Primes in P Manindra Agrawal Neeraj Kayal Nitin Saxena CS300: Technical Paper review by Arpan Agrawal

Introduction Prime numbers are of fundamental importance in mathematics. Many primality testing algorithm exists. This paper presented the first unconditional deterministic polynomial time algorithm.

Previous Work Numerous efforts were made to find an efficient primality testing algorithm. Miller (1975) gave a deterministic polynomial running time algorithm assuming Extended Riemann Hypothesis. Rabin modified it to yield an unconditional but randomized polynomial time algorithm. Several other algorithms were proposed but none could achieve unconditonal deterministic polynomial running time.

Idea Based on the generalization of Fermat’s Little Theorem to polynomial rings over finite fields which states: Let a ∈ Z, n ∈ N, n ≥ 2, and gcd(a, n) = 1. Then n is prime if and only if: (X + a)^n = X^n + a (mod n) (1)

Idea To reduce the number of coefficients in LHS, Eq.(1) modulo (X^r-1) is evaluated which gives: (X + a)^n = X^n + a(mod X^r−1, n) (2) The problem is that some composite numbers may also satisfy eq. (2). But it is proved that for appropriately chosen r, if several a’s satisfy above equation then n must be a prime power.

Idea It is also established that the number of a’s and the appropriate r are bound by a polynomial in logn. Thus, an unconditional deterministic polynomial time primality testing algorithm is achieved.

Highlights Time Complexity is O ∼ (log n)^(15/2). Simpler proof of correctness as compared of other primality testing algorithms. If Agrawal’s conjecture is established, time complexity can be improved to O ~ (logn)^3. The authors were awarded with Clay Research Award(2002), Fulkerson Prize(2006) and Godel Prize(2006) for the paper.

References mality_v6.pdf rize wal

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