounds Bitwise operations Collisions found Example Performance (MiB/s)[13]
MD5 (as reference) 128 128 512 264 − 1 32 64 and,or,xor,rot Yes 335
SHA-0 160 160 512 264 − 1 32 80 and,or,xor,rot Yes -
SHA-1 160 160 512 264 − 1 32 80 and,or,xor,rot Theoretical attack (261)[14] 192
SHA-2 SHA-224
SHA-256
224
256
256 512 264 − 1 32 64 and,or,xor,shr,rot None 139
SHA-384
SHA-512
SHA-512/224
SHA-512/256
384
512
224
256
512 1024 2128 − 1 64 80 and,or,xor,shr,rot None 154
SHA-3 SHA3-224
SHA3-256
SHA3-384
SHA3-512
224
256
384
512
1600
(5×5 array of 64-bit words)
1152
1088
832
576
64 24 and,xor,not,rot None

Tweaks[edit]

Throughout the NIST hash function competition, entrants were permitted to "tweak" their algorithms to address issues that were discovered. Changes that have been made to Keccak are:[15][16]

  • The number of rounds was increased from 12+ℓ to 12+2ℓ to be more conservative about security.
  • The message padding was changed from a more complex scheme to the simple 10*1 pattern described above.
  • The rate r was increased to the security limit, rather than rounding down to the nearest power of 2.

NIST announcement controversy[edit]

In February 2013 at the RSA Conference, and then in August 2013 at CHES, NIST announced they would select different values for the capacity, i.e., the security parameter, for the SHA-3 standard, compared to the submission.[11][12] The changes caused some turmoil.

In September 2013, on the NIST hash-forum mailing list,[17] Daniel J. Bernstein suggested strengthening the security to the 576-bit capacity that was originally proposed as the default Keccak.[18] In late September, the Keccak team responded by stating that they proposed 128-bit security by setting c=256 as an option already in their SHA-3 proposal.[19] But in the light of the uproar in the cryptographic community, they proposed raising the capacity to 512 bits for all instances.[20]

In early October 2013, Bruce Schneier criticized NIST's decision on the basis of its possible detrimental effects on the acceptance of the algorithm, saying

There is too much mistrust in the air. NIST risks publishing an algorithm that no one will trust and no one (except those forced) will use.[21]

Paul Crowley expressed his support of the decision, saying that Keccak supposed to be tunable and there is no reason for different security levels within one primitive. He also added:

Yes, it’s a bit of a shame for the competition that they demanded a certain security level for entrants, then went to publish a standard with a different one. But there’s nothing that can be done to fix that now, except re-opening the competition. Demanding that they stick to their mistake doesn’t improve things for anyone.[22]

There was also some confusion that internal changes were made to Keccak. The Keccak team clarified this, stating that NIST's proposal for SHA-3 is a subset of the Keccak family, for which one can generate test vectors using their reference code submitted to the contest, and that this proposal was the result of a series of discussions between them and the NIST hash team.[23] Also, Bruce Schneier corrected his earlier statement, saying

I misspoke when I wrote that NIST made "internal changes" to the algorithm. That was sloppy of me. The Keccak permutation remains unchanged. What NIST proposed was reducing the hash function's capacity in the name of performance. One of Keccak's nice features is that it's highly tunable.[21]

In November 2013, in the light of the uproar in the cryptographic community, John Kelsey of NIST proposed to go back to the original c=2n proposal for all SHA-2 drop-in replacement instances.[24]

Examples of SHA-3 and Keccak variants[edit]

Hash values of empty string. Actual parameters other than digest size are the same as the submission to NIST.

  • For SHA3-n and Keccak-n, where n is 224, 256, 384, or 512, n is the output length.
  • For SHA3-n, an additional two bits 01 are appended to the message before padding.
  • As mentioned above, capacity is set to double the output length, per the submission to NIST.
  • Rate is set to 1600 bits minus capacity (rate plus capacity must always equal state size, so specifying any two implies the third).
  • The digest is encoded as a hexadecimal string.
Keccak-224("")
0x f71837502ba8e10837bdd8d365adb85591895602fc552b48b7390abd
Keccak-256("")
0x c5d2460186f7233c927e7db2dcc703c0e500b653ca82273b7bfad8045d85a470
Keccak-384("")
0x 2c23146a63a29acf99e73b88f8c24eaa7dc60aa771780ccc006afbfa8fe2479b2dd2b21362337441ac12b515911957ff
Keccak-512("")
0x 0eab42de4c3ceb9235fc91acffe746b29c29a8c366b7c60e4e67c466f36a4304c00fa9caf9d87976ba469bcbe06713b435f091ef2769fb160cdab33d3670680e
SHA3-224("")
0x 6b4e03423667dbb73b6e15454f0eb1abd4597f9a1b078e3f5b5a6bc7
SHA3-256("")
0x a7ffc6f8bf1ed76651c14756a061d662f580ff4de43b49fa82d80a4b80f8434a
SHA3-384("")
0x 0c63a75b845e4f7d01107d852e4c2485c51a50aaaa94fc61995e71bbee983a2ac3713831264adb47fb6bd1e058d5f004
SHA3-512("")
0x a69f73cca23a9ac5c8b567dc185a756e97c982164fe25859e0d1dcc1475c80a615b2123af1f5f94c11e3e9402c3ac558f500199d95b6d3e301758586281dcd26

Even a small change in the message will (with overwhelming probability) result in a mostly different hash, demonstrating the avalanche effect. For example, adding a period to the end of the sentence:

Keccak-224("The quick brown fox jumps over the lazy dog")
0x 310aee6b30c47350576ac2873fa89fd190cdc488442f3ef654cf23fe
Keccak-224("The quick brown fox jumps over the lazy dog.")
0x c59d4eaeac728671c635ff645014e2afa935bebffdb5fbd207ffdeab

Keccak-256("The quick brown fox jumps over the lazy dog")
0x 4d741b6f1eb29cb2a9b9911c82f56fa8d73b04959d3d9d222895df6c0b28aa15
Keccak-256("The quick brown fox jumps over the lazy dog.")
0x 578951e24efd62a3d63a86f7cd19aaa53c898fe287d2552133220370240b572d

Keccak-384("The quick brown fox jumps over the lazy dog")
0x 283990fa9d5fb731d786c5bbee94ea4db4910f18c62c03d173fc0a5e494422e8a0b3da7574dae7fa0baf005e504063b3
Keccak-384("The quick brown fox jumps over the lazy dog.")
0x 9ad8e17325408eddb6edee6147f13856ad819bb7532668b605a24a2d958f88bd5c169e56dc4b2f89ffd325f6006d820b

Keccak-512("The quick brown fox jumps over the lazy dog")
0x d135bb84d0439dbac432247ee573a23ea7d3c9deb2a968eb31d47c4fb45f1ef4422d6c531b5b9bd6f449ebcc449ea94d0a8f05f62130fda612da53c79659f609
Keccak-512("The quick brown fox jumps over the lazy dog.")
0x ab7192d2b11f51c7dd744e7b3441febf397ca07bf812cceae122ca4ded6387889064f8db9230f173f6d1ab6e24b6e50f065b039f799f5592360a6558eb52d760
SHA3-224("The quick brown fox jumps over the lazy dog")
0x d15dadceaa4d5d7bb3b48f446421d542e08ad8887305e28d58335795
SHA3-224("The quick brown fox jumps over the lazy dog.")
0x 2d0708903833afabdd232a20201176e8b58c5be8a6fe74265ac54db0

SHA3-256("The quick brown fox jumps over the lazy dog")
0x 69070dda01975c8c120c3aada1b282394e7f032fa9cf32f4cb2259a0897dfc04
SHA3-256("The quick brown fox jumps over the lazy dog.")
0x a80f839cd4f83f6c3dafc87feae470045e4eb0d366397d5c6ce34ba1739f734d

SHA3-384("The quick brown fox jumps over the lazy dog")
0x 7063465e08a93bce31cd89d2e3ca8f602498696e253592ed26f07bf7e703cf328581e1471a7ba7ab119b1a9ebdf8be41
SHA3-384("The quick brown fox jumps over the lazy dog.")
0x 1a34d81695b622df178bc74df7124fe12fac0f64ba5250b78b99c1273d4b080168e10652894ecad5f1f4d5b965437fb9

SHA3-512("The quick brown fox jumps over the lazy dog")
0x 01dedd5de4ef14642445ba5f5b97c15e47b9ad931326e4b0727cd94cefc44fff23f07bf543139939b49128caf436dc1bdee54fcb24023a08d9403f9b4bf0d450
SHA3-512("The quick brown fox jumps over the lazy dog.")
0x 18f4f4bd419603f95538837003d9d254c26c23765565162247483f65c50303597bc9ce4d289f21d1c2f1f458828e33dc442100331b35e7eb031b5d38ba6460f8

SHA-3 also includes two variable length Extendable-Output Functions, SHAKE128 and SHAKE256, with the numerical component determining their expected security level. These differ in both their capacity and padding rules. The capacity for SHAKE128 is 256 bits, and for SHAKE256 is 512 bits. An additional four bits 1111 are appended to the message before padding, and the output is truncated to the desired length. The first two appended bits are to differentiate SHAKE from SHA3-n, last two appended bits are for the Sakura coding scheme, and will be different for future tree hashing extensions of SHA-3.

References[edit]

  1. ^ "Draft FIPS 202, SHA-3 Standard". 
  2. ^ "Tentative SHA-3 standard (FIPS XXX) development timeline". NIST. Retrieved 2014-01-02. 
  3. ^ a b "NIST Selects Winner of Secure Hash Algorithm (SHA-3) Competition". NIST. Retrieved 2012-10-02. 
  4. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "The Keccak sponge function family: Specifications summary". Retrieved 2011-05-11. 
  5. ^ Cryptographic hash function - Wikipedia's Page on Cryptographic Hashes
  6. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "Sponge Functions". Ecrypt Hash Workshop 2007. 
  7. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "On the Indifferentiability of the Sponge Construction". EuroCrypt 2008. 
  8. ^ Keccak implementation overview Version 3.2 http://keccak.noekeon.org/Keccak-implementation-3.2.pdf
  9. ^ Guo, Xu; Huang, Sinan; Nazhandali, Leyla; Schaumont, Patrick (Aug 2010), "Fair and Comprehensive Performance Evaluation of 14 Second Round SHA-3 ASIC Implementations", NIST 2nd SHA-3 Candidate Conference: 12, retrieved 2011-02-18  Keccak is second only to Luffa, which did not advance to the final round.
  10. ^ NIST, Third-Round Report of the SHA-3 Cryptographic Hash Algorithm Competition, sections 5.1.2.1 (mentioning "tree mode"), 6.2 ("other features", mentioning authenticated encryption), and 7 (saying "extras" may be standardized in the future)
  11. ^ a b John Kelsey. "SHA3, Where We've Been, Where We're Going". RSA Conference 2013. 
  12. ^ a b John Kelsey. "SHA3, Past, Present, and Future". CHES 2013. 
  13. ^ "Crypto++ 5.6.0 Benchmarks". Retrieved 2013-06-13. 
  14. ^ "Cryptanalysis of MD5 & SHA-1" (PDF). Retrieved 2013-04-25. 
  15. ^ "Keccak parameter changes for round 2". 
  16. ^ "Simplifying Keccak's padding rule for round 3". 
  17. ^ "NIST hash forum mailing list". 
  18. ^ "The Keccak SHA-3 submission" (PDF). 2011-01-14. Retrieved 2014-02-08. 
  19. ^ "On 128-bit security". 
  20. ^ "A concrete proposal". 
  21. ^ a b "Schneier on Security: Will Keccak = SHA-3?". 
  22. ^ "LShift: Why I support the US Government making a cryptography standard weaker". 
  23. ^ "Yes, this is Keccak!". 
  24. ^ "Moving Forward with SHA-3". 
  25. ^ "Draft FIPS 202, SHA-3 Standard". 

External links[edit]