# SHA-3

Secure Hash Algorithm
Concepts
hash functions · SHA · DSA
Main standards
SHA-0 · SHA-1 · SHA-2 · SHA-3
SHA-3
(Keccak)
General
Designers Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche.
First published 2015
Series (SHA-0), SHA-1, SHA-2, SHA-3
Certification FIPS PUB 202
Detail
Digest sizes arbitrary
Structure sponge construction
Speed 12.5 cpb on Core 2 [c=1024, r=576].

SHA-3 (Secure Hash Algorithm 3), a subset of the cryptographic primitive family Keccak (/ˈkɛtʃæk/, or /kɛtʃɑːk/),[1][2][3] is a cryptographic hash function designed by Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche, building upon RadioGatún. SHA-3 is a member of the Secure Hash Algorithm family. The SHA-3 standard was released by NIST on August 5, 2015.[4][5] The reference implementation source code was dedicated to public domain via CC0 waiver.[6]

## History

The Keccak algorithm is the work of Guido Bertoni, Joan Daemen (who also co-designed the Rijndael cipher with Vincent Rijmen), Michael Peeters, and Gilles Van Assche. It is based on earlier hash function designs PANAMA and RadioGatún. PANAMA was designed by Daemen and Craig Clapp in 1998. RadioGatún, a successor of PANAMA, was designed by Daemen, Peeters, and Van Assche, and was presented at the NIST Hash Workshop in 2006.[7]

In 2006 NIST started to organize the NIST hash function competition to create a new hash standard, SHA-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, SHA-0 and SHA-1,[8] NIST perceived a need for an alternative, dissimilar cryptographic hash, which became SHA-3.

After a setup period, admissions were to be submitted by the end of 2008. Keccak was accepted as one of the 51 candidates. In July 2009, 14 algorithms were selected for the second round. Keccak advanced to the last round in December 2010.[9]

During the competition, entrants were permitted to "tweak" their algorithms to address issues that were discovered. Changes that have been made to Keccak are:[10][11]

• 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 below.
• The rate r was increased to the security limit, rather than rounding down to the nearest power of 2.

On October 2, 2012, Keccak was selected as the winner of the competition.[1]

In 2014, the NIST published a draft FIPS 202 "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions".[12] FIPS 202 was approved on August 5, 2015.[13]

On August 5, 2015 NIST announced that SHA-3 had become a hashing standard.[14]

## Design

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,[15][16] in which data is "absorbed" into the sponge, then the result is "squeezed" out. In the absorbing phase, message blocks are XORed into a subset of the state, which is then transformed as a whole. In the "squeeze" phase, output blocks are read from the same subset of the state, alternated with state transformations. The size of the part of the state that is written and read is called the "rate" (often denoted r), and the size of the part that is untouched by input/output is called the "capacity" (often denoted c). The capacity determines the security of the scheme. The maximum security level is half the capacity.

In SHA-3, the state consists of a 5 × 5 array of 64-bit words, 1600 bits total. 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.[17][18]

The block transformation is a permutation that uses xor, and and not operations, and designed for easy implementation in both software and hardware. The authors claim 12.5 cycles per byte[19] on an Intel Core 2 CPU. However, in hardware implementations, it is notably faster than all other finalists.[20]

Keccak's authors have proposed additional, not-yet-standardized uses for the function, including an authenticated encryption system and a "tree" hashing scheme for faster hashing on certain architectures.[21]

## The block permutation

This is defined for any power-of-two word size, w = 2 bits. The main SHA-3 submission uses 64-bit words, = 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 and row-major indexing. I.e. i selects the row, j the column, and k the bit.

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 + 2 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[0...4][ j−1][k]) ⊕ parity(a[0...4][ 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 ${\displaystyle {\begin{pmatrix}i\\j\end{pmatrix}}={\begin{pmatrix}3&2\\1&0\end{pmatrix}}^{t}{\begin{pmatrix}0\\1\end{pmatrix}}}$.
π
Permute the 25 words in a fixed pattern. a[ j][2i+3 j] ← a[i][ j].
χ
Bitwise combine along rows, using xx ⊕ (¬y & z). 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, 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

To ensure the message can be evenly divided into r-bit blocks, padding is required. Keccak uses the 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 for the security proof to work for sponges of different rates, that is, different hash variants (multi-rate padding). Without it, different hash variants of the same short message would be the same up to truncation.

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 desired number of bits n is squeezed. For the SHA3 instances, r is always greater than n, thus there is never a need for additional block permutations in the squeezing phase. The leading n bits of the state are the desired hash. However, arbitrary output length may be useful in applications such as optimal asymmetric encryption padding.

## Instances

The NIST standard defines the following instances.

Keccak[capacity](M, d) is defined as the sponge construction using Keccak-f[1600, capacity], message M and output length d. Note, that the appended postfixes are written as bit strings, not hexadecimal digits.

Instance Definition
SHA3-224(M) Keccak[448](M || 01, 224)
SHA3-256(M) Keccak[512](M || 01, 256)
SHA3-384(M) Keccak[768](M || 01, 384)
SHA3-512(M) Keccak[1024](M || 01, 512)
SHAKE128(M, d) Keccak[256](M || 1111, d)
SHAKE256(M, d) Keccak[512](M || 1111, d)

The SHA3 instances are the drop-in replacements for SHA2, with identical security claims. SHAKE instances are so called XOF's, Extendable Output Functions. For example, SHAKE128(M, 256) can be used as a hash function with a 256 bit length and 128 bit overall security.

Note that all instances append some bits to the message. Since 10*1 padding always adds at least two bits, in byte aligned libraries there are always six unused zero bits. Therefore, these appended extra bits never make the padded message longer.

## 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.[22][23] The changes caused some turmoil.

In September 2013, Daniel J. Bernstein suggested on the NIST hash-forum mailing list[24] to strengthen the security to the 576-bit capacity that was originally proposed as the default Keccak.[25] 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.[26] However, in the light of the negative response from the cryptographic community, they proposed raising the capacity to 512 bits for all instances.[27]

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.[28]

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.[29]

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.[30] 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.[28]

In response to the controversy, in November 2013 John Kelsey of NIST proposed to go back to the original c = 2n proposal for all SHA-2 drop-in replacement instances.[31] These changes were confirmed in the April 2014 draft.[32] This proposal was implemented in the final release standard in August 2015.[4]

## Examples of SHA-3 variants

The following hash values are from NIST.gov[33]

SHA3-224("")
6b4e03423667dbb73b6e15454f0eb1abd4597f9a1b078e3f5b5a6bc7
SHA3-256("")
a7ffc6f8bf1ed76651c14756a061d662f580ff4de43b49fa82d80a4b80f8434a
SHA3-384("")
SHA3-512("")
a69f73cca23a9ac5c8b567dc185a756e97c982164fe25859e0d1dcc1475c80a615b2123af1f5f94c11e3e9402c3ac558f500199d95b6d3e301758586281dcd26
SHAKE128("", 256)
7f9c2ba4e88f827d616045507605853ed73b8093f6efbc88eb1a6eacfa66ef26
SHAKE256("", 512)
46b9dd2b0ba88d13233b3feb743eeb243fcd52ea62b81b82b50c27646ed5762fd75dc4ddd8c0f200cb05019d67b592f6fc821c49479ab48640292eacb3b7c4be


Changing a single bit causes each bit in the output to change with 50% probability, demonstrating an avalanche effect:

SHAKE128("The quick brown fox jumps over the lazy dog", 256)
f4202e3c5852f9182a0430fd8144f0a74b95e7417ecae17db0f8cfeed0e3e66e
SHAKE128("The quick brown fox jumps over the lazy dof", 256)
853f4538be0db9621a6cea659a06c1107b1f83f02b13d18297bd39d7411cf10c


## 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)
Max message size
(bits)
Rounds Operations Security bits
(Info)
Example performance[35]
(MiB/s)
First Published
MD5 (as reference) 128 128
(4 × 32)
512 Unlimited[36] 64 And, Xor, Rot, Add (mod 232), Or <64
(collisions found)
335 1992
SHA-0 160 160
(5 × 32)
512 264 − 1 80 And, Xor, Rot, Add (mod 232), Or <80
(collisions found)
- 1993
SHA-1 160 160
(5 × 32)
512 264 − 1 80 <63
(collision found[37])
192 1995
SHA-2 SHA-224
SHA-256
224
256
256
(8 × 32)
512 264 − 1 64 And, Xor, Rot, Add (mod 232), Or, Shr 112
128
139 2001
SHA-384
SHA-512
SHA-512/224
SHA-512/256
384
512
224
256
512
(8 × 64)
1024 2128 − 1 80 And, Xor, Rot, Add (mod 264), Or, Shr 192
256
112
128
154 2001
SHA-3 SHA3-224
SHA3-256
SHA3-384
SHA3-512
224
256
384
512
1600
(5 × 5 × 64)
1152
1088
832
576
Unlimited[38] 24[39] And, Xor, Rot, Not 112
128
192
256
- 2015
SHAKE128
SHAKE256
d (arbitrary)
d (arbitrary)
1344
1088
min(d/2, 128)
min(d/2, 256)
- 2015

## References

1. ^ a b "NIST Selects Winner of Secure Hash Algorithm (SHA-3) Competition". NIST. 2012-10-02. Retrieved 2012-10-02.
2. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "The Keccak sponge function family: Specifications summary". Retrieved 2011-05-11.
3. ^ "Keccak: The New SHA-3 Encryption Standard". Dr. Dobbs.
4. ^ a b http://www.nist.gov/itl/csd/201508_sha3.cfm
5. ^ http://www.nist.gov/manuscript-publication-search.cfm?pub_id=919061
6. ^ KeccakReferenceAndOptimized-3.2.zip mainReference.c "The Keccak sponge function, designed by Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. For more information, feedback or questions, please refer to our website: http://keccak.noekeon.org/Implementation by the designers, hereby denoted as "the implementer". To the extent possible under law, the implementer has waived all copyright and related or neighboring rights to the source code in this file. http://creativecommons.org/publicdomain/zero/1.0/"
7. ^
8. ^ Stevens, Marc; Bursztein, Elie; Karpman, Pierre; Albertini, Ange; Markov, Yarik. "The first collision for full SHA-1" (PDF). Retrieved 23 February 2017.
9. ^
10. ^
11. ^
12. ^ "SHA-3 standardization". NIST. Retrieved 2015-04-16.
13. ^ National Institute of Standards and Technology (Aug 5, 2015). "Federal Information Processing Standards: Permutation-Based Hash and Extendable-Output Functions, etc.". Retrieved 5 Aug 2015.
14. ^
15. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "Sponge Functions". Ecrypt Hash Workshop 2007.
16. ^ Guido Bertoni, Joan Daemen, Michaël Peeters and Gilles Van Assche. "On the Indifferentiability of the Sponge Construction". EuroCrypt 2008.
17. ^ Daemen, Joan, CAESAR submission: Ketje v1 (PDF)
18. ^ Daemen, Joan, CAESAR submission: Keyak v1 (PDF)
19. ^ Keccak implementation overview Version 3.2 http://keccak.noekeon.org/Keccak-implementation-3.2.pdf
20. ^ Guo, Xu; Huang, Sinan; Nazhandali, Leyla; Schaumont, Patrick (Aug 2010), "Fair and Comprehensive Performance Evaluation of 14 Second Round SHA-3 ASIC Implementations" (PDF), 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.
21. ^ 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)
22. ^ John Kelsey. "SHA3, Where We've Been, Where We're Going" (PDF). RSA Conference 2013.
23. ^ John Kelsey. "SHA3, Past, Present, and Future". CHES 2013.
24. ^
25. ^ "The Keccak SHA-3 submission" (PDF). 2011-01-14. Retrieved 2014-02-08.
26. ^
27. ^
28. ^ a b
29. ^
30. ^
31. ^ "Moving Forward with SHA-3" (PDF).
32. ^ NIST Computer Security Division (CSD). "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions" (PDF). NIST.
33. ^
34. ^ "Crypto++ 5.6.0 Benchmarks". Retrieved 2013-06-13.
35. ^ Found on an AMD Opteron 8354 2.2 GHz processor running 64-bit Linux[34]
36. ^ "The MD5 Message-Digest Algorithm". Retrieved 2016-04-18.
37. ^ "Announcing the first SHA1 collision". Retrieved 2017-02-23.
38. ^ "The Sponge Functions Corner". Retrieved 2016-01-27.
39. ^ "The Keccak sponge function family". Retrieved 2016-01-27.