Primality Test: Rabin Miller vs AKS

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto

PrimalityTestProbabilisticvsDeterministicHistory ofPrimalityTesting

Rabin-MillerPrimalityTestFermatsPrimality TestRabin-Miller

AKSPrimalityTestFactsAlgorithm

References

Primality Testing:Rabin-Miller vs AKS

Author: Bruno NetoInstructor: Dr. George Steele

PMAT 427 - Number TheoryWinter 2014

University of Calgary

April 9th, 2014

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Table of Contents

1 Primality TestProbabilistic vs DeterministicHistory of Primality Testing

2 Rabin-Miller Primality TestFermats Primality TestRabin-Miller

3 AKS Primality TestFactsAlgorithm

4 References

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Primality Test

Goal

The goal of a primality test algorithm is to verify whether agiven positive integer is prime or not.

Gauss, disquisitiones arithmeticae, 1801

"The problem of distinguishing prime numbers from compositenumbers and of resolving the latter into their prime factors isknow to be one of the most important and useful inarithmetic(...) Nevertheless, we must confess that all methodsthat have been proposed thus far are either restricted to veryspecial cases or are so laborious and prolix that (...) thesesmethods do not apply at all to large numbers. "

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Probabilistic vs Deterministic

Probabilistic (or Randonmized) Test

Give a positive integerCheck for a random a if it worksRepeat k timesResult: Maybe prime

Ex: Fermat, Rabin-Miller, RSA, Solovay-Strassen

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Probabilistic vs Deterministic

Deterministic Test

Give a positive integerTest the algorithmResult: Guaranteed whether prime or composite

Ex: Trial Division, AKS, Elliptic Curve, "Rabin-Miller"

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

History of Primality Testing

In 17th Century, Fermats Little Theorem says for anyprime integer n and any integer a coprime to n,an1 1(mod n)1976 - Miller used the alternative property to obtain adeterministic polynomial time1980 - Rabin modified Milles test to a randomizedpolynomial time algorithm1984 - Adleman, Pomerance, and Rumely created adeterministic for primality test that runs in(log(n))O(log log log(n)) time1986 - Goldwasser and Kilian proposed a randomizedalgorithm based on Elliptic Curves running in expectedpolynomial time2002 - Agrawal, Kayal, and Saxena proved a deterministicalgorithm in polynomial time

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Fermats Primality Test

Fermats Little Theorem

If n is prime and (a, n) = 1, then an1 1(mod n).

Alternative Statement 1

If an1 6 1(mod n) and (a, n) = 1, then n is definitelycomposite.

Alternative Statement 2

If an1 1(mod n) and (a, n) = 1, then n is probably prime.

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Examples

base 2, 3 < n < 1000

2340 1(mod 341), however 341 = 11 x 312560 1(mod 561), however 561 = 3 x 11 x 172645 1(mod 645), however 645 = 3 x 5 x 43

base 5, 3 < n < 1000

53 1(mod 4), however 4 is composite5123 1(mod 124), however 124 = 4 x 315560 1(mod 561), however 561 is composite5780 1(mod 781), however 781 = 11 x 71

Fermats Pseudoprimes

341, 561 in base 2 and 4, 124, 561, 781 in base 5

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Carmichael Numbers

Carmichael Numbers less then 10,000

561, 1105, 1729, 2465, 2821, 6601, 8911

Theorem(Erdos)

Theorem: Let C (X ) be the number of Carmichael numbersless then X, then

C (X ) < Xexp

(k2log(x)logloglog(X )loglog(X )

)

C (103) = 1

C (1010) = 1547

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

History of Rabin-Miller Primality Test

Gary G. Miller is a professor at Carnegie Mellon University andcreated the algorithm deterministically assuming theGeneralized Riemann Hypothesis in 1976. While Michael O.Rabin, based his method to test the algorithm probabilisticallyusing Millers work in 1980.

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Algorithm

Rabin-Miller Algorithm

1 Input: n odd, k repetitions2 make n 1 = 2sm, m odd and s biggest power of 23 for i in range(1,k)4 a = random [2, n-1]5 while s > 06 If a2

sm 1(mod n), then n is composite7 If a2

sm 1(mod n), then n is probably a prime8 s = s 19 Output: n definitely composite or n probably prime

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Example

Test, after one interation

Enter an odd integer to be tested:4547337172376300111955330758342147474062293202868155909393Enter the number of times you want to test: 1The number is probably a prime

Test, after two interations

Enter an odd integer to be tested:4547337172376300111955330758342147474062293202868155909393Enter the number of times you want to test: 2The number is definitely a composite integer.

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Lemma 1

Lemma: Let a Z, n N, n 2, and (a, n) = 1. Then n isprime if and only if (X + a)n (X n + a) 0 (mod n)

Proof of Lemma 1

Proof: For 0 < i < n, the coefficient of x i in(X + a)n (X n + a) 0 (mod n) is (ni )an1.n prime (ni ) 0 (mod n) all the coefficients are zero.If n is composite. Consider a prime q that is a factor of n andlet qk |n. Then qk does not divide (nq) and is coprime to anqand hence the coefficient of X q is not zero (modn). Thus,((X + a)n(X n + a)) is not identically zero over Zn.

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Algorithm

AKS psedocode Algorithm

Input: integer n > 11 If (n = ab) for a N and b > 1, output COMPOSITE2 Find the smallest r such that or (n) > log2n.3 If 1 < (a, n) < n for some a r , output COMPOSITE4 If n r , output PRIME5 For a = 1 to b(r)log(n)c do6 if((X + a)n 6 X n + a (mod X r 1, n)), output

COMPOSITE7 output PRIME

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

References

1 Agrawal, M., Kayal, N., and Saxena, N., PRIMES is in P,Kanpur-208016, India, 2002

2 Perrenoud, M., Randomized and Deterministic PrimalityTest, Ecole Polytechnique Federale de Lausanne, 2009

3 Jin, T., Research and Implementation on AKS Algorithm,University of Bath, 2005

4 Mcintosh, C., Finding Prime Numbers: Miller Rabin andBeyond, Furman University, Eletronic Journal ofUndergraduate Mathematics, Vol. 12, 1-4, 2007

PrimalityTesting:

Rabin-Millervs AKS

Bruno Neto




References

Thank you!

Primality TestProbabilistic vs DeterministicHistory of Primality Testing

Rabin-Miller Primality TestFermat's Primality TestRabin-Miller

AKS Primality TestFactsAlgorithm

References

Primality Test: Rabin Miller vs AKS

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