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Concept# Collision resistance

Summary

In cryptography, collision resistance is a property of cryptographic hash functions: a hash function H is collision-resistant if it is hard to find two inputs that hash to the same output; that is, two inputs a and b where a ≠ b but H(a) = H(b). The pigeonhole principle means that any hash function with more inputs than outputs will necessarily have such collisions; the harder they are to find, the more cryptographically secure the hash function is.
The "birthday paradox" places an upper bound on collision resistance: if a hash function produces N bits of output, an attacker who computes only 2N/2 (or ) hash operations on random input is likely to find two matching outputs. If there is an easier method to do this than brute-force attack, it is typically considered a flaw in the hash function.
Cryptographic hash functions are usually designed to be collision resistant. However, many hash functions that were once thought to be collision resistant were later broken. MD5 and SHA-1 in particular both have published techniques more efficient than brute force for finding collisions. However, some hash functions have a proof that finding collisions is at least as difficult as some hard mathematical problem (such as integer factorization or discrete logarithm). Those functions are called provably secure.
A family of functions {hk : {0, 1}m(k) → {0, 1}l(k)} generated by some algorithm G is a family of collision-resistant hash functions, if |m(k)| > |l(k)| for any k, i.e., hk compresses the input string, and every hk can be computed within polynomial time given k, but for any probabilistic polynomial algorithm A, we have
Pr [k ← G(1n), (x1, x2) ← A(k, 1n) s.t. x1 ≠ x2 but hk(x1) = hk(x2)] < negl(n),
where negl(·) denotes some negligible function, and n is the security parameter.
Collision resistance is desirable for several reasons.
In some digital signature systems, a party attests to a document by publishing a public key signature on a hash of the document.

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Birthday problem

In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share a birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%. The birthday paradox is a veridical paradox: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the birthday comparisons will be made between every possible pair of individuals.

Collision attack

In cryptography, a collision attack on a cryptographic hash tries to find two inputs producing the same hash value, i.e. a hash collision. This is in contrast to a where a specific target hash value is specified. There are roughly two types of collision attacks: Classical collision attack Find two different messages m1 and m2 such that hash(m1) = hash(m2). More generally: Chosen-prefix collision attack Given two different prefixes p1 and p2, find two appendages m1 and m2 such that hash(p1 ∥ m1) = hash(p2 ∥ m2), where ∥ denotes the concatenation operation.

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