PRIMES is in P AgrawalKayalSaxena Presented by Xiaosi

PRIMES is in P Agrawal-Kayal-Saxena Presented by: Xiaosi Zhou

Outline • Introduction 1. What is PRIMES 2. Algorithms for PRIMES before AKS • AKS algorithm 1. Basic idea 2. Notation and Preliminaries 3. The algorithm and its correctness 4. Time complexity analysis 5. Conclusions

Introduction • What is PRIMES: v The decision problem of efficiently determining whether or not a given integer n is prime. v Efficiently means in polynomial time, i. e, O(logn) - the size of the input. v Referred to as primality testing problem.

Algorithms before AKS (1) • The ancient method v Try dividing n by every number v If any m divides n then n is composite otherwise prime v Inefficient--

Algorithms before AKS (2) • Fermat Little Theorem — incorrect testing v For any prime number n, and any number a which has no common divisors with n, Efficient — O(logn) v Counterexample: , but 4 is composite v However, it became the basis of many efficient primality tests. v

Algorithms before AKS (3) • In 1975, Pratt showed that PRIMES is in NP. • In 1976, Miller obtained a deterministic polynomial-time algorithm based on Fermat’s Little Theorem assuming Extended Riemann Hypothesis (ERH). • In 1977, Solovay and Strassen came up with a randomized algorithm which has a probability of error that can be made arbitrarily small for all inputs. • Rabin modified Miller’s algorithm to yield an unconditional but randomized polynomial-time algorithm. • In 1986, Goldwasser and Killian proposed a randomized algorithm based on elliptic curves, running in expected polynomial-time on almost all inputs. • In 1992, Adleman and Huang modified the Goldwasser-Killian algorithm to obtain a randomized polynomial time algorithm that always produced a certificate of primality.

AKS algorithm • There does exist a polynomial-time algorithm for proving primality before AKS algorithm. • But what is surprising is that AKS algorithm is a relatively simple deterministic algorithm which relies on no unproved assumptions.

AKS algorithm – the idea • This test is based on the generalization of Fermat’s Little Theorem. • Theorem: Suppose that a and p are relatively prime integers with p > 1. p is prime if and only if • The theorem suggests a simple test: given input p, choose an a and test whether the above congruence is satisfied. • Too many coefficients to check, O(n)

The idea (Cont’d) • A simpler condition to reduce the coefficients, test if the following equation is satisfied • This must hold if p is prime • The problem now is that some composites n may satisfy the equation for a few values of a and r. • n must be a prime power if the equation holds for several a’s and an appropriately chosen r.

Notation and Preliminaries • denotes the order of a modulo r, which is the smallest number k such that • is Euler’s totient function giving the number of numbers less than r that are relatively prime to r.

AKS algorithm Input: integer n > 1. 1. If , output COMPOSITE; 2. Find the smallest r such that 3. If 1 < (a, n) < n for some , output COMPOSITE; 4. If , output PRIME; 5. For a=1 to do if ( ), output COMPOSITE; 6. Output PRIME;

Correctness (1) • Theorem. The algorithm returns PRIME if and only if n is prime. Proof. [if] If n is prime, steps 1 and 3 can never return COMPOSITE. By the modified Fermat Little Theorem, the for loop also cannot return COMPOSITE. Therefore the algorithm will identify n as PRIME either in step 4 or in step 6.

Correctness (2) Proof. [only if] If the algorithm returns PRIME in step 4 then n must be prime since otherwise step 3 would have found a nontrivial factor of n. How about the algorithm returns PRIME in step 6 ? We need more lemmas.

Correctness (3) Let p be a prime divisor of n. Also, let Two sets: and

Correctness (4) • Define two groups based on the two sets. 1. The first group G is the set of all residues of numbers in I modulo r. We have |G| = t > 2. The second group U is the set of all non-zero residues of polynomials in P modulo h(X) and p, where h(X) is one irreducible factor of degree of

Correctness (5) • Lemma. If n is not a power of p then • Lemma. If the algorithm returns PRIME then n is prime. Proof. We have t=|G| and Therefore, for some k>0. If k>1 then the algorithm will return COMPOSITE in step 1. Thus, n=p. QED

Time complexity We use the symbol Ex. for Theorem. The asymptotic time complexity of the algorithm is

Time complexity (Cont’d) Input: integer n > 1. 1 1. If , output COMPOSITE; 2 2. Find the smallest r such that 3 3. If 1 < (a, n) < n for some , output COMPOSITE; 4 4. If , output PRIME; 5. For a=1 to do if ( ), output 5 COMPOSITE; 6. Output PRIME;

Conclusions • AKS algorithm is an unconditional deterministic polynomial-time algorithm for primality testing. • The complexity of the original algorithm of AKS is , and can be improved to by improving the estimate for r. This algorithm can be further reduced to if one additional number theoretical conjecture can be proved.

Thank you very much!