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**Agrawal-Kayal-Saxena Presented by: Xiaosi Zhou**

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

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**Outline Introduction 1. What is PRIMES AKS algorithm 1. Basic idea**

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

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**Introduction What is PRIMES:**

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

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**Algorithms before AKS (1)**

The ancient method Try dividing n by every number If any m divides n then n is composite otherwise prime Inefficient--

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**Algorithms before AKS (2)**

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

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**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.

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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.

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**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)

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**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.

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**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.

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**AKS algorithm Input: integer n > 1. If , output COMPOSITE;**

Find the smallest r such that If 1 < (a, n) < n for some , output COMPOSITE; If , output PRIME; For a=1 to do if ( ), output COMPOSITE; Output PRIME;

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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.

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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 non-trivial factor of n. How about the algorithm returns PRIME in step 6 ? We need more lemmas.

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

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**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

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**Correctness (5) Lemma. 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

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**Time complexity We use the symbol for Ex.**

Theorem. The asymptotic time complexity of the algorithm is

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**Time complexity (Cont’d)**

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

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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.

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Thank you very much!

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