 Oneway function

Unsolved problems in computer science Do oneway functions exist? In computer science, a oneway function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being onetoone is not considered sufficient of a function for it to be called oneway (see Theoretical Definition, below).
The existence of such oneway functions is still an open conjecture. In fact, their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science.^{[1]}^{:ex. 2.2, page 70} Existence of a proof that P and NP are not equal would not directly imply the existence of oneway functions.^{[2]}
In applied contexts, the terms "easy" and "hard" are usually interpreted relative to some specific computing entity; typically "cheap enough for the legitimate users" and "prohibitively expensive for any malicious agents". Oneway functions, in this sense, are fundamental tools for cryptography, personal identification, authentication, and other data security applications. While the existence of such functions too is an open conjecture, there are several candidates that have withstood decades of intense scrutiny. Some of them are essential ingredients of most telecommunications, ecommerce, and ebanking systems around the world.
Contents
Theoretical definition
A function f: {0, 1}^{*} → {0, 1}^{*} is oneway if f can be computed by a polynomial time algorithm, but for every randomized polynomial time algorithm A,
for every positive polynomial p(n) and sufficiently large n, assuming that x is chosen from the uniform distribution on {0, 1}^{n} and the randomness of A. In this definition, it is crucial that A may run in time polynomial in x.
Note that, by this definition, the function must be "hard to invert" in the averagecase, rather than worstcase sense; while in most of complexity theory (e.g., NPhardness) the term "hard" is meant in the worstcase.^{[dubious – discuss]}
Note also that just making a function "lossy" (not onetoone) does not make it a oneway function. In this context, inverting a function means identifying some preimage element of a given value, which does not require the existence of an inverse function. For example, f(x) = x^{2} is not invertible (for example f(2) = f(2) = 4) but is also not oneway, since given any value, you can compute one of its preimage elements in polynomial time by taking its square root.
Related concepts
A trapdoor oneway function or trapdoor permutation is a special kind of oneway function. Such a function is hard to invert unless some secret information, called the trapdoor, is known.
A oneway permutation is a oneway function that is also a permutation—that is, a oneway function that is both injective and surjective. Oneway permutations are an important cryptographic primitive, and it is not known that their existence is implied by the existence of oneway functions.
A collisionfree hash function f is a oneway function that is also collisionresistant; that is, no randomized polynomial time algorithm can find a collision—distinct values x, y such that f(x) = f(y)—with nonnegligible probability.^{[3]}
Theoretical implications of oneway functions
If f is a oneway function, then the inversion of f would be a problem whose output is hard to compute (by definition) but easy to check (just by computing f on it). Thus, the existence of a oneway function implies that P≠NP. However, it is not known whether P≠NP implies the existence of oneway functions.
The existence of a oneway function implies the existence of many other useful concepts, including:
 Pseudorandom generators
 Pseudorandom function families
 Bit commitment schemes
 Privatekey encryption schemes secure against adaptive chosenciphertext attack
 Message authentication codes
 Digital signature schemes (secure against adaptive chosenmessage attack)
Candidates for oneway functions
Following are several candidates for oneway functions (as of April 2009). Clearly, it is not known whether these functions are indeed oneway; but extensive research has so far failed to produce an efficient inverting algorithm for any of them.
Multiplication and factoring
The function f takes as inputs two prime numbers p and q in binary notation and returns their product. This function can be computed in O(n^{2}) time where n is the total length (number of digits) of the inputs. Inverting this function requires finding the factors of a given integer N. The best factoring algorithms known for this problem run in time , which is only pseudopolynomial in log N, 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 oneway for arbitrary p,q>1, since the product will have 2 as a factor with probability 3/4.
Modular squaring and square roots
The function f takes two positive integers x and N, where N is the product of two primes p and q, and outputs the remainder of x^{2} divided by N. Inverting this function requires computing square roots modulo N; that is, given y and N, find some x such that x^{2} mod N = y. It can be shown that the latter problem is computationally equivalent to factoring N (in the sense of polynomialtime reduction) The Rabin cryptosystem is based on the assumption that this Rabin function is oneway.
Discrete exponential and logarithm
The function f takes a prime number p and an integer x between 0 and p−1; and returns the remainder of 2^{x} divided by p. This discrete exponential function can be easily computed in time O(n^{3}) where n is the number of bits in p. Inverting this function requires computing the discrete logarithm modulo p; namely, given a prime p and an integer y between 0 and p−1, find x such that 2^{x} = y. As of 2009, there is no published algorithm for this problem that runs in polynomial time. The ElGamal encryption scheme is based on this function.
Cryptographically secure hash functions
There are a number of cryptographic hash functions that are fast to compute like SHA 256. Some of the simpler versions have fallen to sophisticated analysis, but the strongest versions continue to offer fast, practical solutions for oneway computation. Most of the theoretical support for the functions are more techniques for thwarting some of the previously successful attacks.
Other candidates
Other candidates for oneway functions have been based on the hardness of the decoding of random linear codes, the subset sum problem (NaccacheStern knapsack cryptosystem), or other problems.
Universal oneway function
There is an explicit function which has been demonstrated to be oneway if and only if oneway functions exist.^{[4]} Since this function was the first combinatorial complete oneway function to be demonstrated, it is known as the "universal oneway function". The problem of determining the existence of oneway functions is thus reduced to the problem of proving that this specific function is oneway.
See also
References
 ^ Oded Goldreich (2001). Foundations of Cryptography: Volume 1, Basic Tools, (draft available from author's site). Cambridge University Press. ISBN 0521791723.
 ^ Goldwasser, S. and Bellare, M. "Lecture Notes on Cryptography". Summer course on cryptography, MIT, 1996–2001
 ^ Russell, A. Necessary and Sufficient Conditions for CollisionFree Hashing. Journal of Cryptology vol. 8, no. 2., pp. 87–99. 1992.
 ^ Leonid Levin (2003). The Tale of OneWay Functions. ACM. arXiv:cs.CR/0012023.
Further reading
 Jonathan Katz and Yehuda Lindell (2007). Introduction to Modern Cryptography. CRC Press. ISBN 1584885513.
 Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 053494728X. Section 10.6.3: Oneway functions, pp. 374–376.
 Christos Papadimitriou (1993). Computational Complexity (1st ed.). Addison Wesley. ISBN 0201530821. Section 12.1: Oneway functions, pp. 279–298.
Categories: Cryptography
 Cryptographic primitives
 Unsolved problems in computer science
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