Primes in P

1. Primes in P • Deterministic polynomial-time algorithm of Agrawal, Kayal and Saxena • Presented by Vladimir Braverman

2. History • Sieve of Eratosthenes 240BC • Randomized, polynomial-time algorithm of Miller-Rabin, 1980 (practically used) • Adleman, Pomerance and Rumeley (APR) a deterministic algorithm, an almost polynomial time NloglogN , 1983 • Agrawal, Kayal and Saxena, deterministic, polinomial-time algorithm 2002

3. Impact • Data security • Number Theory • Theory of algorithms • Doesn’t have a practical value yet

4. The Heroes • Neeraj Kayal and Nitin Saxena are PhD students (22 years old). • Indian Institute of Technology, Kanpur • Agrawal was recently awarded the Clay Research Award, one of the coveted honors in mathematical research.

5. Number Theory Finite Fields Group Theory Primes in P Polynomial identity idea Algorithms in Number Theory

6. Content • Background • Algorithms in number theory • Rings • Number Theory • Basic idea: Polynomial identity • Algorithm • Pseudo code • Correctness • Running time

7. Algorithms in number theory • Lemma 1 • 0 < a,b < n • ab can be calculated in O(log3 (n)) • Maximal integer less than b/a can be calculated in O(log2 (n)) • a mod b can be calculated in O(log2 (n)) • Gcd(a,b) can be calculated in O(log3 (n))

8. Algebra • A ring is a set together with addition and multiplication • 1. Additive associativity: • 2. Additive commutativity: • 3. Additive identity. • 4. Additive inverse • 5. Multiplicative associativity: • 6. Left and right distributivity: • A ring is therefore an Abelian group under addition and a semigroup under multiplication.

9. Algebra (cont) • A polynomialf is defined to be a formal expression of the form • where the coefficients a0, ... , an are elements of some ringR and X is considered to be a formal symbol. • Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal.

10. Algebra (end) • Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive low and the rules • Xa = aX for all elements a of the ring R • XkXl = Xk+l for all natural numbers k and l. • The set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X].

11. Algorithms in number theory (cont) • Lemma 2 Let 1 < a,b, r and n = max(a,r). Than ab mod r can be calculated in O(log2 (n) + log(b) log2(r)) • Lemma 3 Let 1< a,r < n. The r coefficients of (x-a)n mod (xr-1) in Zn[x] can be computed in O(r2 log3 (n)) time.

12. Algorithms in number theory (cont) • Lemma 4 Let n > 2 be an integer. There is an algorithm that decides in O(√n log2 (n)) whether n is prime. Algorithm: • r:=2; s:=4 (s = r2) • While s <= n • Do If n mod r = 0 return NO • Else r:=r+1; s:=2r-1 • Endif • EndWhile • Return Yes

13. Algorithms in number theory (cont) • Lemma 4 (cont) • The correctness follows from the fact that n is not prime iff there is an integer r <= √n that divides n. • The while-loop makes at most √n iterations each one taking (log2 n) time.

14. Algorithms in number theory (end) • Lemma 5 Let n > 2 be an integer. There is an algorithm that computes the largest prime factor of n in O(√n log2 (n)). • Lemma 6 Let n > 2 be an integer. There is an algorithm that decides in O(log4 (n) log (log (n)) ) whether there exist integers a,b such that n = ab.

15. Number theory • Lemma 7 If p is prime number than for any i < p (pi) = 0 mod p Proof (pi) = p(p-1)…(p-i+1) / i! Since gcd(p, i!) = 1 the lemma is proved

16. Number theory(cont) • Lemma 8 If p is prime number than for any a, ap = a mod p Proof (induction) • a = 1 • a -> a+1 (a+1)p =  (pi) ai (a+1)p = ap+1 = a+1 mod p

17. Polynomial identity • Theorem 1 Let a, n be integers • If n is a prime number, then (x-a)n = xn-a in the ring Zn[x] • If gcd(a, n) =1 and n is not a prime number, then (x-a)n <> xn-a in the ring Zn[x]

18. Proof of Theorem 1 (x-a)n = (ni) xi(-a)n-I. • Prime case • If n is prime, (ni) = 0 mod n, according to Lemma 7 • Therefore (x-a)n = xn – an = xn – a mod n, according to Lemma 8 • n is not a prime and gcd(n,a) = 1 q – prime factor of n, qk | n, k – maximal gcd (a,q) = 1 and gcd (an-q, qk) = 1

19. Proof of Theorem 1(Cont) Lemma: qk doesn’t divide (nq) Proof: (by contradiction) • Suppose, (nq) = b qk. Than, n(n-1)…(n-q+1)/q! = b qk. n = (q-1)! b qk+1 /(n-1)…(n-q+1). • For each 0 < j < q , q doesn’t divide (n-j). Indeed, since q | n and (n-j) = 0 mod q we have j = 0 mod q. • Therefore, (q-1)! b /(n-1)…(n-q+1) is an integer and qk+1 | n that contradicts with maximal property of k

20. Proof of Theorem 1(Cont) • (x-a)p The coefficient of xp is (nq)(-1)n-qan-q • If it is divisible by n, than (nq)an-q = bn Hence, (nq)an-q/qk = bn / qk • Right-hand side is integer and gcd(an-q,qk) = 1 • Therefore qk | (nq) that contradicts Lemma • We proved that (x-a)n <> xn-a in the ring Zn[x]

21. Improved polynomial identity • Theorem 2 Let n be an integer and let q, r be prime numbers and • gcd (m,n) = 1 for all m < r • q divides r-1 • q ≥ 2√r log(n) + 2 • n(r-1)/q≠ 1 mod r • (x-a)a = (xn-a) mod (xr-1) in Zn[x] for a < 2√r log(n) + 1 Then n is a power of prime number.

22. Lemma 9 There exist constants 0 < c1 < c2 such that for any large n, there exists a prime number r c1 log6(n) < r < c2 log6(n) r-1 has a prime factor q, q ≥ 2√r log(n) + 2 n(r-1)/q≠ 1 mod r

23. Algorithm Prime(n) • r := 2; found := false; • While r < n and found = false • Do if gcd(r,n) ≠ 1 then return NO • If r is a prime and r > 2 • Then q := largest prime factor of r-1 • If q ≥ 2√r log(n) + 2 and n(r-1)/q ≠ 1 mod r • Then found = true • Endif • Endif • If found = false then r++ endif • Endwhile

24. Algorithm Prime(n) (Cont) • For a:= 1 to 2√r log(n) + 1 • Do if (x-a)a ≠ (xn-a) mod (xr-1) in Zn[x] • Then return NO • Endif • Endfor • If n = ab for some integers a,b ≥ 2 • Then return NO • Else return YES • Endif

25. Correctness of the algorithm (Sketch of the proof ) • If n is prime, • Algorithm doesn’t return NO during While loop, since gcd(n,r) = 1 • By Theorem 1, (x-a)a ≠ (xn-a) mod (xr-1) in Zn[x] Thus, the algorithm doesn’t return NO during For loop • N is not in the form ab, therefore YES is returned • If n is not a prime, then, according to Theorem 2, the algorithm returns NO

26. Running time (Sketch of the proof ) • While loop

27. Running time(Cont) • For loop • Testing if n is perfect power • O(log4(n) log(log(n))) , by Lemma 6

28. References • Agrawal, Kayal and Saxena. Primes in P • M. Smid. Primality testing in polynomial time • F. Bornemann. PRIMES Is in P: A breakthrough for “Everyman”

29. END