SHA-3

SHA-3 is the latest member of the Secure Hash Algorithm family of standards, released by NIST on August 5, 2015. Although part of the same series of standards, SHA-3 is internally different from the MD5-like structure of SHA-1 and SHA-2.
SHA-3 is a subset of the broader cryptographic primitive family Keccak, designed by Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche, building upon RadioGatún. Keccak's authors have proposed additional uses for the function, not standardized by NIST, including a stream cipher, an authenticated encryption system, a "tree" hashing scheme for faster hashing on certain architectures, and AEAD ciphers Keyak and Ketje.
Keccak is based on a novel approach called sponge construction. Sponge construction is based on a wide random function or random permutation, and allows inputting any amount of data, and outputting any amount of data, while acting as a pseudorandom function with regard to all previous inputs. This leads to great flexibility.
NIST does not plan to withdraw SHA-2 or remove it from the revised Secure Hash Standard. The purpose of SHA-3 is that it can be directly substituted for SHA-2 in current applications if necessary, and to significantly improve the robustness of NIST's overall hash algorithm toolkit.
For small message sizes, the creators of the Keccak algorithms and the SHA-3 functions suggest using the faster function KangarooTwelve with adjusted parameters and a new tree hashing mode without extra overhead.

History

The Keccak algorithm is the work of Guido Bertoni, Joan Daemen, Michaël 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. The reference implementation was released to the public domain.
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 publicly demonstrated. Because of the successful attacks on MD5, SHA-0 and SHA-1,
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.
During the 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 to 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.
In 2014, the NIST published a draft FIPS 202 "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions". FIPS 202 was approved on August 5, 2015.
On August 5, 2015, NIST announced that SHA-3 had become a hashing standard.

Weakening controversy

In early 2013 NIST announced they would select different values for the "capacity", the overall strength vs. speed parameter, for the SHA-3 standard, compared to the submission. The changes caused some turmoil.
The hash function competition called for hash functions at least as secure as the SHA-2 instances. It means that a d-bit output should have d/2-bit resistance to collision attacks and d-bit resistance to preimage attacks, the maximum achievable for d bits of output. Keccak's security proof allows an adjustable level of security based on a "capacity" c, providing c/2-bit resistance to both collision and preimage attacks. To meet the original competition rules, Keccak's authors proposed. The announced change was to accept the same d/2-bit security for all forms of attack and standardize. This would have sped up Keccak by allowing an additional d bits of input to be hashed each iteration. However, the hash functions would not have been drop-in replacements with the same preimage resistance as SHA-2 any more; it would have been cut in half, making it vulnerable to advances in quantum computing, which effectively would cut it in half once more.
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 addition to and not included in the SHA-3 specifications. This would have provided at least a SHA3-224 and SHA3-256 with the same preimage resistance as their SHA-2 predecessors, but SHA3-384 and SHA3-512 would have had significantly less preimage resistance than their SHA-2 predecessors. In late September, the Keccak team responded by stating that they had proposed 128-bit security by setting as an option already in their SHA-3 proposal. Although the reduced capacity was justifiable in their opinion, in the light of the negative response, they proposed raising the capacity to bits for all instances. This would be as much as any previous standard up to the 256-bit security level, while providing reasonable efficiency, but not the 384-/512-bit preimage resistance offered by SHA2-384 and SHA2-512. The authors stated that "claiming or relying on security strength levels above 256 bits is meaningless".
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:
He later retracted his earlier statement, saying:
Paul Crowley, a cryptographer and 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:
There was some confusion that internal changes may have been made to Keccak, which were cleared up by the original team, 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.
In response to the controversy, in November 2013 John Kelsey of NIST proposed to go back to the original proposal for all SHA-2 drop-in replacement instances. The reversion was confirmed in subsequent drafts and in the final release.

Design

SHA-3 uses the sponge construction, 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 using a permutation function . In the "squeeze" phase, output blocks are read from the same subset of the state, alternated with the state transformation function. The size of the part of the state that is written and read is called the "rate", and the size of the part that is untouched by input/output is called the "capacity". The capacity determines the security of the scheme. The maximum security level is half the capacity.
Given an input bit string, a padding function, a permutation function that operates on bit blocks of width, a rate and an output length, we have capacity and the sponge construction. This yields a bit string of length as follows:

pad the input N using the pad function, yielding a padded bit string P with a length divisible by
break P into n consecutive r-bit pieces P₀,..., P_n−1
initialize the state S to a string of b zero bits
absorb the input into the state: for each block P_i:
* extend P_i at the end by a string of c zero bits, yielding one of length b
* XOR that with S
* apply the block permutation f to the result, yielding a new state S
initialize Z to be the empty string
while the length of Z is less than d:
* append the first r bits of S to Z
* if Z is still less than d bits long, apply f to S, yielding a new state S
truncate Z to d bits

The fact that the internal state S contains c additional bits of information in addition to what is output to Z prevents the length extension attacks that SHA-2, SHA-1, MD5 and other hashes based on the Merkle–Damgård construction are susceptible to.
In SHA-3, the state S consists of a array of w-bit words, b = 5 × 5 × w = 5 × 5 × 64 = 1600 bits total. Keccak is also defined for smaller power-of-2 word sizes w down to 1 bit. Small state sizes can be used to test cryptanalytic attacks, and intermediate state sizes can be used in practical, lightweight applications.
For SHA3-224, SHA3-256, SHA3-384, and SHA3-512 instances, r is greater than d, so there is no need for additional block permutations in the squeezing phase; the leading d bits of the state are the desired hash. However, SHAKE128 and SHAKE256 allow an arbitrary output length, which is useful in applications such as optimal asymmetric encryption padding.

Padding

To ensure the message can be evenly divided into r-bit blocks, padding is required. SHA-3 uses the pattern 10...01 in its padding function: a 1 bit, followed by zero or more 0 bits and a final 1 bit.
The maximum of zero bits occurs when the last message block is bits long. Then another block is added after the initial 1 bit, containing zero bits before the final 1 bit.
The two 1 bits will be added even if the length of the message is already divisible by r. In this case, another block is added to the message, containing a 1 bit, followed by a block of zero bits and another 1 bit. This is necessary so that a message with length divisible by r ending in something that looks like padding does not produce the same hash as the message with those bits removed.
The initial 1 bit is required so messages differing only in a few additional 0 bits at the end do not produce the same hash.
The position of the final 1 bit indicates which rate r was used, which is required for the security proof to work for different hash variants. Without it, different hash variants of the same short message would be the same up to truncation.

The block permutation

The block transformation f, which is Keccak-f for SHA-3, is a permutation that uses XOR, AND and NOT operations, and is designed for easy implementation in both software and hardware.
It is defined for any power-of-two word size, bits. The main SHA-3 submission uses 64-bit words,.
The state can be considered to be a array of bits. Let be bit of the input, using a little-endian bit numbering convention and row-major indexing. I.e. selects the row, the column, and 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 rounds of five steps:
;
;
;
;
;

Speed

The speed of SHA-3 hashing of long messages is dominated by the computation of f = Keccak-f and XORing S with the extended P_i, an operation on b = 1600 bits. However, since the last c bits of the extended P_i are 0 anyway, and XOR with 0 is a NOP, it is sufficient to perform XOR operations only for r bits. The lower r is, the less efficient but more secure the hashing becomes since fewer bits of the message can be XORed into the state before each application of the computationally expensive f.
The authors report the following speeds for software implementations of Keccak-f plus XORing 1024 bits, which roughly corresponds to SHA3-256:

57.4 cpb on IA-32, Intel Pentium 3
41 cpb on IA-32+MMX, Intel Pentium 3
20 cpb on IA-32+SSE, Intel Core 2 Duo or AMD Athlon 64
12.6 cpb on a typical x86-64-based machine
6–7 cpb on IA-64

For the exact SHA3-256 on x86-64, Bernstein measures 11.7–12.25 cpb depending on the CPU. SHA-3 has been criticized for being slow on instruction set architectures which do not have instructions meant specially for computing Keccak functions faster – SHA2-512 is more than twice as fast as SHA3-512, and SHA-1 is more than three times as fast on an Intel Skylake processor clocked at 3.2 GHz. The authors have reacted to this criticism by suggesting to use SHAKE128 and SHAKE256 instead of SHA3-256 and SHA3-512, at the expense of cutting the preimage resistance in half. With this, performance is on par with SHA2-256 and SHA2-512.
However, in hardware implementations, SHA-3 is notably faster than all other finalists, and also faster than SHA-2 and SHA-1.
As of 2018, ARM's ARMv8 architecture includes special instructions which enable Keccak algorithms to execute faster and IBM's z/Architecture includes a complete implementation of SHA-3 and SHAKE in a single instruction. There have also been extension proposals for RISC-V to add Keccak-specific instructions.

Instances

The NIST standard defines the following instances, for message M and output length d:

With the following definitions

Keccak = sponge, pad10^*1, r]
Keccak-f = Keccak-p
c is the capacity
r is the rate = 1600 − c
N is the input bit string

SHA-3 instances are drop-in replacements for SHA-2, intended to have identical security properties.
SHAKE will generate as many bits from its sponge as requested, thus being extendable-output functions. For example, SHAKE128 can be used as a hash function with a 256 character bitstream with 128-bit security strength. Arbitrarily large lengths can be used as pseudo-random number generators. Alternately, SHAKE256 can be used as a hash function with a 128-bit length and 128-bit resistance.
All instances append some bits to the message, the rightmost of which represent the domain separation suffix. The purpose of this is to ensure that it is not possible to construct messages that produce the same hash output for different applications of the Keccak hash function. The following domain separation suffixes exist:

Suffix	Meaning
00	cSHAKE
01	SHA-3
...11	RawSHAKE
1111	SHAKE

Additional instances

In December 2016 NIST published a new document, NIST SP.800-185, describing additional SHA-3-derived functions:

Instance	Description
cSHAKE128	A version of SHAKE supporting explicit domain separation via customization parameters.
cSHAKE256	A version of SHAKE supporting explicit domain separation via customization parameters.
KMAC128	A keyed hash function based on Keccak. Can also be used without a key as a regular hash function.
KMAC256	A keyed hash function based on Keccak. Can also be used without a key as a regular hash function.
KMACXOF128	A keyed hash function based on Keccak. Can also be used without a key as a regular hash function.
KMACXOF256	A keyed hash function based on Keccak. Can also be used without a key as a regular hash function.
TupleHash128	A function for hashing tuples of strings. The output of this function depends on both the contents and the sequence of input strings.
TupleHash256	A function for hashing tuples of strings. The output of this function depends on both the contents and the sequence of input strings.
TupleHashXOF128	A function for hashing tuples of strings. The output of this function depends on both the contents and the sequence of input strings.
TupleHashXOF256	A function for hashing tuples of strings. The output of this function depends on both the contents and the sequence of input strings.
ParallelHash128	A function designed to exploit parallelism in modern processors for faster hashing. Unlike KangarooTwelve, does not use reduced-round Keccak.
ParallelHash256	A function designed to exploit parallelism in modern processors for faster hashing. Unlike KangarooTwelve, does not use reduced-round Keccak.
ParallelHashXOF128	A function designed to exploit parallelism in modern processors for faster hashing. Unlike KangarooTwelve, does not use reduced-round Keccak.
	A function designed to exploit parallelism in modern processors for faster hashing. Unlike KangarooTwelve, does not use reduced-round Keccak.

X is the main input bit string. It may be of any length, including zero.
L is an integer representing the requested output length in bits.
N is a function-name bit string, used by NIST to define functions based on cSHAKE. When no function other than cSHAKE is desired, N is set to the empty string, in which case it is the same as SHAKE.
S is a customization bit string. The user selects this string to define a variant of the function. When no customization is desired, S is set to the empty string.
K is a key bit string of any length, including zero.
B is the block size in bytes for parallel hashing. It may be any integer such that 0 < B < 2²⁰⁴⁰.
Later developments

KangarooTwelve

In 2016 the same team that made the SHA-3 functions and the Keccak algorithm introduced faster reduced-rounds alternatives which can exploit the availability of parallel execution by using tree hashing: KangarooTwelve and MarsupilamiFourteen.
These functions differ from ParallelHash, the FIPS standardized Keccak-based parallelizable hash function, with regard to the parallelism, in that they are faster than ParallelHash for small message sizes.
The reduced number of rounds is justified by the huge cryptanalytic effort focused on Keccak which did not produce practical attacks on anything close to twelve-round Keccak. These higher-speed algorithms are not part of SHA-3, and thus are not FIPS compliant; but because they use the same Keccak permutation they are secure for as long as there are no attacks on SHA-3 reduced to 12 rounds.
KangarooTwelve is a higher-performance reduced-round version of Keccak which claims to have 128 bits of security while having performance as high as 0.55 cycles per byte on a Skylake CPU. This algorithm is specified in IETF RFC 9861.
MarsupilamiFourteen, a slight variation on KangarooTwelve, uses 14 rounds of the Keccak permutation and claims 256 bits of security. Note that 256-bit security is not more useful in practice than 128-bit security, but may be required by some standards. 128 bits are already sufficient to defeat brute-force attacks on current hardware, so having 256-bit security does not add practical value, unless the user is worried about significant advancements in the speed of classical computers. For resistance against quantum computers, see below.
KangarooTwelve and MarsupilamiFourteen are Extendable-Output Functions, similar to SHAKE, therefore they generate closely related output for a common message with different output length. Such property is not exhibited by hash functions such as SHA-3 or ParallelHash.

The Farfalle construction

In 2016, the Keccak team released a different construction called Farfalle construction, and Kravatte, an instance of Farfalle using the Keccak-p permutation, as well as two authenticated encryption algorithms Kravatte-SANE and Kravatte-SANSE

Sakura tree hashing

RawSHAKE is the basis for the Sakura coding for tree hashing, which has not been standardized yet. Sakura uses a suffix of 1111 for single nodes, equivalent to SHAKE, and other generated suffixes depending on the shape of the tree.

Security against quantum attacks

There is a general result that quantum computers can perform a structured preimage attack in, while a classical brute-force attack needs 2^d. A structured preimage attack implies a second preimage attack and thus a collision attack. A quantum computer can also perform a birthday attack, thus break collision resistance, in . Noting that the maximum strength can be, this gives the following upper bounds on the quantum security of SHA-3:

It has been shown that the Merkle–Damgård construction, as used by SHA-2, is collapsing and, by consequence, quantum collision-resistant, but for the sponge construction used by SHA-3, the authors provide proofs only for the case when the block function f is not efficiently invertible; Keccak-f, however, is efficiently invertible, and so their proof does not apply.

Examples of SHA-3 variants

The following hash values are from NIST.gov:
SHA3-224
6b4e03423667dbb73b6e15454f0eb1abd4597f9a1b078e3f5b5a6bc7
SHA3-256
a7ffc6f8bf1ed76651c14756a061d662f580ff4de43b49fa82d80a4b80f8434a
SHA3-384
0c63a75b845e4f7d01107d852e4c2485c51a50aaaa94fc61995e71bbee983a2ac3713831264adb47fb6bd1e058d5f004
SHA3-512
a69f73cca23a9ac5c8b567dc185a756e97c982164fe25859e0d1dcc1475c80a615b2123af1f5f94c11e3e9402c3ac558f500199d95b6d3e301758586281dcd26
SHAKE128
7f9c2ba4e88f827d616045507605853ed73b8093f6efbc88eb1a6eacfa66ef26
SHAKE256
46b9dd2b0ba88d13233b3feb743eeb243fcd52ea62b81b82b50c27646ed5762fd75dc4ddd8c0f200cb05019d67b592f6fc821c49479ab48640292eacb3b7c4be
Changing a single bit causes each bit in the output to change with 50% probability, demonstrating an avalanche effect:
SHAKE128
f4202e3c5852f9182a0430fd8144f0a74b95e7417ecae17db0f8cfeed0e3e66e
SHAKE128
853f4538be0db9621a6cea659a06c1107b1f83f02b13d18297bd39d7411cf10c

Comparison of SHA functions

In the table below, internal state means the number of bits that are carried over to the next block.
Optimized implementation using AVX-512VL of SHA3-256 do achieve about 6.4 cycles per byte for large messages, and about 7.8 cycles per byte when using AVX2 on Skylake CPUs. Performance on other x86, Power and ARM CPUs depending on instructions used, and exact CPU model varies from about 8 to 15 cycles per byte, with some older x86 CPUs up to 25–40 cycles per byte.

Implementations

Below is a list of cryptography libraries that support SHA-3:

ARMv8 six-core SoC CPU cores have support for accelerating SHA-3 using specialized instructions from ARMv8.2-SHA crypto extension set. These instructions do not implement the full SHA3 procedure; instead, they implement smaller operations: EOR3, RAX, XAR, BCAX.
OpenSSL includes various assembly-language implementations of SHA-3. Most improvement comes from optimization of scalar code and not much is gained from SIMD.

On 32-bit x86, only a MMX implementation is provided as "non-MMX-capable processors is an extinct breed". Comments note that it runs at about twice the speed as the scalar x86 code generated by gcc-5.x.
On 64-bit x86, there are generic, AVX2, AVX-512, and AVX-512VL implementations. The plain version is about 15% faster than code generated by gcc-5.x, the AVX2 version another 15% faster than the generic version on Intel processors. The AVX-512 versions show about 50% improvement over generic on Skylake-X.
On ARMv4 and above, the handwritten scalar code provides a doubling to tripling of performance over compiler output. The Thumb-2 scalar code provides about 10% benefit. The NEON code provides another doubling of performance, though even in the best case it takes 24 cycles for each byte processed. On ARMv8, using NEON instructions do not provide any benefit, except in the case of ARMv8.2-SHA where vector registers are lightly used for the hardware-assisted instructions.
On PowerPC, the scalar assembly version is about twice as fast as gcc-4.x code. POWER8 has a 2x64-bit vector rotate which is helpful, but in POWER9 the faster scalar issue capacity makes the scalar version once again faster.

The IBM z/Architecture supports SHA-3 since 2017 as part of the Message-Security-Assist Extension 6. The processors support a complete implementation of the entire SHA-3 and SHAKE algorithms via the KIMD and KLMD instructions using a hardware assist engine built into each core.

Parallel variants

It is easier to accelerate parallel variants of SHA-3 such as ParallelHash128. One such implementation for SSSE3 is found in Crypto++.

Usage in protocols

uses the Keccak-256 hash function.