DOCSLIB.ORG

Primality Testing and the Miller-Rabin Algorithm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Primality Testing and the Miller-Rabin Algorithm J. David Primality Testing and the Miller-Rabin Taylor Algorithm Introduction FA 2018 Cryptography Seminar First Attempt Second Attempt J. David Taylor October 12, 2018 Bob needs a pair of large prime numbers! If p; q are composite, then Bob will need to know how to factor them, and the cipher will be less secure. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm J. David Taylor Suppose Bob wants to implement RSA. Introduction First Attempt Second Attempt If p; q are composite, then Bob will need to know how to factor them, and the cipher will be less secure. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm J. David Taylor Suppose Bob wants to implement RSA. Introduction First Attempt Bob needs a pair of large prime numbers! Second Attempt and the cipher will be less secure. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm J. David Taylor Suppose Bob wants to implement RSA. Introduction First Attempt Bob needs a pair of large prime numbers! Second Attempt If p; q are composite, then Bob will need to know how to factor them, Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm J. David Taylor Suppose Bob wants to implement RSA. Introduction First Attempt Bob needs a pair of large prime numbers! Second Attempt If p; q are composite, then Bob will need to know how to factor them, and the cipher will be less secure. If Bob can efficiently distinguish prime numbers from composite numbers, then Bob can choose large numbers at random and check which are prime. Bob wants an efficient algorithm that detects composite numbers. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm The prime number theorem tells us that J. David Taylor N The number of primes ≤ N ≈ Introduction log N First Attempt . Second Attempt Bob wants an efficient algorithm that detects composite numbers. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm The prime number theorem tells us that J. David Taylor N The number of primes ≤ N ≈ Introduction log N First Attempt . Second Attempt If Bob can efficiently distinguish prime numbers from composite numbers, then Bob can choose large numbers at random and check which are prime. Why Primality Testing? Primality Testing and the Miller-Rabin Algorithm The prime number theorem tells us that J. David Taylor N The number of primes ≤ N ≈ Introduction log N First Attempt . Second Attempt If Bob can efficiently distinguish prime numbers from composite numbers, then Bob can choose large numbers at random and check which are prime. Bob wants an efficient algorithm that detects composite numbers. Example: Let n = 31987937737479355332620068643713101490952335301. The congruence ( mod n) 2n−1 ≡ 1281265953551359064133601216247151836053160074 tells Bob that n is not prime! Fermat's Little Theorem Primality Testing and the Miller-Rabin Algorithm Theorem p J. David Let p be a prime number, then a ≡ a mod p for every integer Taylor a. Introduction First Attempt Second Attempt The congruence ( mod n) 2n−1 ≡ 1281265953551359064133601216247151836053160074 tells Bob that n is not prime! Fermat's Little Theorem Primality Testing and the Miller-Rabin Algorithm Theorem p J. David Let p be a prime number, then a ≡ a mod p for every integer Taylor a. Introduction First Attempt Example: Let Second Attempt n = 31987937737479355332620068643713101490952335301. Fermat's Little Theorem Primality Testing and the Miller-Rabin Algorithm Theorem p J. David Let p be a prime number, then a ≡ a mod p for every integer Taylor a. Introduction First Attempt Example: Let Second Attempt n = 31987937737479355332620068643713101490952335301. The congruence ( mod n) 2n−1 ≡ 1281265953551359064133601216247151836053160074 tells Bob that n is not prime! For example, 2 is a witness for 6, but 3 isn't. Idea: Try numbers less than n until you find a witness or try all of them Witnesses Primality Testing and the Miller-Rabin Algorithm J. David Taylor Let a; n be integers. We say that a is a witness for (the Introduction compositeness of) n if an 6≡ a mod n. First Attempt Second Attempt Idea: Try numbers less than n until you find a witness or try all of them Witnesses Primality Testing and the Miller-Rabin Algorithm J. David Taylor Let a; n be integers. We say that a is a witness for (the Introduction compositeness of) n if an 6≡ a mod n. First Attempt Second For example, 2 is a witness for 6, but 3 isn't. Attempt Witnesses Primality Testing and the Miller-Rabin Algorithm J. David Taylor Let a; n be integers. We say that a is a witness for (the Introduction compositeness of) n if an 6≡ a mod n. First Attempt Second For example, 2 is a witness for 6, but 3 isn't. Attempt Idea: Try numbers less than n until you find a witness or try all of them (Korselt) Theorem: A positive composite integer n is Carmichael iff n is square-free and p − 1jn − 1 for every prime pjn. First seven Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, and 8911. (Alfred, Granville, Pomerance) Theorem: There are infinitely many Carmichael numbers Carmichael Numbers Primality Testing and the Miller-Rabin Algorithm Composite numbers with no witnesses are called J. David Carmichael numbers. Taylor Introduction First Attempt Second Attempt First seven Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, and 8911. (Alfred, Granville, Pomerance) Theorem: There are infinitely many Carmichael numbers Carmichael Numbers Primality Testing and the Miller-Rabin Algorithm Composite numbers with no witnesses are called J. David Carmichael numbers. Taylor Introduction (Korselt) Theorem: A positive composite integer n is First Attempt Carmichael iff n is square-free and p − 1jn − 1 for every Second prime pjn. Attempt (Alfred, Granville, Pomerance) Theorem: There are infinitely many Carmichael numbers Carmichael Numbers Primality Testing and the Miller-Rabin Algorithm Composite numbers with no witnesses are called J. David Carmichael numbers. Taylor Introduction (Korselt) Theorem: A positive composite integer n is First Attempt Carmichael iff n is square-free and p − 1jn − 1 for every Second prime pjn. Attempt First seven Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, and 8911. Carmichael Numbers Primality Testing and the Miller-Rabin Algorithm Composite numbers with no witnesses are called J. David Carmichael numbers. Taylor Introduction (Korselt) Theorem: A positive composite integer n is First Attempt Carmichael iff n is square-free and p − 1jn − 1 for every Second prime pjn. Attempt First seven Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, and 8911. (Alfred, Granville, Pomerance) Theorem: There are infinitely many Carmichael numbers Let a be an integer coprime to p. Then aq ≡ 1 mod p, or one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. Essentially, a2k q = ap−1 ≡ 1 mod p, so either q kills a or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction First Attempt Second Attempt Then aq ≡ 1 mod p, or one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. Essentially, a2k q = ap−1 ≡ 1 mod p, so either q kills a or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction Let a be an integer coprime to p. First Attempt Second Attempt or one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. Essentially, a2k q = ap−1 ≡ 1 mod p, so either q kills a or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction Let a be an integer coprime to p. Then First Attempt q Second a ≡ 1 mod p, Attempt Essentially, a2k q = ap−1 ≡ 1 mod p, so either q kills a or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction Let a be an integer coprime to p. Then First Attempt q Second a ≡ 1 mod p, or Attempt one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. so either q kills a or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction Let a be an integer coprime to p. Then First Attempt q Second a ≡ 1 mod p, or Attempt one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. Essentially, a2k q = ap−1 ≡ 1 mod p, or some number in the list is a non-trivial square root of 1. Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J. David Let p = 2k q + 1 be an odd prime number with q odd. Taylor Introduction Let a be an integer coprime to p. Then First Attempt q Second a ≡ 1 mod p, or Attempt one of aq; a2q; a4q;:::; a2k−1q is congruent to −1 mod p. Essentially, a2k q = ap−1 ≡ 1 mod p, so either q kills a Miller-Rabin Criterion Primality Testing and the Miller-Rabin Algorithm J.
Recommended publications
  • Primality Testing and Integer Factorisation

    Primality Testing and Integer Factorisation

    Primality Testing and Integer Factorisation Richard P. Brent, FAA Computer Sciences Laboratory Australian National University Canberra, ACT 2601 Abstract The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the Rivest-Shamir-Adelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60-decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several recent algorithms for primality testing and factorisation, give examples of their use and outline some applications. 1. Introduction It has been known since Euclid’s time (though first clearly stated and proved by Gauss in 1801) that any natural number N has a unique prime power decomposition α1 α2 αk N = p1 p2 ··· pk (1.1) αj (p1 < p2 < ··· < pk rational primes, αj > 0). The prime powers pj are called αj components of N, and we write pj kN. To compute the prime power decomposition we need – 1. An algorithm to test if an integer N is prime. 2. An algorithm to find a nontrivial factor f of a composite integer N. Given these there is a simple recursive algorithm to compute (1.1): if N is prime then stop, otherwise 1. find a nontrivial factor f of N; 2.
  • An Analysis of Primality Testing and Its Use in Cryptographic Applications

    An Analysis of Primality Testing and Its Use in Cryptographic Applications

    An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability.
  • Primes and Primality Testing

    Primes and Primality Testing

    Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100.
  • Primality Testing for Beginners

    Primality Testing for Beginners

    STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
  • Miller–Rabin Test

    Miller–Rabin Test

    Lecture 31: Miller–Rabin Test Miller–Rabin Test Recall In the previous lecture we considered an efficient randomized algorithm to generate prime numbers that need n-bits in their binary representation This algorithm sampled a random element in the range f2n−1; 2n−1 + 1;:::; 2n − 1g and test whether it is a prime number or not By the Prime Number Theorem, we are extremely likely to hit a prime number So, all that remains is an algorithm to test whether the random sample we have chosen is a prime number or not Miller–Rabin Test Primality Testing Given an n-bit number N as input, we have to ascertain whether N is a prime number or not in time polynomial in n Only in 2002, Agrawal–Kayal–Saxena constructed a deterministic polynomial time algorithm for primality testing. That is, the algorithm will always run in time polynomial in n. For any input N (that has n-bits in its binary representation), if N is a prime number, the AKS primality testing algorithm will return 1; otherwise (if, the number N is a composite number), the AKS primality testing algorithm will return 0. In practice, this algorithm is not used for primality testing because this turns out to be too slow. In practice, we use a randomized algorithm, namely, the Miller–Rabin Test, that successfully distinguishes primes from composites with very high probability. In this lecture, we will study a basic version of this Miller–Rabin primality test. Miller–Rabin Test Assurance of Miller–Rabin Test Miller–Rabin outputs 1 to indicate that it has classified the input N as a prime.
  • A Potentially Fast Primality Test

    A Potentially Fast Primality Test

    A Potentially Fast Primality Test Tsz-Wo Sze February 20, 2007 1 Introduction In 2002, Agrawal, Kayal and Saxena [3] gave the ¯rst deterministic, polynomial- time primality testing algorithm. The main step was the following. Theorem 1.1. (AKS) Given an integer n > 1, let r be an integer such that 2 ordr(n) > log n. Suppose p (x + a)n ´ xn + a (mod n; xr ¡ 1) for a = 1; ¢ ¢ ¢ ; b Á(r) log nc: (1.1) Then, n has a prime factor · r or n is a prime power. The running time is O(r1:5 log3 n). It can be shown by elementary means that the required r exists in O(log5 n). So the running time is O(log10:5 n). Moreover, by Fouvry's Theorem [8], such r exists in O(log3 n), so the running time becomes O(log7:5 n). In [10], Lenstra and Pomerance showed that the AKS primality test can be improved by replacing the polynomial xr ¡ 1 in equation (1.1) with a specially constructed polynomial f(x), so that the degree of f(x) is O(log2 n). The overall running time of their algorithm is O(log6 n). With an extra input integer a, Berrizbeitia [6] has provided a deterministic primality test with time complexity 2¡ min(k;b2 log log nc)O(log6 n), where 2kjjn¡1 if n ´ 1 (mod 4) and 2kjjn + 1 if n ´ 3 (mod 4). If k ¸ b2 log log nc, this algorithm runs in O(log4 n). The algorithm is also a modi¯cation of AKS by verifying the congruent equation s (1 + mx)n ´ 1 + mxn (mod n; x2 ¡ a) for a ¯xed s and some clever choices of m¡ .¢ The main drawback of this algorithm¡ ¢ is that it requires the Jacobi symbol a = ¡1 if n ´ 1 (mod 4) and a = ¡ ¢ n n 1¡a n = ¡1 if n ´ 3 (mod 4).
  • 15-251: Great Theoretical Ideas in Computer Science Recitation 14 Solutions PRIMES ∈ NP

    15-251: Great Theoretical Ideas in Computer Science Recitation 14 Solutions PRIMES ∈ NP

    15-251: Great Theoretical Ideas In Computer Science Recitation 14 Solutions PRIMES 2 NP The set of PRIMES of all primes is in co-NP, because if n is composite and k j n, we can verify this in polynomial time. In fact, the AKS primality test means that PRIMES is in P. We'll just prove PRIMES 2 NP. ∗ (a) We know n is prime iff φ(n) = n−1. Additionally, if n is prime then Zn is cyclic with order n−1. ∗ Given a generator a of Zn and a (guaranteed correct) prime factorization of n − 1, can we verify n is prime? If a has order n − 1, then we are done. So we must verify a has order dividing n − 1: an−1 ≡ 1 mod n and doesn't have smaller order, i.e. for all prime p j n − 1: a(n−1)=p 6≡ 1 mod n a is smaller than n, and there are at most log(n) factors each smaller than n, so our certificate is polynomial in n's representation. Similarly, each arithmetic computation is polynomial in log(n), and there are O(log(n)) of them. 1 (b) What else needs to be verified to use this as a primality certificate? Do we need to add more information? We also need to verify that our factorization (p1; α1); (p2; α2);:::; (pk; αk) of n − 1 is correct, two conditions: α1 α2 αk 1. p1 p2 : : : pk = n − 1 can be verified by multiplying and comparing, polynomial in log(n). 2.
  • Pseudoprimes and Classical Primality Tests

    Pseudoprimes and Classical Primality Tests

    Pseudoprimes and Classical Primality Tests Sebastian Pancratz 2 March 2009 1 Introduction This brief introduction concerns various notions of pseudoprimes and their relation to classical composite tests, that is, algorithms which can prove numbers composite, but might not identify all composites as such and hence cannot prove numbers prime. In the talk I hope to also present one of the classical algorithms to prove a number n prime. In contrast to modern algorithms, these are often either not polynomial-time algorithms, heavily depend on known factorisations of numbers such as n − 1 or are restricted to certain classes of candidates n. One of the simplest composite tests is based on Fermat's Little Theorem, stating that for any prime p and any integer a coprime to p we have ap−1 ≡ 1 (mod p). Hence we can prove an integer n > 1 composite by nding an integer b coprime to n with the property that bn−1 6≡ 1 (mod n). Denition 1. Let n > 1 be an odd integer and b > 1 with (b, n) = 1. We call n a pseudo-prime to base b if bn−1 ≡ 1 (mod n), that is, if the pair n, b satises the conclusion of Fermat's Little Theorem. Denition 2. If n is a positive odd composite integer which is a pseudo-prime to all bases b, we call n a Carmichael number. Using this criterion, we can easily check that e.g. 561 = 3 × 11 × 17 is a Carmichael number. It has been shown that there are innitely many Carmichael numbers [1].
  • Primality Testing and Sub-Exponential Factorization

    Primality Testing and Sub-Exponential Factorization

    Primality Testing and Sub-Exponential Factorization David Emerson Advisor: Howard Straubing Boston College Computer Science Senior Thesis May, 2009 Abstract This paper discusses the problems of primality testing and large number factorization. The first section is dedicated to a discussion of primality test- ing algorithms and their importance in real world applications. Over the course of the discussion the structure of the primality algorithms are devel- oped rigorously and demonstrated with examples. This section culminates in the presentation and proof of the modern deterministic polynomial-time Agrawal-Kayal-Saxena algorithm for deciding whether a given n is prime. The second section is dedicated to the process of factorization of large com- posite numbers. While primality and factorization are mathematically tied in principle they are very di⇥erent computationally. This fact is explored and current high powered factorization methods and the mathematical structures on which they are built are examined. 1 Introduction Factorization and primality testing are important concepts in mathematics. From a purely academic motivation it is an intriguing question to ask how we are to determine whether a number is prime or not. The next logical question to ask is, if the number is composite, can we calculate its factors. The two questions are invariably related. If we can factor a number into its pieces then it is obviously not prime, if we can’t then we know that it is prime. The definition of primality is very much derived from factorability. As we progress through the known and developed primality tests and factorization algorithms it will begin to become clear that while primality and factorization are intertwined they occupy two very di⇥erent levels of computational di⇧culty.
  • Primality Testing

    Primality Testing

    Syracuse University SURFACE Electrical Engineering and Computer Science - Technical Reports College of Engineering and Computer Science 6-1992 Primality Testing Per Brinch Hansen Syracuse University, School of Computer and Information Science, [email protected] Follow this and additional works at: https://surface.syr.edu/eecs_techreports Part of the Computer Sciences Commons Recommended Citation Hansen, Per Brinch, "Primality Testing" (1992). Electrical Engineering and Computer Science - Technical Reports. 169. https://surface.syr.edu/eecs_techreports/169 This Report is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in Electrical Engineering and Computer Science - Technical Reports by an authorized administrator of SURFACE. For more information, please contact [email protected]. SU-CIS-92-13 Primality Testing Per Brinch Hansen June 1992 School of Computer and Information Science Syracuse University Suite 4-116, Center for Science and Technology Syracuse, NY 13244-4100 Primality Testing1 PER BRINCH HANSEN Syracuse University, Syracuse, New York 13244 June 1992 This tutorial describes the Miller-Rabin method for testing the primality of large integers. The method is illustrated by a Pascal algorithm. The performance of the algorithm was measured on a Computing Surface. Categories and Subject Descriptors: G.3 [Probability and Statistics: [Proba­ bilistic algorithms (Monte Carlo) General Terms: Algorithms Additional Key Words and Phrases: Primality testing CONTENTS INTRODUCTION 1. FERMAT'S THEOREM 2. THE FERMAT TEST 3. QUADRATIC REMAINDERS 4. THE MILLER-RABIN TEST 5. A PROBABILISTIC ALGORITHM 6. COMPLEXITY 7. EXPERIMENTS 8. SUMMARY ACKNOWLEDGEMENTS REFERENCES 1Copyright@1992 Per Brinch Hansen Per Brinch Hansen: Primality Testing 2 INTRODUCTION This tutorial describes a probabilistic method for testing the primality of large inte­ gers.
  • 1 Problem Summary & Motivation 2 Previous Work

    1 Problem Summary & Motivation 2 Previous Work

    Primality Testing Report CS6150 December 9th, 2011 Steven Lyde, Ryan Lindeman, Phillip Mates 1 Problem Summary & Motivation Primes are used pervasively throughout modern society. E-commerce, privacy, and psuedorandom number generators all make heavy use of primes. There currently exist many different primality tests, each with their distinct advantages and disadvantages. These tests fall into two general categories, probabilistic and deterministic. Probabilistic algorithms determine with a certain probability that a number is prime, while deterministic algorithms prove whether a number is prime or not. Probabilistic algorithms are fast but contain error. Deterministic algorithms are much slower but produce proofs of correctness with no potential for error. Furthermore, some deterministic algorithms don't have proven upper bounds on their run time complexities, relying on heuristics. How can we combine these algorithms to create a fast, deterministic primality proving algorithm? Is there room for improvement of existing algorithms by hybridridizing them? This is an especially interesting question due to the fact that AKS is relatively new and unexplored. These are the questions we will try to answer in this report. 2 Previous Work For this project we focused on three algorithms, Miller-Rabin, Elliptic Curve Primality Proving (ECPP), and AKS. What is the state of the art for these different implementations? 2.1 ECPP Originally a purely theoretical construct, elliptic curves have recently proved themselves useful in many com- putational applications. Their elegance and power provide considerable leverage in primality proving and factorization studies [6]. From our reference [6], the following describes Elliptic curve fundamentals. Consider the following two or three variable homogeneous equations of degree-3: ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0 ax3 + bx2y + cxy2 + dy3 + ex2z + fxyz + gy2z + hxz2 + iyz2 + jz3 = 0 For these to be degree-3 polynomials, we must ensure at least a, b, c, d, or j is nonzero.
  • Prime Numbers and Discrete Logarithms

    Prime Numbers and Discrete Logarithms

    Lecture 11: Prime Numbers And Discrete Logarithms Lecture Notes on “Computer and Network Security” by Avi Kak ([email protected]) February 25, 2021 12:20 Noon ©2021 Avinash Kak, Purdue University Goals: • Primality Testing • Fermat’s Little Theorem • The Totient of a Number • The Miller-Rabin Probabilistic Algorithm for Testing for Primality • Python and Perl Implementations for the Miller-Rabin Primal- ity Test • The AKS Deterministic Algorithm for Testing for Primality • Chinese Remainder Theorem for Modular Arithmetic with Large Com- posite Moduli • Discrete Logarithms CONTENTS Section Title Page 11.1 Prime Numbers 3 11.2 Fermat’s Little Theorem 5 11.3 Euler’s Totient Function 11 11.4 Euler’s Theorem 14 11.5 Miller-Rabin Algorithm for Primality Testing 17 11.5.1 Miller-Rabin Algorithm is Based on an Intuitive Decomposition of 19 an Even Number into Odd and Even Parts 11.5.2 Miller-Rabin Algorithm Uses the Fact that x2 =1 Has No 20 Non-Trivial Roots in Zp 11.5.3 Miller-Rabin Algorithm: Two Special Conditions That Must Be 24 Satisfied By a Prime 11.5.4 Consequences of the Success and Failure of One or Both Conditions 28 11.5.5 Python and Perl Implementations of the Miller-Rabin 30 Algorithm 11.5.6 Miller-Rabin Algorithm: Liars and Witnesses 39 11.5.7 Computational Complexity of the Miller-Rabin Algorithm 41 11.6 The Agrawal-Kayal-Saxena (AKS) Algorithm 44 for Primality Testing 11.6.1 Generalization of Fermat’s Little Theorem to Polynomial Rings 46 Over Finite Fields 11.6.2 The AKS Algorithm: The Computational Steps 51 11.6.3 Computational Complexity of the AKS Algorithm 53 11.7 The Chinese Remainder Theorem 54 11.7.1 A Demonstration of the Usefulness of CRT 58 11.8 Discrete Logarithms 61 11.9 Homework Problems 65 Computer and Network Security by Avi Kak Lecture 11 Back to TOC 11.1 PRIME NUMBERS • Prime numbers are extremely important to computer security.