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The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". The algorithm was the first one which is able to determine in polynomial time, whether a given number is
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
or composite without relying on mathematical conjectures such as the generalized Riemann hypothesis. The proof is also notable for not relying on the field of
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. In 2006 the authors received both the Gödel Prize and
Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e ...
for their work.


Importance

AKS is the first primality-proving algorithm to be simultaneously ''general'', ''polynomial-time'', ''deterministic'', and ''unconditionally correct''. Previous algorithms had been developed for centuries and achieved three of these properties at most, but not all four. * The AKS algorithm can be used to verify the primality of any general number given. Many fast primality tests are known that work only for numbers with certain properties. For example, the Lucas–Lehmer test works only for
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
s, while Pépin's test can be applied to
Fermat number In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a natural number, positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers ...
s only. * The maximum running time of the algorithm can be bounded by a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
over the number of digits in the target number. ECPP and APR conclusively prove or disprove that a given number is prime, but are not known to have polynomial time bounds for all inputs. * The algorithm is guaranteed to distinguish deterministically whether the target number is prime or composite. Randomized tests, such as Miller–Rabin and Baillie–PSW, can test any given number for primality in polynomial time, but are known to produce only a probabilistic result. * The correctness of AKS is not conditional on any subsidiary unproven
hypothesis A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educated guess o ...
. In contrast, Miller's version of the Miller–Rabin test is fully deterministic and runs in polynomial time over all inputs, but its correctness depends on the truth of the yet-unproven generalized Riemann hypothesis. While the algorithm is of immense theoretical importance, it is not used in practice, rendering it a galactic algorithm. For 64-bit inputs, the Baillie–PSW test is deterministic and runs many orders of magnitude faster. For larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is ''far'' superior to AKS. Additionally, ECPP can output a primality certificate that allows independent and rapid verification of the results, which is not possible with the AKS algorithm.


Concepts

The AKS primality test is based upon the following theorem: Given an integer n\ge 2 and integer a
coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to n, n is prime if and only if the
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
holds within the polynomial ring (\mathbb Z/n\mathbb Z) /math>. Note that X denotes the indeterminate which generates this polynomial ring. This theorem is a generalization to polynomials of
Fermat's little theorem In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , t ...
. In one direction it can easily be proven using the
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and a ...
together with the following property of the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
: : \equiv 0 \pmod for all 0 if n is prime. While the relation () constitutes a primality test in itself, verifying it takes exponential time: the brute force approach would require the expansion of the (X + a)^n polynomial and a reduction \pmod of the resulting n + 1 coefficients. The congruence is an equality in the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
(\mathbb Z/n\mathbb Z) /math>. Evaluating in a quotient ring of (\mathbb Z/n\mathbb Z) /math> creates an upper bound for the degree of the polynomials involved. The AKS evaluates the equality in (\mathbb Z/n\mathbb Z) (X^r -1), making the
computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...
dependent on the size of r. For clarity, this is expressed as the congruence which is the same as: for some polynomials f and g. Note that all primes satisfy this relation (choosing g=0 in () gives (), which holds for n prime). This congruence can be checked in polynomial time when r is polynomial to the digits of n. The AKS algorithm evaluates this congruence for a large set of a values, whose size is polynomial to the digits of n. The proof of validity of the AKS algorithm shows that one can find an r and a set of a values with the above properties such that if the congruences hold then n is a power of a prime.


History and running time

In the first version of the above-cited paper, the authors proved the asymptotic time complexity of the algorithm to be \tilde(\log(n)^) (using Õ from
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
)—the twelfth power of the number of digits in ''n'' times a factor that is polylogarithmic in the number of digits. However, this upper bound was rather loose; a widely-held conjecture about the distribution of the Sophie Germain primes would, if true, immediately cut the worst case down to \tilde(\log(n)^6). In the months following the discovery, new variants appeared (Lenstra 2002, Pomerance 2002, Berrizbeitia 2002, Cheng 2003, Bernstein 2003a/b, Lenstra and Pomerance 2003), which improved the speed of computation greatly. Owing to the existence of the many variants, Crandall and Papadopoulos refer to the "AKS-class" of algorithms in their scientific paper "On the implementation of AKS-class primality tests", published in March 2003. In response to some of these variants, and to other feedback, the paper "PRIMES is in P" was updated with a new formulation of the AKS algorithm and of its proof of correctness. (This version was eventually published in ''
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
''.) While the basic idea remained the same, ''r'' was chosen in a new manner, and the proof of correctness was more coherently organized. The new proof relied almost exclusively on the behavior of
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th prim ...
s over
finite fields In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
. The new upper bound on time complexity was \tilde(\log(n)^), later reduced using additional results from sieve theory to \tilde(\log(n)^). In 2005, Pomerance and Lenstra demonstrated a variant of AKS that runs in \tilde(\log(n)^) operations,H. W. Lenstra Jr. and Carl Pomerance,
Primality testing with Gaussian periods
, preliminary version July 20, 2005.
leading to another updated version of the paper.H. W. Lenstra Jr. and Carl Pomerance,
Primality testing with Gaussian periods
", version of April 12, 2011.
Agrawal, Kayal and Saxena proposed a variant which would run in \tilde(\log(n)^) if Agrawal's conjecture were true; however, a heuristic argument by Pomerance and Lenstra suggested that it is probably false.


The algorithm

The algorithm is as follows: : Input: integer . # Check if ''n'' is a perfect power: if for integers and , then output ''composite''. # Find the smallest ''r'' such that . If ''r'' and ''n'' are not coprime, then output ''composite''. # For all 2 ≤ ''a'' ≤ min (''r'', ''n''−1), check that ''a'' does not divide ''n'': If ''a'', ''n'' for some 2 ≤ ''a'' ≤ min (''r'', ''n''−1), then output ''composite''. # If ''n'' ≤ ''r'', then output ''prime''. # For to \left\lfloor \sqrt\log_2(n) \right\rfloor do #: if (''X''+''a'')''n'' ≠ ''X''''n''+''a'' (mod ''X''''r'' − 1,''n''), then output ''composite''; # Output ''prime''. Here ord''r''(''n'') is the
multiplicative order In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative orde ...
of ''n''
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 ...
''r'', log2 is the
binary logarithm In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, th ...
, and \varphi(r) is
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
of ''r''. Step 3 is shown in the paper as checking 1 < gcd(''a'',''n'') < ''n'' for all ''a'' ≤ ''r''. It can be seen this is equivalent to trial division up to ''r'', which can be done very efficiently without using gcd. Similarly the comparison in step 4 can be replaced by having the trial division return ''prime'' once it has checked all values up to and including \left\lfloor \sqrt \right\rfloor. Once beyond very small inputs, step 5 dominates the time taken. The essential reduction in complexity (from exponential to polynomial) is achieved by performing all calculations in the finite ring : R = (\mathbb Z/n\mathbb Z) (X^r -1) consisting of n^r elements. This ring contains only the r
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s \ , and the
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s are in \mathbb Z/n\mathbb Z which has n elements, all of them codable within \log_2(n) bits. Most later improvements made to the algorithm have concentrated on reducing the size of ''r,'' which makes the core operation in step 5 faster, and in reducing the size of ''s'', the number of loops performed in step 5.Daniel J. Bernstein,
Proving Primality After Agrawal-Kayal-Saxena
, version of January 25, 2003.
Typically these changes do not change the computational complexity, but can lead to many orders of magnitude less time taken; for example, Bernstein's final version has a theoretical speedup by a factor of over 2 million.


Proof of validity outline

For the algorithm to be correct, all steps that identify ''n'' must be correct. Steps 1, 3, and 4 are trivially correct, since they are based on direct tests of the divisibility of ''n''. Step 5 is also correct: since (2) is true for any choice of ''a'' coprime to ''n'' and ''r'' if ''n'' is prime, an inequality means that ''n'' must be composite. The difficult part of the proof is showing that step 6 is true. Its proof of correctness is based on the upper and lower bounds of a
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
in \mathbb_ /math> constructed from the (''X'' + ''a'') binomials that are tested in step 5. Step 4 guarantees that these binomials are \left\lfloor \sqrt\log_2(n) \right\rfloor distinct elements of \mathbb_n /math>. For the particular choice of ''r'', the bounds produce a contradiction unless ''n'' is prime or a power of a prime. Together with the test of step 1, this implies that ''n'' is always prime at step 6.


Example 1: ''n'' = 31 is prime

Where PolynomialMod is a term-wise modulo reduction of the polynomial. e.g. PolynomialMod +2x2+3x3, 3= x+2x2+0x3 See AKS Talk page for a discussion on why 'Example 2: n is not Prime past Step 4' is missing.


References


Further reading

*


External links

*
R. Crandall, Apple ACG, and J. Papadopoulos (March 18, 2003): On the implementation of AKS-class primality tests
(PDF)
Article by Bornemann, containing photos and information about the three Indian scientists
(PDF)

* ttp://www.scottaaronson.com/writings/prime.pdf The Prime Facts: From Euclid to AKS by
Scott Aaronson Scott Joel Aaronson (born May 21, 1981) is an American Theoretical computer science, theoretical computer scientist and Schlumberger Centennial Chair of Computer Science at the University of Texas at Austin. His primary areas of research are ...
(PDF)
The PRIMES is in P little FAQ
by Anton Stiglic


2006 Fulkerson Prize Citation

The AKS "PRIMES in P" Algorithm Resource
{{Authority control Primality tests Finite fields Articles with example pseudocode