SHA-3

As of April  2014, NIST has updated Draft FIPS Publication 202, SHA-3 Standard separate from the Secure Hash Standard (SHS). The content of this article is subject to change once the final standard is published.

SHA-3
(Keccak)

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

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

History

On October 2, 2012, Keccak was selected as the winner of the NIST hash function competition. 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, NIST perceived a need for an alternative, dissimilar cryptographic hash, which became SHA-3.

In 2014, the NIST has published a draft FIPS 202 "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions". The standardization process is in progress as of January 2015.

Design

SHA-3 uses the sponge construction, in which message blocks are XORed into a subset of the state, which is then transformed as a whole. 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 on an Intel Core 2 CPU. However, in hardware implementations, it is notably faster than all other finalists.

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. 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 (from w = 8, 200 bits, to w = 32, 800 bits) can be used in practical, lightweight applications.

The block permutation

This is defined for any power-of-two word size, w = 2^ℓ 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 (5i + j) × w + k of the input, using a little-endian bit numbering 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 5w (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] ← 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][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][2^m−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

The sponge construction for hash functions. p_i are input, z_i 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 = 25w − r 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. In April 2014, NIST published a draft that confirms these values.

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"). The current draft includes adding bits 01 to the message before the applying the padding. This provides domain separation from the SHAKEs, the other sponge modes included in the draft. For byte granularity data, this never increases the message size, since we have six unused bits anyways.

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).

Tweaks

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:

• 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

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. The changes caused some turmoil.

In September 2013, Daniel J. Bernstein suggested on the NIST hash-forum mailing list to strengthen the security to the 576-bit capacity that was originally proposed as the default Keccak. 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. However, in the light of the uproar in the cryptographic community, they proposed raising the capacity to 512 bits for all instances.

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.

Paul Crowley, a senior developer at an independent software development company, expressed his support of the decision, saying that Keccak is 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.

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. 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.

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. These changes were confirmed in the April 2014 draft.

Examples of SHA-3 and Keccak variants

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 and 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, the RHASH implementation has published the following outputs with inputs differing only in a period:

Using RHash implementation
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

The RHash Implementation is not the same as either the final SHA3 submission, nor is it like the FIPS 202 draft from April. RHash does not use the same bit order when absorbing bits. It does still match the same result published by NIST for the empty string case, because no bits are absorbed on an empty string. The final SHA3 submission for Keccak appends a 1 bit directly to start padding, while both the FIPS 202 draft and the RHash implementation use three bits "011" at the end of the message to begin padding.

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.

Comparison of SHA functions

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

Comparison of SHA functions

Algorithm and variant		Output size (bits)	Internal state size (bits)	Block size (bits)	Rounds	Operations	Security against collision attacks (bits)	Security against length extension attacks (bits)	Performance on Skylake (median cpb)		First published
									Long messages	8 bytes
MD5 (as reference)		128	128 (4 × 32)	512	4 (16 operations in each round)	And, Xor, Or, Rot, Add (mod 232)	≤ 18 (collisions found)	0	4.99	55.00	1992
SHA-0		160	160 (5 × 32)	512	80	And, Xor, Or, Rot, Add (mod 232)	< 34 (collisions found)	0	≈ SHA-1	≈ SHA-1	1993
SHA-1							< 63 (collisions found)		3.47	52.00	1995
SHA-2	SHA-224 SHA-256	224 256	256 (8 × 32)	512	64	And, Xor, Or, Rot, Shr, Add (mod 232)	112 128	32 0	7.62 7.63	84.50 85.25	2004 2001
	SHA-384	384	512 (8 × 64)	1024	80	And, Xor, Or, Rot, Shr, Add (mod 264)	192	128	5.12	135.75	2001
	SHA-512	512					256	0	5.06	135.50	2001
	SHA-512/224 SHA-512/256	224 256					112 128	288 256	≈ SHA-384	≈ SHA-384	2012
SHA-3	SHA3-224 SHA3-256 SHA3-384 SHA3-512	224 256 384 512	1600 (5 × 5 × 64)	1152 1088 832 576	24	And, Xor, Rot, Not	112 128 192 256	448 512 768 1024	8.12 8.59 11.06 15.88	154.25 155.50 164.00 164.00	2015
	SHAKE128 SHAKE256	d (arbitrary) d (arbitrary)		1344 1088			min(d/2, 128) min(d/2, 256)	256 512	7.08 8.59	155.25 155.50

External links

• The Keccak web site
• A Cryptol implementation of Keccak
• A Java implementation of Keccak
• VHDL source code developed by the Cryptographic Engineering Research Group (CERG) at George Mason University
• Erlang NIF implementation based on the NIST reference code
• Keccak implemented in PureBasic