Rabin Miller Primality Test

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Motivation Algorithm Conclusion References

Rabin Miller Probabilistic Primality

Test

Akhilesh Chaganti,Shafiuddin Rehan AhmedGautham Raj G,Pattabi Ramiah

Indian Institute of Technology Hyderabad

November 29, 2010

-Typeset by LATEX-
http://goforward/http://find/http://goback/


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Contents

1 Motivation

Cryptography - Encryption mechanismsThe properties of numbersThe element of randomization


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Contents

1 Motivation


2 Algorithm

definitionInterpretation of the algorithmPseudocodeExpected Running Time

Error BoundsImplementationWitnessing and Non Witnessing


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Contents

1 Motivation


2 Algorithm



3 Conclusion

M i i Al i h C l i R f


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Contents

1 Motivation


2 Algorithm



3 Conclusion

4 References

M ti ti Al ith C l i R f


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Cryptography - Encryption mechanisms

Cryptography is a very important and heavily growing branch ofcomputer science which is based on number theory. The famousproblem of this field is the public key encryption mechanismswhich is used extensively in network security.



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The main way of implementing this encryption mechanism is tointroduce the notion of a one-way function. It can interpreted as afunction which is easy to compute and very hard to compute backits inverse.



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The main way of implementing this encryption mechanism is tointroduce the notion of a one-way function. It can interpreted as afunction which is easy to compute and very hard to compute backits inverse.

An important implementation of the public key encryption systemis based on a conjecture of multiplying two prime numbers as aone way function



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The properties of numbers

Till date it is observed that the multiplication of two very largeprime numbers is almost a one-way process i.e. it is easy to get theproduct of numbers but computing the prime factors for a given(large) number is hard. This provides motivation for the primalitytesting of a (large) number which can be inturn used in encryption

mechanisms.



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mechanisms.

Is the problem of primality testing easy (deterministic polynomialtime algorithm)??



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Moti ation lgorithm Conclusion References



mechanisms.


Yes. In 2002, 3 Indians(@ IIT Kanpur) proposed an algorithm (TheAKS algorithm) which tests the primality of the given number n in

the order of O(log152 n)



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g



mechanisms.


Yes. In 2002, 3 Indians(@ IIT Kanpur) proposed an algorithm (TheAKS algorithm) which tests the primality of the given number n in

the order of O(log152 n)

NOTE: The standard measure of the complexity of the number

theoritic algorithms is reffered to bit complexity.



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The element of randomization

Since, there is no real time efficient algorithm for primality testingwe see the need for randomization.There are several randomized

algorithms testing primality of number with certain confidence(with a controllably small probability of error.)



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The element of randomization

Since, there is no real time efficient algorithm for primality testingwe see the need for randomization.There are several randomized

algorithms testing primality of number with certain confidence(with a controllably small probability of error.)

One such algorithm is

Rabin Miller Probabilistic Primality test.



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Rabin-Miller Primality Test

Definition

Given (b,n), where n is the number to test for the primality, and bis randomly chosen from Zn-{0}. Let n-1 = 2

qm, where m is an

odd integer. if eitherbm 1(mod n) or

there is an integer i in [0,q-1] such that bm2i

-1(mod n)

then return probabily prime else return composite

NOTE : Rabin-Miller is a Monte Carlo algorithm: there is anonzero probability of error.



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Interpretation of the Algorithm

The algorithm enables one to probabilistically test the primality ofa number n as follows:

If n fails the test(i.e. results in n is composite) for any b in[1,n-1] then n is definitely composite (although, interestingly,no factor is provided by the algorithm).

A composite number has at most 1 chance in 4k of passing allk of a series of k tests, where b is chosen randomly from [ 1 ,n - 1 ] for each test. Therefore, if a suspected prime, n, passesk of k tests, we can conclude with a certainty of at least 1 -

(1/4)k

that n is prime.

NOTE: Rabin-Miller Test is co-RP-algorithm for testing theprimality of the given number

PRIMES

co-RP



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Pseudocode

Pseudocode

Input: n 4, an odd integer to be tested for primality;Input: k, a parameter that determines the accuracy of the testOutput: composite if n is composite, otherwise probably primewrite n 1 as 2sd with d odd by factoring powers of 2 from n 1

LOOP: repeat k times:pick a randomly in the range [2, n 2]x ad mod nif x = 1 or x = n 1 then do next LOOPfor r = 1 .. s1

x x2 mod nif x = 1 then return compositeif x = n 1 then do next LOOPreturn composite

return probabily prime



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Expected Running Time

Running Time

Using modular exponentiation by repeated squaring, the runningtime of this algorithm is O(k log3 n), where k is the number ofdifferent values of a we test; thus this is a reasonably efficientpolynomial-time algorithm when compared the deterministic one.



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Coming back to the bound. . .

The algorithm enables one to probabilistically test the primality ofa number n as follows:

If n fails the test(i.e. results in n is composite) for any b in

[1,n-1] then n is definitely composite (although, interestingly,no factor is provided by the algorithm).

A composite number has at most 1 chance in 4k of passing allk of a series of k tests, where b is chosen randomly from [ 1 ,n - 1 ] for each test. Therefore, if a suspected prime, n, passesk of k tests, we can conclude with a certainty of at least 1 -(1/4)k that n is prime.



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Implementation on a known prime

We will now demonstrate this algorithm on a number that wedefinitely know is prime.Lets take n = 29.



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So n-1 = 28, which can be written as 22.7So, here q = 2 and m = 7. Take b = 10.



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So n-1 = 28, which can be written as 22.7So, here q = 2 and m = 7. Take b = 10.Applying the algorithm107 17 (mod 29) which is not congruent to 1 or -1.So,we continue the test.



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So n-1 = 28, which can be written as 22.7So, here q = 2 and m = 7. Take b = 10.Applying the algorithm107 17 (mod 29) which is not congruent to 1 or -1.So,we continue the test.

(107)2 -1(mod 29) Inconclusive state



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Implementation on a known composite

We will now demonstrate this algorithm on a number that wedefinitely know is composite.Lets take n = 13 7 = 221.



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We will now demonstrate this algorithm on a number that wedefinitely know is composite.Lets take n = 13 7 = 221.n-1 = 220 = 22 55, here q = 2 and m = 55. Take b = 5.



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We will now demonstrate this algorithm on a number that wedefinitely know is composite.Lets take n = 13 7 = 221.n-1 = 220 = 22 55, here q = 2 and m = 55. Take b = 5.Applying this algorithm

for i = 0, 555 112 (mod 221) = 1 or -1 (mod 221).So, lets take i = 1, (555)2 168 (mod 221) = -1 (mod 221)



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for i = 0, 555 112 (mod 221) = 1 or -1 (mod 221).So, lets take i = 1, (555)2 168 (mod 221) = -1 (mod 221)After using all the values of i, we are still in a false case. So wecan definitely say its a composite number.



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However, if b = 21.Then (21)55 200 (mod 221) which is not congruent to 1 or -1.So we continue the test. We obtain (2155)2 -1 (mod 221) Thatis for a composite number 221 which is inconclusive.



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However, if b = 21.Then (21)55 200 (mod 221) which is not congruent to 1 or -1.So we continue the test. We obtain (2155)2 -1 (mod 221) Thatis for a composite number 221 which is inconclusive.We call such a base a non-witness to the compositeness of 221.



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Witnessing and non witnessing bases

Non-Witness

The base b in [1,n-1] results in inconclusive for a compositenumber, then b is a non-witness to the compositness of n.

example

For 221, we have six non-witnesses namely 1 , 21 , 47 , 174 , 200and 220.

We denote number of non-witnesses of n by nw(n). So, in ourexample nw(221) = 6.



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Rabin stated that the number witness to compostiness has a lowerbound.

Theorem

If n > 4 is composite, then the number of bases, b, in [1, n-1] suchthat b is a witness to the compositeness of n is at least 3 (n-1) / 4.

The following corollary describing the upper bound onnon-witnesses to compositeness follows directly from theorem 1.

Corollary

If n > 4 is composite then at most (n-1) / 4 of the bases, b, in [1,

n-1] are non-witnesses to the compositeness of n.

Thus, we can get a bound on probability of passing the statementfalse, even when the number is actually composite as 1/4.



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Conclusion

conclusions and Remarks

Although, there are number of primality tests, Rabin-Millertest is the most commonly used.

Compared to any other test which provides the confidence asmuch as Rabin-Miller test, it is the test which can beimplemented with ease and that to in O(n3)

Rabin-Miller test also produces pseudo primes but notabsolute pseudo primes like Fermat and Euler tests for

primality. So definitely better than the above testThere are also deterministic variants of the test assuming thetruth of unproven Generalized Riemann Hypothesis.



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References

[Motwani R.,Raghavan P.]Randomized Algorithms , ISBN 0-521-61390-6

[Brian C. Higgins]The Rabin-Miller Primality Test : Some results on number ofnon-witnesses to compositeness.

[Rabin Miller Primality test]

http://en.wikipedia.org/
http://en.wikipedia.org/wiki/Rabin-Miller_primality_testhttp://en.wikipedia.org/wiki/Rabin-Miller_primality_testhttp://goforward/http://find/http://goback/

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Rabin Miller Primality Test