The AKS primality test (also known as the Agrawal–Kayal–Saxena primality test and the cyclotomic AKS test) is a deterministic primality-proving algorithm 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 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. In 2006 the authors received both the Gödel Prize and Fulkerson Prize 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 numbers, while Pépin's test can be applied to Fermat numbers only.
  • The maximum running time of the algorithm can be bounded by a polynomial 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 unproved hypothesis. 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-unproved 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 <math>n\ge 2</math> and integer <math>a</math> coprime to <math>n</math>, <math>n</math> is prime if and only if the polynomial congruence relation

holds within the polynomial ring <math>(\mathbb Z/n\mathbb Z)[X]</math>. leading to another updated version of the paper. Agrawal, Kayal and Saxena proposed a variant which would run in <math>\tilde{O}(\log(n)^{3})</math> 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: 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 <math>\mathbb{Z}_{n}[x]</math> constructed from the (X + a) binomials that are tested in step 5. Step 4 guarantees that these binomials are <math>\left\lfloor \sqrt{\varphi(r)}\log_2(n) \right\rfloor</math> distinct elements of <math>\mathbb{Z}_n[x]</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.

References

Further reading

  • 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)
  • Andrew Granville: It is easy to determine whether a given integer is prime
  • The Prime Facts: From Euclid to AKS, by Scott Aaronson (PDF)
  • The PRIMES is in P little FAQ by Anton Stiglic
  • 2006 Gödel Prize Citation
  • 2006 Fulkerson Prize Citation
  • The AKS "PRIMES in P" Algorithm Resource