View source for SWIFFT

{{Short description|Cryptographic hash function}}
{{Infobox cryptographic hash function
| name           = SWIFFT
| image          =
| caption        =
<!-- General -->
| designers      = Vadim Lyubashevsky, Daniele Micciancio, Chris Peikert, Alon Rosen
| publish date   = 2008
| series         =
| derived from   =
| derived to     =
| related to     = FFT-based algorithms
| certification  =
<!-- Detail -->
| digest size    =
| structure      =
| rounds         =
| cryptanalysis  =
}}

In [[cryptography]], '''SWIFFT''' is a collection of [[Provably secure cryptographic hash function|provably secure]] [[cryptographic hash function|hash functions]]. It is based on the concept of the [[fast Fourier transform]] (FFT). SWIFFT is not the first hash function based on the FFT, but it sets itself apart by providing a mathematical proof of its security. It also uses the [[Lenstra-Lenstra-Lovász lattice basis reduction algorithm|LLL basis reduction algorithm]]. It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/[[Ideal lattice cryptography|ideal lattices]] in the ''worst case''. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other [[provably secure cryptographic hash function|cryptographic hash functions]].

Unlike many other provably secure hash functions, the algorithm is quite fast, yielding a throughput of 40&nbsp;Mbit/s on a 3.2&nbsp;GHz Intel Pentium 4. Although SWIFFT satisfies many desirable cryptographic and statistical properties, it was not designed to be an "all-purpose" cryptographic hash function. For example, it is not a [[pseudorandom function]], and would not be a suitable instantiation of a [[random oracle]]. The algorithm is less efficient than most traditional hash functions that do not give a proof of their collision-resistance. Therefore, its practical use would lie mostly in applications where the proof of collision-resistance is particularly valuable, such as digital signatures that must remain trustworthy for a long time.

A modification of SWIFFT called [[SWIFFTX]] was proposed as a candidate for SHA-3 function to the [[NIST hash function competition]]<ref>{{cite web
| url=https://www.eecs.harvard.edu/~alon/PAPERS/lattices/swifftx.pdf
| title=SWIFFTX: A Proposal for the SHA-3 Standard
| author=Daniele Micciancio
| author2=Yuriy Arbitman
| author3=Gil Dogon
| author4=Vadim Lyubashevsky
| author5=Chris Peikert
| author6=Alon Rosen
| date=2008-10-30
| accessdate=2017-03-03}}</ref> and was rejected in the first round.<ref>{{cite web
 |url         = http://csrc.nist.gov/groups/ST/hash/sha-3/Round2/submissions_rnd2.html
 |title       = Second Round Candidates
 |publisher   = [[National Institute of Standards and Technology]]
 |date        = 2009-07-16
 |accessdate  = 2017-03-03
 |url-status     = bot: unknown
 |archiveurl  = https://web.archive.org/web/20170604091329/http://csrc.nist.gov/groups/ST/hash/sha-3/Round2/submissions_rnd2.html
 |archivedate = 2017-06-04
}}</ref>

==The algorithm==
The algorithm is as follows:<ref>{{cite web
| url=https://www.eecs.harvard.edu/~alon/PAPERS/lattices/swifft.pdf
| title=SWIFFT: A Modest Proposal for FFT Hashing
| author1=Vadim Lyubashevsky
| author2=Daniele Micciancio
| author3=Chris Peikert
| author4=Alon Rosen
| date=2008-02-21
| accessdate=2017-03-03}}</ref>
#Let the [[polynomial]] variable be called {{Mvar|&alpha;}}.
#'''Input''': message {{Mvar|M}} of length {{Math|''mn''}}
#Convert {{Mvar|M}} to a collection of polynomials {{Math|''p''<sub>1</sub>, &mldr;, ''p''<sub>''m''</sub>}} in a certain [[polynomial ring]] {{Mvar|R}} with binary coefficients.
#Compute the Fourier coefficients of each {{Math|''p''<sub>''i''</sub>}} using SWIFFT.
#Define the Fourier coefficients of {{Math|''a''<sub>''i''</sub>}}, so that they are fixed and depend on a family of SWIFFT.
#Point-wise multiply the Fourier coefficients {{Math|''p''<sub>''i''</sub>}} with the Fourier coefficients of {{Math|''a''<sub>''i''</sub>}} for each {{Mvar|i}}.
#Use inverse FFT to obtain {{Mvar|m}} polynomials {{Math|''f''<sub>''i''</sub>}} of degree {{Math|< 2''n''}}.
#Compute <math>f = \sum_{i=1}^m (f_i)</math> modulo {{Mvar|p}} and {{Math|&alpha;<sup>''n''</sup> + 1}}.
#Convert {{Mvar|f}} to {{Math|''n'' log(''p'')}} bits and '''output''' it.
The [[Fast Fourier transform|FFT]] operation in step 4 is easy to invert, and is performed to achieve [[Confusion and diffusion|diffusion]], that is, to mix the input bits. The [[linear combination]] in step 6 achieves [[Confusion and diffusion|confusion]], since it compresses the input. This is just a high level description of what the algorithm does, some more advanced optimizations are used to finally yield a high performing algorithm.

===Example===
Assuming the parameters {{Math|1=(''n'', ''m'', ''p'') = (64, 16, 257)}}, any fixed compression function in the family takes a binary input of length {{Math|1=''mn'' = 1024}} bits (128 bytes) to an output in the range {{Math|&integers;{{su|b=''p''|p=''n''}}}}, which has size {{Math|1=''p''<sup>''n''</sup> = 257<sup>64</sup>}}. An output in {{Math|&integers;{{su|b=''p''|p=''n''}}}} can easily be represented using 528 bits (66 bytes).

==Algebraic description==
The SWIFFT functions can be described as a simple algebraic expression over some [[polynomial ring]] {{Mvar|R}}. A family of these functions depends on three main parameters: let {{Mvar|n}} be a power of 2, let {{Math|''m'' > 0}} be a small integer, and let {{Math|''p'' > 0}} be a modulus (not necessarily [[Prime number|prime]], but is convenient to choose it prime). Define {{Mvar|R}} to be the ring  {{Math|1=''R'' = &integers;<sub>''p''</sub>[&alpha;]/(&alpha;<sup>''n''</sup> + 1)}}, i.e., the ring of polynomials in {{Mvar|&alpha;}} having integer coefficients, modulo {{Mvar|p}} and {{Math|&alpha;<sup>''n''</sup> + 1}}. An element of {{Mvar|R}} can be written as a polynomial of degree {{Math|< ''n''}} having coefficients in {{Math|&integers;<sub>''p''</sub>}}. A certain function in the SWIFFT family is specified by {{Mvar|m}} fixed elements {{Math|''a''<sub>1</sub>, &mldr;, ''a''<sub>''m''</sub> &in; ''R''}} of the ring <math>R</math>, that are called multipliers. The function corresponds to the following formula over {{Mvar|R}}:

: <math> \sum_{i=1}^m (a_i \cdot x_i) </math>

The {{Math|''x''<sub>1</sub>, &mldr;, ''x''<sub>''m''</sub> &in; ''R''}} are polynomials with binary coefficients, and corresponding to the binary input of length {{Math|''mn''}}.

==Computing the polynomial product==
To compute the above expression, the main problem is to compute the polynomial products {{Math|''a''<sub>''i''</sub> &sdot; ''x''<sub>''i''</sub>}}. A fast way to compute these products is given by the [[convolution theorem]]. This says that under certain restrictions, {{Math|1={{mathcal|F}} {{mset|''f'' &lowast; ''g''}} = {{mathcal|F}} {{mset|''f''}} &sdot; {{mathcal|F}} {{mset|''g''}}}}, where {{Math|{{mathcal|F}}}} denotes the [[Fourier transform]] and {{Math|&sdot;}} denotes the pointwise product. In the general case of the convolution theorem, {{Math|&lowast;}} does not denote multiplication but [[convolution]]. It can, however, be shown that polynomial multiplication is a convolution.

===Fast Fourier transform===
The [[fast Fourier transform]] is used to find the [[Fourier coefficients]] of each polynomial, which are then multiplied point-wise with the respective Fourier coefficients of the other polynomial. The resulting coefficients are then transformed to a polynomial of degree {{Math|< 2''n''}} using an inverse fast Fourier transform.

===Number-theoretic transform===
Instead of the normal Fourier transform, SWIFFT uses the [[number-theoretic transform]]. The number-theoretic transform uses roots of unity in {{Math|&integers;<sub>''p''</sub>}} instead of complex roots of unity. In order for this to work, {{Math|&integers;<sub>''p''</sub>}} needs to be a [[finite field]] and primitive {{Math|2''n''}}<sup>th</sup> roots of unity need to exist in this field. This can be done by taking {{Mvar|p}} prime such that {{Math|2''n''}} divides {{Math|''p'' &minus; 1}}.

==Parameter choice==
The parameters {{Mvar|m}}, {{Mvar|p}}, and {{Mvar|n}} are subject to the following restrictions:
* {{Mvar|n}} must be a power of 2,
* {{Mvar|p}} must be prime,
* {{Math|''p'' &minus; 1}} must be a multiple of {{Math|2''n''}}, and
* {{Math|log(''p'') < ''m''}} (otherwise the output will not be smaller than the input).

A possible choice is {{Math|1=(''n'',''m'',''p'') = (64,16,257)}}, which achieves a throughput of about 40&nbsp;Mbit/s, security of about {{Math|2<sup>106</sup>}} operations for finding collisions, and a digest size of 512 bits.

==Statistical properties==
*  '''Universal hashing.''' The SWIFFT family of functions is [[Universal hashing|universal]]. This means that for any fixed distinct {{Mvar|x}} and {{Mvar|y}}, the probability (over the random choice of {{Mvar|f}} from the family) that {{Math|1=''f''(''x'') = ''f''(''y'')}} is the inverse of the size of the range.
* '''Regularity.''' The SWIFFT family of compression functions is regular. A function {{Mvar|f}} is said to be regular if, for an input {{Mvar|x}} chosen uniformly at random from the domain, the output {{Math|''f''(''x'')}} is distributed uniformly over the range.
* '''Randomness extractor.''' SWIFFT is a [[randomness extractor]]. For hash tables and related applications, it is usually desirable for the outputs of the hash function to be distributed uniformly (or as close to uniformly as possible), even when the inputs are not uniform. Hash functions that give such guarantees are known as [[randomness extractor]]s, because they ''distill'' the non-uniform randomness of the input down to an (almost) uniformly distributed output. Formally, randomness extraction is actually a property of a family of functions, from which one function is chosen at random (and obliviously to the input).

==Cryptographic properties and security==
* SWIFFT is not [[Pseudorandom function family|pseudorandom]], due to linearity. For any function {{Mvar|f}} from our family and any two inputs {{Mvar|x}} and {{Mvar|y}} such that {{Math|''x'' + ''y''}} is also a valid input, we have that {{Math|1=''f''(''x'') + ''f''(''y'') = ''f''(''x'' + ''y'')}}. This relation is very unlikely to hold for a random function, so an adversary can easily distinguish our functions from a random function.
* It is not claimed by the authors that SWIFFT functions behave like a [[random oracle]]. A function is said to behave like a random oracle if it acts like a truly random function. This differs from pseudorandomness in that the function is fixed and public.
* The SWIFFT family is [[Provably secure cryptographic hash function|provably]] collision resistant (in an asymptotic sense), under a relatively mild assumption about the [[Worst-case complexity|worst-case]] difficulty of finding short vectors in cyclic/[[Ideal lattice|ideal lattices]]. This implies that the family is also second preimage resistant.

===Theoretical security===
SWIFFT is an example of a [[provably secure cryptographic hash function]]. As with most security proofs, the security proof of SWIFFT relies on a [[Polynomial-time reduction|reduction]] to a certain difficult-to-solve mathematical problem. Note that this means that the security of SWIFFT relies strongly on the difficulty of this mathematical problem.

The reduction in the case of SWIFFT is to the problem of finding short vectors in cyclic/ideal lattices. It can be proven that the following holds: Suppose we have an algorithm that, for a random version of SWIFFT given by {{Mvar|f}}, can find collisions in {{Mvar|f}} within some feasible time {{Mvar|T}}, and with probability {{Mvar|p}}. It is allowed that the algorithm only works in a small but noticeable fraction of the SWIFFT family. Then we can find also an algorithm {{Math|''f''<sub>2</sub>}} which can ''always'' find a short vector in ''any'' ideal lattice over the ring {{Math|1=&integers;<sub>''p''</sub>[&alpha;]/(&alpha;<sup>''n''</sup> + 1)}} in some feasible time {{Math|''T''<sub>2</sub>}}, depending on {{Mvar|T}} and {{Mvar|p}}. This means that finding collisions in SWIFFT is at least as difficult as the worst-case scenario of finding short vectors in a lattice over {{Math|1=&integers;<sub>''p''</sub>[&alpha;]/(&alpha;<sup>''n''</sup> + 1)}}. At the moment{{When|date=August 2024}}, the fastest algorithms for finding short vectors are all exponential in {{Mvar|n}}. Note that this ensures that there is no significant set of "weak instances" where the security of SWIFFT is weak. This guarantee is not given by most other provably secure hash functions.

===Practical security===
Known working attacks are the [[Birthday_problem#The_generalized_birthday_problem|generalized birthday attack]], which takes 2<sup>106</sup> operations, and ''inversion attacks'' which takes 2<sup>448</sup> operations for a standard parameter choice. This is usually considered to be enough to render an attack by an adversary infeasible.

==References==
<references />

{{Cryptography hash}}

[[Category:Cryptographic hash functions]]