In
computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, a one-way function is a
function that is easy to compute on every input, but hard to
invert given the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of a random input. Here, "easy" and "hard" are to be understood in the sense of
computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, specifically the theory of
polynomial time problems. This has nothing to do with whether the function is
one-to-one; finding any one input with the desired image is considered a successful inversion. (See , below.)
The existence of such one-way functions is still an open
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
. Their existence would prove that the
complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science.
Oded Goldreich
Oded Goldreich (; born 1957) is a professor of computer science at the faculty of mathematics and computer science of the Weizmann Institute of Science, Israel. His research interests lie within the theory of computation and are, specifically, ...
(2001). Foundations of Cryptography: Volume 1, Basic Tools
draft available
from author's site). Cambridge University Press. . See als
The converse is not known to be true, i.e. the existence of a proof that P ≠ NP would not directly imply the existence of one-way functions.
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". One-way functions, in this sense, are fundamental tools for
cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
,
personal identification,
authentication
Authentication (from ''authentikos'', "real, genuine", from αὐθέντης ''authentes'', "author") is the act of proving an Logical assertion, assertion, such as the Digital identity, identity of a computer system user. In contrast with iden ...
, and other
data security
Data security or data protection means protecting digital data, such as those in a database, from destructive forces and from the unwanted actions of unauthorized users, such as a cyberattack or a data breach.
Technologies
Disk encryption
...
applications. While the existence of one-way functions in this sense is also an open conjecture, there are several candidates that have withstood decades of intense scrutiny. Some of them are essential ingredients of most
telecommunications
Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
,
e-commerce
E-commerce (electronic commerce) refers to commercial activities including the electronic buying or selling products and services which are conducted on online platforms or over the Internet. E-commerce draws on technologies such as mobile co ...
, and
e-banking systems around the world.
Theoretical definition
A function ''f'' :
* →
* is one-way if ''f'' can be computed by a polynomial-time algorithm, but any polynomial-time
randomized algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
that attempts to compute a pseudo-inverse for ''f'' succeeds with
negligible probability. (The * superscript means any number of repetitions, see
Kleene star
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repe ...
.) That is, for all randomized algorithms
, all positive integers ''c'' and all sufficiently large ''n'' = length(''x''),
:
where the probability is over the choice of ''x'' from the
discrete uniform distribution on
''n'', and the randomness of
.
Note that, by this definition, the function must be "hard to invert" in the
average-case, rather than worst-case sense. This is different from much of complexity theory (e.g.,
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
ness), where the term "hard" is meant in the worst-case. That is why even if some candidates for one-way functions (described below) are known to be
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''.
Somewhat more precisely, a problem is NP-complete when:
# It is a decision problem, meaning that for any ...
, it does not imply their one-wayness. The latter property is only based on the lack of known algorithms to solve the problem.
It is not sufficient to make a function "lossy" (not one-to-one) to have a one-way function. In particular, the function that outputs the string of ''n'' zeros on any input of length ''n'' is ''not'' a one-way function because it is easy to come up with an input that will result in the same output. More precisely: For such a function that simply outputs a string of zeroes, an algorithm ''F'' that just outputs any string of length ''n'' on input ''f''(''x'') will "find" a proper preimage of the output, even if it is not the input which was originally used to find the output string.
Related concepts
A one-way permutation is a one-way function that is also a permutation—that is, a one-way function that is
bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
. One-way permutations are an important
cryptographic primitive Cryptographic primitives are well-established, low-level cryptography, cryptographic algorithms that are frequently used to build cryptographic protocols for computer security systems. These routines include, but are not limited to, one-way hash fun ...
, and it is not known if their existence is implied by the existence of one-way functions.
A
trapdoor one-way function or trapdoor permutation is a special kind of one-way function. Such a function is hard to invert unless some secret information, called the ''trapdoor'', is known.
A collision-free hash function ''f'' is a one-way function that is also ''collision-resistant''; that is, no
randomized polynomial time algorithm can find a
collision—distinct values ''x'', ''y'' such that ''f''(''x'') = ''f''(''y'')—with non-negligible probability.
Theoretical implications of one-way functions
If ''f'' is a one-way 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 one-way function implies that
FP ≠
FNP, which in turn implies that P ≠ NP. However, P ≠ NP does not imply the existence of one-way functions.
The existence of a one-way function implies the existence of many other useful concepts, including:
*
Pseudorandom generator
In theoretical computer science and cryptography, a pseudorandom generator (PRG) for a class of statistical tests is a deterministic procedure that maps a random seed to a longer pseudorandom string such that no statistical test in the class c ...
s
*
Pseudorandom function families
*
Bit commitment schemes
* Private-key encryption schemes secure against
adaptive chosen-ciphertext attack
*
Message authentication codes
*
Digital signature schemes (secure against adaptive chosen-message attack)
Candidates for one-way functions
The following are several candidates for one-way functions (as of April 2009). Clearly, it is not known whether
these functions are indeed one-way; 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 "easily" computed in
''O''(''b''2) 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
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 , 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 ).
The Rabin function (modular squaring)
The Rabin function,
or squaring
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
, where and are primes is believed to be a collection of one-way functions. We write
:
to denote squaring modulo : 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 (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 and are of the same bit length. The
Rabin cryptosystem is based on the assumption that this Rabin function is one-way.
Discrete exponential and logarithm
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
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
''n''. Denote its
group operation by multiplication. Consider a
primitive element and another element . The discrete logarithm problem is to find the positive integer ''k'', where , such that:
:
The integer ''k'' that solves the equation is termed the discrete logarithm of ''β'' to the base ''α''. One writes .
Popular choices for the group ''G'' in discrete logarithm
cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
are the cyclic groups
(Z''p'')''×'' (e.g.
ElGamal encryption,
Diffie–Hellman key exchange, and the
Digital Signature Algorithm
The Digital Signature Algorithm (DSA) is a Public-key cryptography, public-key cryptosystem and Federal Information Processing Standards, Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular e ...
) and cyclic subgroups of
elliptic curves over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s (''see''
elliptic curve cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys to provide equivalent security, compared to cryptosystems based on modula ...
).
An elliptic curve is a set of pairs of elements of a
field satisfying . 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
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
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 , but if only ''R'' and ''P'' are known, it is assumed to be hard to compute ''k''.
Cryptographically secure hash functions
There are a number of
cryptographic hash function
A cryptographic hash function (CHF) is a hash algorithm (a map (mathematics), map of an arbitrary binary string to a binary string with a fixed size of n bits) that has special properties desirable for a cryptography, cryptographic application: ...
s that are fast to compute, such as
SHA-256
SHA-2 (Secure Hash Algorithm 2) is a set of cryptographic hash functions designed by the United States National Security Agency (NSA) and first published in 2001. They are built using the Merkle–Damgård construction, from a one-way compressi ...
. 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
Other candidates for one-way functions include the hardness of the decoding of random
linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of Code word (communication), codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although t ...
s, the hardness of certain
lattice problems, and the
subset sum problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' ...
(
Naccache–Stern knapsack cryptosystem).
Universal one-way function
There is an explicit function ''f'' that has been proved to be one-way, if and only if one-way functions exist.
In other words, if any function is one-way, then so is ''f''. Since this function was the first combinatorial complete one-way function to be demonstrated, it is known as the "universal one-way function". The problem of finding a one-way function is thus reduced to provingperhaps non-constructivelythat one such function exists.
There also exists a function that is one-way if polynomial-time bounded
Kolmogorov complexity is mildly hard on average. Since the existence of one-way functions implies that polynomial-time bounded Kolmogorov complexity is mildly hard on average, the function is a universal one-way function.
See also
*
One-way compression function
*
Cryptographic hash function
A cryptographic hash function (CHF) is a hash algorithm (a map (mathematics), map of an arbitrary binary string to a binary string with a fixed size of n bits) that has special properties desirable for a cryptography, cryptographic application: ...
*
Geometric cryptography
*
Trapdoor function
References
Further reading
* Jonathan Katz and Yehuda Lindell (2007). ''Introduction to Modern Cryptography''. CRC Press. .
* Section 10.6.3: One-way functions, pp. 374–376.
* Section 12.1: One-way functions, pp. 279–298.
{{DEFAULTSORT:One-Way Function
Cryptographic primitives
Unsolved problems in computer science