Security of cryptographic hash functions: Difference between revisions

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In [[cryptography]], [[cryptographic hash functions]] can be divided into two main categories. In the first category are those functions whose designs are based on a mathematical problemproblems, and thus theirwhose security thus follows from rigorous mathematical proofs, [[Computational complexity theory|complexity theory]] and [[Reduction (complexity)|formal reduction]]. These functions are called '''Provablyprovably Securesecure Cryptographiccryptographic Hashhash Functionsfunctions'''. However this does not mean that such a function could not be broken. To construct themthese is very difficult, and only a few examples werehave been introduced. TheTheir practical use is limited.

In the second category are functions thatwhich are not based on mathematical problems, but on an ad -hoc basisconstructions, wherein which the bits of the message are mixed to produce the hash. TheyThese are then believed to be hard to break, but no such formal proof is given. Almost all widely-spread hash functions fallin widespread use reside in this category. Some of these functions are already broken, and are no longer in use. ''See'' [[Hash function security summary]].

== Types of Securitysecurity of Hash Functionshash functions==

Generally, the ''basic'' security of [[cryptographic hash functions]] can be seen from three different angles: pre-image resistance, second pre-image resistance and, collision resistance., and pseudo-randomness.

* '''Pre-image resistance''': given a hash {{Mvar|h}}, it should be ''hard'' to find any message {{Mvar|m}} such that {{Math|1=''h'' = hash(''m'')}}. This concept is related to that of onethe [[one-way function]]. Functions that lack this property are vulnerable to [[pre-image attacksattack]]s.

* '''Second pre-image resistance''': given an input m1{{Math|''m''₁}}, it should be ''hard'' to find another input, m2{{Math|''m''₂ (not≠ equal to m1)''m''₁}} such that {{Math|1=hash(m1''m''₁) = hash(m2''m''₂)}}. This property is sometimes referred to as weak collision resistance. Functions that lack this property are vulnerable to [[second pre-image attacksattack]]s.

* '''Collision resistance''': it should be ''hard'' to find two different messages m1{{Math|''m''₁}} and m2{{Math|''m''₂}} such that {{Math|1=hash(m1''m''₁) = hash(m2''m''₂)}}. Such a pair is called a (cryptographic) hash collision. This property is sometimes referred to as strong collision resistance. It requires a hash value at least twice as long as what is required for pre-image resistance,; otherwise, collisions may be found by a [[birthday attack]].

* '''Pseudo-randomness''': it should be ''hard'' to distinguish a pseudo-random number generator based on the hash function from true random number generator; for example, it passes usual [[randomness tests]].

=== The Meaningmeaning of "Hard" ''hard''===

The basic question is the meaning of "'''hard'''". There are two approaches to answer this question. First is the intuitive/practical approach: "''hard'' means that it is almost certainly beyond the reach of any adversary who must be prevented from breaking the system for as long as the security of the system is deemed important." The second approach is theoretical and is based on the [[computational complexity theory]]: if problem ''A'' is hard, then there exists a formal [[Reduction (complexity)|security reduction]] from a problem which is widely considered unsolvable in [[polynomial time]], such as [[integer factorization]] or the [[discrete logarithm]] problem.

However, non-existence of a polynomial time algorithm does not automatically ensure that the system is secure. The difficulty of a problem also depends on its size. For example, [[RSA public key cryptography|RSA public-key cryptography]] (which relies on the difficulty of [[integer factorization]]) is considered secure only with keys that are at least 2048 bits long, whereas keys for the [[ElGamal encryption|ElGamal cryptosystem]] (which relies on the difficulty of the [[discrete logarithm]] problem) are commonly in the range of 256–512 bits.

The basic question is the meaning of "'''hard'''". There are two approaches to answer this question. First is the intuitive/practical approach: "hard means that it is almost certainly beyond the reach of any adversary who must be prevented from breaking the system for as long as the security of the system is deemed important."

===Password case===

The second approach is theoretical and is based on the [[computational complexity theory]]. If problem A is hard, there exists a formal [[Reduction (complexity)|security reduction]] from a problem which is widely supposed to be unsolvable in [[polynomial time]], such as [[integer factorization]] problem or [[discrete logarithm]] problem.

If the set of inputs to the hash is relatively small or is ordered by likelihood in some way, then a brute force search may be practical, regardless of theoretical security. The likelihood of recovering the preimage depends on the input set size and the speed or cost of computing the hash function. A common example is the use of hashes to store [[password]] validation data. Rather than store the plaintext of user passwords, an access control system typically stores a hash of the password. When a person requests access, the password they submit is hashed and compared with the stored value. If the stored validation data is stolen, then the thief will only have the hash values, not the passwords. However, most users choose passwords in predictable ways, and passwords are often short enough so that all possible combinations can be tested if fast hashes are used.<ref>{{cite web

| url=https://arstechnica.com/information-technology/2012/12/25-gpu-cluster-cracks-every-standard-windows-password-in-6-hours/

| title=25-GPU cluster cracks every standard Windows password in <6 hours

| date=2012-12-10

| first=Dan

| last=Goodin

| publisher=[[Ars Technica]]

| access-date=2020-11-23}}</ref> Special hashes called [[key derivation function]]s have been created to slow searches. ''See'' [[Password cracking]].

==Cryptographic hash functions==

However, non-existence of a polynomial time algorithm does not automatically ensure that the system is secure. It is equally important to choose the parameters sensibly (e.g. a length of the numbers that the system works with). For instance, factoring 21 is easy although in general [[integer factorization]] is considered a hard problem.

{{Main article|Cryptographic hash function}}

Most hash functions are built on an ad -hoc basis, where the bits of the message are nicely mixed to produce the hash. Various [[bitwise operation]]s (e.g. rotations), [[Modular arithmetic|modular additions]], and [[One-way compression function|compression functions]] are used in iterative mode to ensure high complexity and pseudo-randomness of the output. In this way, the security is very hard to prove and the proof is usually not done. Only a few years ago{{When|date=August 2024}}, one of the most popular hash functions, [[SHA-1]], was shown to be less secure than its length suggested: collisions could be found in only 2⁵¹<ref>{{Cite web| title=Classification and Generation of Disturbance Vectors for Collision Attacks against SHA-1 | url=http://eprint.iacr.org/2008/469.pdf | archive-url=https://web.archive.org/web/20090115213127/http://eprint.iacr.org/2008/469.pdf | archive-date=2009-01-15}}</ref> tests, rather than the brute-force number of 2⁸⁰. The search for a "good" hash function has thus become a hot topic.

== Classical Hash Functions - Practical Approach to Security ==

Most hash functions are built on an ad hoc basis, where the bits of the message are nicely mixed to produce the hash. Various [[bitwise operation]]s (e.g. rotations), [[Modular arithmetic|modular additions]] and [[One-way compression function|compression functions]] are used in iterative mode to ensure high complexity and pseudo-randomness of the output. In this way, the security is very hard to prove and the proof is usually not done. Only a few years ago, one of the most popular hash functions, [[SHA-1]], was shown to be less secure than its length suggested: collisions could be found in only 2⁵¹<ref>http://eprint.iacr.org/2008/469.pdf</ref> tests, rather than the brute-force number of 2⁸⁰. The search for a "good" hash function has thus become a hot topic.

In other words, most of the hash functions in use nowadays are not provably collision-resistant. These hashes are not based on purely mathematical functions. This approach results generally in more effective hashing functions, but with the risk that a weakness of such a function will be eventually used to find collisions. TheOne famous case is [[MD5]].

=== More practical provablyProvably secure hash functions ===

Meaning of "hard to find collision" in these cases means "almost certainly beyond the reach of any adversary who must be prevented from breaking the system for as long as the security of the system is deemed important." The meaning of the term is therefore somewhat dependent on the application, since the effort that a malicious agent may put into the task is usually proportional to his expected gain. However, since the needed effort usually grows very quickly with the digest length, even a thousandfold advantage in processing power can be neutralized by adding a few dozen bits to the latter.

In this approach, we base the security of a hash function is based on some hard mathematical problem, and weit proveis proved that finding collisions of the hash functionsfunction is as hard as breaking the underlying problem. This gives mucha somewhat stronger notion of security than just relying on complex mixing of bits as in the classical approach.

A cryptographic hash function has ''provable security against collision attacks'' if finding collisions is provably [[Polynomial-time reduction|polynomial-time reducible]] from a problem ''P'' which is supposed to be unsolvable in polynomial time. The function is then called provably secure, or just provable.

== Provably Secure Hash Functions ==

In this approach, we base the security of hash function on some hard mathematical problem and we prove that finding collisions of the hash functions is as hard as breaking the underlying problem. This gives much stronger security than just relying on complex mixing of bits as in the classical approach.

AIt cryptographicmeans hashthat functionif hasfinding collisions would be feasible in polynomial time by algorithm ''A'provable', securitythen againstone collisioncould find and use polynomial time algorithm attacks''R'' if(reduction findingalgorithm) collisionsthat iswould provablyuse [[Polynomial-timealgorithm reduction|polynomial-time''A'' reducible]]to fromsolve problem ''P'', which is widely supposed to be unsolvable in polynomial time. The functionThat is thena calledcontradiction. provablyThis secure,means orthat justfinding provablecollisions cannot be easier than solving ''P''.

However, this only indicates that finding collisions is difficult in ''some'' cases, as not all instances of a computationally hard problem are typically hard. Indeed, very large instances of NP-hard problems are routinely solved, while only the hardest are practically impossible to solve.

It means that if finding collisions would be feasible in polynomial time by algorithm A, we could find and use polynomial time algorithm R (reduction algorithm) that would use algorithm A to solve problem P, which is widely supposed to be unsolvable in polynomial time. That is a contradiction. This means, that finding collisions cannot be easier than solving P.

=== Hard problems ===

Hash functions with the proof of their security are based on mathematical functions.

Examples of problems, that are assumed to be not solvable in polynomial time include

* [[Discrete logarithm|Discrete Logarithm Problem]]

* [[Quadratic residue|Finding Modular Squaresquare Rootsroots]]

* [[Integer factorization|Integer Factorization Problem]]

* [[Subset sum problem|Subset Sum Problemsum]]

=== Downsides of Provablethe Approachprovable approach===

=== Hard problems ===

* Current{{When|date=August 2024}} collision-resistant hash algorithms that have provable security [[Reduction (complexity)|reductions]] are too inefficient to be used in practice. In comparison to classical hash functions, they tend to be relatively slow and do not always meet all of criteria traditionally expected of cryptographic hashes. [[Very smooth hash]] is an example.

Examples of problems, that are assumed to be not solvable in polynomial time

* Constructing a hash function with provable security is much more difficult than using a classical approach where weone just hopehopes that the complex mixing of bits in the hashing algorithm is strong enough to prevent adversary from finding collisions.

* [[Discrete logarithm|Discrete Logarithm Problem]]

* The proof is often a reduction to a problem with asymptotically hard [[Worst-case complexity|worst-case]] or [[average-case complexity]]. Worst-case complexity measures the difficulty of solving pathological cases rather than typical cases of the underlying problem. Even a reduction to a problem with hard average-case complexity offers only limited security, as there still can be an algorithm that easily solves the problem for a subset of the problem space. For example, early versions of [[Fast Syndrome Based Hash]] turned out to be insecure. This problem was solved in the latest version.

* [[Quadratic residue|Finding Modular Square Roots]]

* [[Integer factorization|Integer Factorization Problem]]

* [[Subset sum problem|Subset Sum Problem]]

[[SWIFFT]] is an example of a hash function that circumvents these security problems. It can be shown that, for any algorithm that can break SWIFFT with probability P{{Mvar|p}} within an estimated time T{{Mvar|t}}, weone can find an algorithm that solves the ''worst -case'' scenario of a certain difficult mathematical problem within time T{{Math|''t''&prime;}} depending on T{{Mvar|t}} and P{{Mvar|p}}.{{citation needed|date=June 2014}}

=== Downsides of Provable Approach ===

* Current collision-resistant hash algorithms that have provable security [[Reduction (complexity)|reductions]] are too inefficient to be used in practice. In comparison to classical hash functions, they tend to be relatively slow and do not always meet all of criteria traditionally expected of cryptographic hashes. [[Very smooth hash]] is an example.

* Constructing a hash function with provable security is much more difficult than using a classical approach where we just hope that the complex mixing of bits in the hashing algorithm is strong enough to prevent adversary from finding collisions.

* The proof is often a reduction to a problem with asymptotically hard [[Worst-case complexity|worst-case]] or [[average-case complexity]]. Worst-case measures the difficulty of solving pathological cases rather than typical cases of the underlying problem. Even a reduction to a problem with hard average complexity offers only limited security as there still can be an algorithm that easily solves the problem for a subset of the problem space. For example, early versions of [[Fast Syndrome Based Hash]] turned out to be insecure. This problem was solved in the latest version.

===Example of (impractical) provably secure hash function===

[[SWIFFT]] is an example of a hash function that circumvents these security problems. It can be shown that for any algorithm that can break SWIFFT with probability P within an estimated time T, we can find an algorithm that solves the ''worst case'' scenario of a certain difficult mathematical problem within time T' depending on T and P.

HashLet {{Math|1=hash(''m'') = ''x''^''m'' mod ''n''}}, where ''{{Mvar|n''}} is a hard -to -factor composite number, and ''{{Mvar|x''}} is some prespecified base value. A collision {{Math|''x''^''m1m''₁ congruent to&equiv; ''x''^''m2m''₂ (mod ''n'')}} reveals a multiple {{Math|''m1m''₁ -&minus; m2''m''₂}} of the [[multiplicative order]] of ''{{Mvar|x''}} modulo {{Mvar|n}}. SuchThis information can be used to factor ''{{Mvar|n''}} in polynomial time, assuming certain properties of ''{{Mvar|x''}}.

But the algorithm is quite inefficient because it requires on average 1.5 multiplications modulo ''{{Mvar|n''}} per message-bit.

=== Example of (unpractical) Provably Secure Hash Function ===

Hash(''m'') = ''x''^''m'' mod ''n'' where ''n'' is hard to factor composite number, and ''x'' is some prespecified base value. A collision ''x''^''m1'' congruent to ''x''^''m2'' reveals a multiple ''m1 - m2'' of the order of ''x''. Such information can be used to factor ''n'' in polynomial time assuming certain properties of ''x''.

===More practical provably secure hash functions===

But the algorithm is quite inefficient because it requires on average 1.5 multiplications modulo ''n'' per message-bit.

* [[Very smooth hash|VSH - Very Smooth Hash function]]—Very -Smooth aHash—a provably secure collision-resistant hash function assuming the hardness of finding nontrivial modular square roots modulo composite number <math>{{Mvar|n</math>}} (this is proven to be as hard as [[Integer factorization|factoring]] <math>{{Mvar|n</math>}}).

=== More practical provably secure hash functions ===

* [[Very smooth hash|VSH - Very Smooth Hash function]] - a provably secure collision-resistant hash function assuming the hardness of finding nontrivial modular square roots modulo composite number <math>n</math> (this is proven to be as hard as [[Integer factorization|factoring]] <math>n</math>).

* [[MuHASH]]

* [[Elliptic curve only hash|ECOH]]—[[Elliptic -curve only hash|Elliptic Curve Only hash function]] - based—based on the concept of [[Elliptic curve|elliptic curves]], Subsetthe Sum[[subset Problemsum problem]], and summation of polynomials. The security proof of the collision resistance was based on weakened assumptionsassumptionsm, and eventually a second pre-image attack was found.

* [[Fast Syndrome Based Hash|FSB]]—[[Fast -Syndrome Based Hash|Fast Syndrome-Based hash function]] - it—it can be proven that breaking FSB is at least as difficult as solving a''regular certainsyndrome NP-completedecoding'', problemwhich is known asto Regularbe Syndrome DecodingNP-complete.

* [[SWIFFT]] - SWIFFT—SWIFFT is based on the [[Fastfast Fourier transform]] and is provably collision resistant, under a relatively mild assumption about the worst-case difficulty of finding short vectors in cyclic/[[Ideal lattice cryptography|ideal lattices]].

* [[Chaum, van Heijst, Pfitzmann hash function]] - A—a compression function where finding collisions is as hard as solving the [[discrete logarithm|discrete logarithm problem]] in a finite group {{Math|''F''<mathsub>F_{2p2''p''+1}</mathsub>}}.

* [[Knapsack-based hash functions]] - A—a family of hash functions based on the [[Knapsackknapsack problem]].

* [[The Zémor-Tillich hash function]] - A—a family of hash functions that relyrelies on the arithmetic of the group of matrices [[Special linear group|SL2SL₂]]. Finding collisions is at least as difficult as finding factorization of certain elements in this group. This is supposed to be hard, at least [[PSPACE-complete]].{{Dubious|PSPACE-complete?!|date=January 2019}} For this hash, an attack was eventually discovered with a time complexity close to {{Math|2<mathsup>2^{''n''/2}</mathsup>}}. This beat by far the birthday bound and ideal pre-image complexities, which are {{Math|2<mathsup>2^{3n3''n''/2}</mathsup>}} and {{Math|2<mathsup>2^{3n}3''n''</mathsup>}} for the Zémor-Tillich hash function. As the attacks include a birthday search in a reduced set of size <math>2n</math>{{Math|2''n''}}, they indeed do not destroy the idea of provable security ofor invalidate the scheme but rather suggest that the initial parameters were too small.<ref>{{Citation

| last1 = Petit | first1 = C.

| last2 = Quisquater | first2 = J.-J.

Line 67 ⟶ 76:

| contribution = Hard and easy Components of Collision Search in the Zémor-Tillich hash function:new Attacks and Reduced Variants with Equivalent Security

| contribution-url = http://www-rocq.inria.fr/secret/Jean-Pierre.Tillich/publications/CTRSA.pdf }}</ref>

* [[Hash functions from Sigma Protocols]] - there—there exists a general way of constructing a provably secure hash, specifically from a any (suitable) [[Proof_of_knowledgeProof of knowledge#Sigma_protocolsSigma protocols|sigma protocol]]. A faster version of [[Very smooth hash|VSH]] (called VSH*) could be obtained in this way.

==References==