One-way function

The function f takes as inputs two prime numbers p and q in binary notation and returns their product. This function can be "easily" computed in O(b²) time, where b is the total number of bits of the inputs. Inverting this function requires finding the factors of a given integer N. The best factoring algorithms known run in ${\displaystyle O\left(\exp {\sqrt[{3}]{{\frac {64}{9}}b(\log b)^{2}}}\right)}$time, where b is the number of bits needed to represent N.

This function can be generalized by allowing p and q to range over a suitable set of semiprimes. Note that f is not one-way for randomly selected integers p, q > 1, since the product will have 2 as a factor with probability 3/4 (because the probability that an arbitrary p is odd is 1/2, and likewise for q, so if they're chosen independently, the probability that both are odd is therefore 1/4; hence the probability that p or q is even, is 1 − 1/4 = 3/4).

The Rabin function,[1]^: 57 or squaring modulo ${\displaystyle N=pq}$, where p and q are primes is believed to be a collection of one-way functions. We write

${\displaystyle \operatorname {Rabin} _{N}(x)\triangleq x^{2}{\bmod {N}}}$

to denote squaring modulo N: a specific member of the Rabin collection. It can be shown that extracting square roots, i.e. inverting the Rabin function, is computationally equivalent to factoring N (in the sense of polynomial-time reduction). Hence it can be proven that the Rabin collection is one-way if and only if factoring is hard. This also holds for the special case in which p and q are of the same bit length. The Rabin cryptosystem is based on the assumption that this Rabin function is one-way.

Modular exponentiation can be done in polynomial time. Inverting this function requires computing the discrete logarithm. Currently there are several popular groups for which no algorithm to calculate the underlying discrete logarithm in polynomial time is known. These groups are all finite abelian groups and the general discrete logarithm problem can be described as thus.

Let G be a finite abelian group of cardinality n. Denote its group operation by multiplication. Consider a primitive element α ∈ G and another element β ∈ G. The discrete logarithm problem is to find the positive integer k, where 1 ≤ k ≤ n, such that:

${\displaystyle \alpha ^{k}=\underbrace {\alpha \cdot \alpha \cdot \ldots \cdot \alpha } _{k\;\mathrm {times} }=\beta }$

The integer k that solves the equation α^k = β is termed the discrete logarithm of β to the base α. One writes k = log_α β.

Popular choices for the group G in discrete logarithm cryptography are the cyclic groups (Z_p)^× (e.g. ElGamal encryption, Diffie–Hellman key exchange, and the Digital Signature Algorithm) and cyclic subgroups of elliptic curves over finite fields (see elliptic curve cryptography).

An elliptic curve is a set of pairs of elements of a field satisfying y² = x³ + ax + b. The elements of the curve form a group under an operation called "point addition" (which is not the same as the addition operation of the field). Multiplication kP of a point P by an integer k (i.e., a group action of the additive group of the integers) is defined as repeated addition of the point to itself. If k and P are known, it is easy to compute R = kP, but if only R and P are known, it is assumed to be hard to compute k.

There are a number of cryptographic hash functions that are fast to compute, such as SHA 256. Some of the simpler versions have fallen to sophisticated analysis, but the strongest versions continue to offer fast, practical solutions for one-way computation. Most of the theoretical support for the functions are more techniques for thwarting some of the previously successful attacks.

Other candidates for one-way functions include the hardness of the decoding of random linear codes, the hardness of certain lattice problems, and the subset sum problem (Naccache–Stern knapsack cryptosystem).