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{{Short description|Cryptographic hash function}}
{{Infobox cryptographic hash function
| name           = Very Smooth Hash (VSH)
| image          = 
| caption        = 
<!-- General -->
| designers      = [[Scott Contini]], [[Arjen Lenstra|Arjen K. Lenstra]], [[Ron Steinfeld]]
| publish date   = 2005
| series         = 
| derived from   = 
| derived to     = VSH*
| related to     = 
| certification  = 
<!-- Detail -->
| digest size    = 1024 bits and up
| structure      = 
| rounds         = 
| cryptanalysis  = 
}}
In [[cryptography]], '''Very Smooth Hash (VSH)''' is a {{not a typo|provably}} secure [[cryptographic hash function]] invented in 2005 by Scott Contini, Arjen Lenstra, and Ron Steinfeld.<ref name="main">{{Citation
  | last1 = Contini
  | first1 = S.
  | last2 = Lenstra
  | first2 = A.
  | last3 = Steinfeld
  | first3 = R.
  | title = VSH, an Efficient and Provable Collision-Resistant Hash Function.
  | date = 2005-06-23
  | url = https://eprint.iacr.org/2005/193}}
</ref> [[Provably secure cryptographic hash function|Provably secure]] means that finding collisions is as difficult as some known hard mathematical problem. Unlike other {{not a typo|provably}} secure [[Collision resistance|collision-resistant]] hashes, VSH is efficient and usable in practice. [[Big O notation|Asymptotically]], it only requires a single multiplication per {{Math|log(''n'')}} message-bits and uses RSA-type arithmetic. Therefore, VSH can be useful in embedded environments where code space is limited.

Two major variants of VSH were proposed. For one, finding a [[Collision resistance|collision]] is {{not a typo|provably}} as difficult as finding a nontrivial [[modular square root]] of a very [[smooth number]] modulo&nbsp;{{Mvar|n}}. The other one uses a prime modulus {{Mvar|p}} (with no [[Trapdoor function|trapdoor]]), and its security proof relies on the hardness of finding [[Discrete logarithm|discrete logarithms]] of very smooth numbers modulo&nbsp;{{Mvar|p}}. Both versions have similar efficiency.

VSH is not suitable as a substitute for a [[random oracle]], but can be used to build a {{not a typo|provably}} secure randomized trapdoor hash function. This function can replace the [[trapdoor function]] used in the [[Cramer–Shoup cryptosystem|Cramer–Shoup signature scheme]], maintaining its provable security while speeding up verification time by about 50%.

== VSN and VSSR ==
All cryptographic hash functions that are now{{When|date=August 2024}} widely used are not based on hard mathematical problems. Those few functions that are constructed on hard mathematical problems are called [[Provably secure cryptographic hash function|provably secure]]. [[Collision resistance|Finding collisions]] is then known to be as hard as solving the hard mathematical problem. For the basic version of Very Smooth Hash, this hard problem is to find [[Modular square root|modular square roots]] (VSSR) of certain special numbers (VSN).<ref name="main" /> This is assumed to be as hard as [[Integer factorization|factoring integers]].

For fixed constants {{Mvar|c}} and {{Mvar|n}}, an integer {{Mvar|m}} is a ''Very Smooth Number'' (VSN) if the largest prime factor of {{Mvar|m}} is at most&nbsp;{{Math|log(''n'')<sup>''c''</sup>}}.

An integer {{Mvar|b}} is a ''Very Smooth Quadratic Residue modulo'' {{Mvar|n}} if the largest prime in {{Mvar|b}}'s factorization is at most {{Math|log(''n'')<sup>''c''</sup>}} and there exists an integer {{Mvar|x}} such that {{Math|''b'' &equiv; ''x''<sup>2</sup> (mod ''n'')}}. The integer {{Mvar|x}} is then said to be a [[Quadratic residue|modular square root]] of&nbsp;{{Mvar|b}}.

We are interested only in non-trivial square roots, those where {{Math|''x''<sup>2</sup> &ge; ''n''}}. If {{Math|''x''<sup>2</sup> < ''n''}}, then the root can be easily computed using algorithms from fields of [[Characteristic (algebra)|characteristic]]&nbsp;0, such as [[Real number|the real field]]. Therefore, they are not suitable in cryptographic primitives.

''Very Smooth Number Nontrivial Modular Square Root'' (VSSR) is the following problem: Let {{Mvar|n}} be the product of two unknown primes of approximately the same size, let {{Math|''k'' &le; (log(''n''))<sup>''c''</sup>}}, and let {{Math|1=(''p''<sub>1</sub>,''p''<sub>2</sub>,''p''<sub>3</sub>,&mldr;) = (2,3,5,&mldr;)}} be the sequence of primes. Given {{Mvar|n}}, find an integer {{Mvar|x}} [[Coprime integers|coprime]] to {{Mvar|n}} such that <math>\textstyle x^2 \equiv \prod_{i=0}^k p_i^{e_i} \pmod{n}</math> and at least one of {{Math|''e''<sub>0</sub>,&mldr;,''e''<sub>''k''</sub>}} is odd.

The ''VSSR assumption'' is that there is no [[PP (complexity)|probabilistic polynomial]] (in {{Math|log(''n'')}}) time algorithm which solves VSSR with [[Negligible function|non-negligible]] probability. This is considered a useless assumption in practice because it does not tell for what size of moduli VSSR is computationally hard. Instead the ''computational VSSR assumption'' is used. It says that solving VSSR is assumed to be as hard as [[Integer factorization|factoring]] a hard-to-factor {{Mvar|s}}-bit modulus, where {{Mvar|s}} is somewhat smaller than the size of {{Mvar|n}}.

===Examples of VSN and VSSR===
Let the parameters be fixed as {{Math|1=''c'' = 5}} and {{Math|1=''n'' = 31}}.

Then {{Math|1=''m''<sub>1</sub> = 35 = 5 &middot; 7}} is a Very Smooth Number with respect to these parameters because {{Math|log(31)<sup>5</sup> &approx; 7.37}} is greater than all of {{Math|''m''<sub>1</sub>}}'s prime factors. On the other hand, {{Math|1=''m''<sub>2</sub> = 55 = 5 &middot; 11}} is not a VSN under these parameters.

The integer {{Math|9}} is a Very Smooth Quadratic Residue modulo {{Mvar|n}} because it is a Very Smooth Number (under {{Math|''c'',''n''}}), and {{Math|3<sup>2</sup> &equiv; 9 (mod ''n'')}}. This is a trivial modular square root, because {{Math|9 < ''n''}} and so the modulus is not involved when squaring.

The integer <math>b_2 = 15</math> is also Very Smooth Quadratic Residue modulo <math>n</math>. All prime factors are smaller than 7.37 and the Modular Square Root is <math>x_2 =  20</math> since <math>20^2 = 400 \equiv 15</math> (mod <math>n</math>). This is thus a non-trivial root. The VSSR problem is to find <math>x_2</math> given <math>b_2</math> and <math>n</math>. And we suppose that this is computationally as hard as factoring <math>n</math>.

The integer {{Math|1=b = 15}} is also a Very Smooth Quadratic Residue modulo {{Mvar|n}}. All of its prime factors are smaller than 7.37, and the modular square root is {{Math|1=''x'' = 20}}, since {{Math|1=20<sup>2</sup> = 400 &equiv; 15 (mod ''n'')}}. This is thus a non-trivial square root. The VSSR problem is to find {{Mvar|x}} given {{Mvar|b}} and {{Mvar|n}}. This is believed to be computationally as hard as factoring {{Mvar|n}}.

== VSH algorithm, basic versions ==
Let {{Mvar|n}} be a large RSA composite and let {{Math|1=(''p''<sub>1</sub>,''p''<sub>2</sub>,''p''<sub>3</sub>,&mldr;) = (2,3,5,&mldr;)}} be the sequence of primes. Let {{Mvar|k}}, the block length, be the largest integer such that <math>\textstyle \prod_{i = 1}^k p_i < n</math>. Let {{Mvar|m}} be an {{Mvar|&ell;}}-bit message to be hashed consisting of bits {{Math|(''m''<sub>1</sub>,&mldr;,''m''<sub>&ell;</sub>)}} and assume that {{Math|&ell; < 2<sup>''k''</sup>}}. To compute the hash of {{Mvar|m}}:
# Set {{Math|1=''x''<sub>0</sub> = 1}}.
# Let {{Mvar|L}}, the smallest integer greater than or equal to {{Math|&ell;/''k''}}, be the number of blocks.
# Let {{Math|1=''m''<sub>''i''</sub> = 0}} for {{Math|&ell; < ''i'' < ''Lk''}} (padding).
# Let <math>\textstyle \ell = \sum_{i=1}^k l_i 2^{i-1}</math> with {{Math|&ell;<sub>''i''</sub> &in; {{mset|0,1}}}} be the binary representation of the message length {{Mvar|&ell;}} and define {{Math|1=''m''<sub>''Lk''+''i''</sub> = &ell;<sub>''i''</sub>}} for {{Math|1 &le; ''i'' &le; ''k''}}.
# For {{Math|1=''j'' = 0, 1, &mldr;, ''L''}} in succession, compute <math>x_{j+1} = x_j^2 \prod_{i=1}^k p_i^{m_{jk+i}}\mod n</math>
# Return {{Math|''x''<sub>''L''+1</sub>}}.
The function in step 5 is called the compression function.

== Properties of VSH ==
* The message length does not need to be known in advance.
* Finding a collision in VSH is as hard as solving VSSR. Thus VSH is (strongly) [[collision-resistant]], which also implies second preimage resistance.  VSH has not been proven to be preimage-resistant.
* The compression function is not collision-resistant.  Nonetheless, the hash function VSH is collision-resistant based on the VSSR assumption. An altered version of VSH, called VSH*, uses a collision-resistant compression function and is about 5 times quicker when hashing short messages.
* Since the output length of VSH is the length of a secure RSA modulus, VSH seems quite suitable in practice for constructing "hash-then-sign" RSA signatures for arbitrarily long messages. However, such a signature must be designed carefully to ensure its security. The naïve approach could be easily broken under a [[Ciphertext indistinguishability|chosen-plaintext attack]].
* The cost of each iteration is less than the cost of 3 modular multiplications. The basic version of VSH altogether requires one multiplication per {{Math|&Omega;(log(''n'') / log(log(''n'')))}} message bits.

== Variants of VSH ==
Several improvements, speedups, and more efficient variants of VSH have been proposed.<ref name="main" /> None of them changes the underlying concept of the function. These improvements are called:

* Cubing VSH (instead of squaring)
* VSH with increased number of small primes
* VSH with precomputed products of primes
* Fast VSH
* Fast VSH with increased block length

== VSDL and VSH-DL variant ==
The VSH-DL is a discrete logarithm variant of VSH that has no [[Trapdoor function|trapdoor]]; its security depends on the difficulty of finding discrete logarithms modulo a prime {{Mvar|p}}.<ref name="main" />

''Very Smooth Number Discrete Logarithm'' (VSDL) is a problem where, given a very smooth number, the task is to find its [[discrete logarithm]] modulo some number {{Mvar|n}}.

As in previous section, {{Math|''p''<sub>''i''</sub>}} denotes the {{Mvar|i}}th prime. Let furthermore {{Mvar|c}} be a fixed constant and {{Math|''p'',''q''}} be primes with {{Math|1=''p'' = 2''q'' + 1}} and let {{Math|''k'' &le; (log ''p'')<sup>''c''</sup>}}. VSDL is the following problem: given {{Mvar|p}}, find integers {{Math|''e''<sub>1</sub>,&mldr;,''e''<sub>''k''</sub>}} such that <math>2^{e_1} \equiv \prod_{i=2}^k p_i^{e_i} \pmod p</math> with {{Math|{{abs|''e''<sub>''i''</sub>}} < ''q''}} for {{Math|1=''i'' = 1, &mldr;, ''k''}}  and at least one of {{Math|''e''<sub>1</sub>,&mldr;,''e''<sub>''k''</sub>}} is non-zero.

The ''VSDL assumption'' is that there is no [[PP (complexity)|probabilistic polynomial]] (in {{Math|log(''p'')}}) time algorithm which solves VSDL with [[Negligible function|non-negligible]] probability. There is a strong connection between the hardness of VSDL and the hardness of computing discrete logarithms modulo {{Mvar|p}}, which is reminiscent of, but somewhat weaker than, the connection between VSSR and integer factorization.

==Security of VSH==
Strong [[collision resistance]] is the only property proven for VSH. This does not imply preimage-resistance or other important hash function properties, and the authors state that "VSH should not be used to model [[random oracle]]s," and cannot be substituted into constructions that depend upon them ([[RSA (algorithm)|RSA signatures]], some [[Message authentication code|MACs]]).<ref name="main" /> VSH should not be considered a general-purpose hash function as usually understood in [[security engineering]].

===Multiplicative property===
VSH is multiplicative: Let {{Mvar|x}}, {{Mvar|y}}, and {{Mvar|z}} be three bitstrings of equal length, where {{Mvar|z}} consists only of zero bits and the strings satisfy {{Math|1=''x'' AND ''y'' = ''z''}}. Then {{Math|''H''(''z'')''H''(''x'' OR ''y'') &equiv; ''H''(''x'')''H''(''y'') (mod ''n'')}}. As a result, VSH succumbs to a classical time-memory trade-off attack that applies to multiplicative and additive hashes.

This fact can be used to construct a preimage attack against VSH of {{Mvar|&ell;}} bits which has {{Math|2<sup>&ell;/2</sup>}} complexity rather than {{Math|2<sup>&ell;</sup>}} as expected.

===Attack against truncated version===
VSH produces a very long hash (typically 1024 bits). There are no indications that a truncated VSH hash offers security that is commensurate with the hash length.

There exists a partial collision attack on VSH truncated to {{Mvar|&ell;}} least significant bits.<ref name="attack">{{Citation
  | last1 = Saarinen
  | first1 = M.-J. O.
  | title = Progress in Cryptology - INDOCRYPT 2006
 | chapter = Security of VSH in the real world
  | series = Lecture Notes in Computer Science
 | year = 2006
  | volume = 4329
 | pages = 95–103
 | doi = 10.1007/11941378_8
  | isbn = 978-3-540-49767-7
 | chapter-url = http://www.tcs.hut.fi/~mjos/doc/saarinen_vsh.pdf
  | url-status = }}{{deadlink|date=September 2023}}
</ref>

The complexity of this attack against VSH is:
* Pre-computing the table offline: {{Math|2<sup>&ell;/3</sup>}} time and space.
* Finding collisions: {{Math|2<sup>&ell;/3</sup>}} iterations.
* Total cost: roughly {{Math|2<sup>&ell;/3</sup>}}, rather than {{Math|2<sup>&ell;/2</sup>}} as expected from a hash function with good [[pseudorandomness]] properties.

This probably{{According to whom|date=August 2024}} rules out the applicability of VSH in digital signature schemes which produce signatures shorter than the VSH hash result, such as elliptic-curve signature schemes.

==See also==
* [[Cryptographic hash functions]]
* [[Provably secure cryptographic hash function]]

==References==
{{reflist}}

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[[Category:Cryptographic hash functions]]