DOCSLIB.ORG

Computer Security Ch8:Introduction to Number Theory & Forouzan’S Book, CH9

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computer Security Ch8:Introduction to Number Theory & Forouzan’S Book, CH9 Computer Security Ch8:Introduction to Number Theory & Forouzan’s Book, CH9 Howon Kim 2019.4 Agenda Prime Numbers Fermat’s & Euler’s Theorems Testing for Primality The Chinese Remainder Theorem Discrete Logarithms 2 Agenda Prime Numbers Fermat’s & Euler’s Theorems Carmichael Number Testing for Primality The Chinese Remainder Theorem Discrete Logarithms 3 Three groups of positive integers Note A prime is divisible only by itself and 1. Ref: Forouzan Book, CH9 9.4 Prime Numbers prime numbers only have divisors of 1 and self they cannot be written as a product of other numbers note: 1 is prime, but is generally not of interest. So, 2 is the smallest prime. 1 is considered to be neither prime nor composite. eg. 2,3,5,7 are prime, 4,6,8,9,10 are not prime numbers are central to number theory • What is the smallest prime? Solution The smallest prime is 2, which is divisible by 2 (itself) and 1. • List the primes smaller than 10. Solution There are four primes less than 10: 2, 3, 5, and 7. It is interesting to note that the percentage of primes in the range 1 to 10 is 40%. The percentage decreases as the range increases. 5 Cardinality of Primes Infinite Number of Primes Note There is an infinite number of primes. Number of Primes 6 Prime Numbers list of prime number less than 2000 is: 907 1993 7 Ex) Checking for Primesness Given a number n, how can we determine if n is a prime? The answer is that we need to see if the number is divisible by all primes less than We know that this method is inefficient, but it is a good start. 8 Ex) Checking for Primesness Is 97 a prime? Solution The floor of 97 = 9. The primes less than 9 are 2, 3, 5, and 7. We need to see if 97 is divisible by any of these numbers. It is not, so 97 is a prime. Is 301 a prime? Solution The floor of 301 = 17. We need to check 2, 3, 5, 7, 11, 13, and 17. The numbers 2, 3, and 5 do not divide 301, but 7 does. Therefore 301 is not a prime. 9 Prime Factorisation to factor a number n is to write it as a product of other numbers: n=a x b x c note that factoring a number is relatively hard compared to multiplying the factors together to generate the number Any integer a>1 can be factored in a unique way as: aa a 12 t a p12 p pt where p1<p2<…<pt are prime numbers 91 = 7 x 13 4 2 2 3600 = 2 x 3 x 5 2 11011 = 7 x 11 x 13 It is expressed as the following form: ap a p,where each ap 0 pP 10 Relatively Prime Numbers & GCD two numbers a,b are relatively prime if have no common divisors apart from 1 Two numbers a and b are relatively prime if gcd(a, b) = 1. eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers 2 1 2 1 2 1 1 0 eg. 300=2 x3 x5 18=2 x3 hence GCD(18,300)=2 x3 x5 =6 11 Agenda Prime Numbers Fermat’s & Euler’s Theorems Carmichael Number Testing for Primality The Chinese Remainder Theorem Discrete Logarithms 12 Fermat's Theorem If p is prime and a is a positive integer not divisible by p (gcd(a,p)=1), then p-1 a = 1 (mod p) also known as Fermat’s Little Theorem An alternative form of Fermat’s theorem is also useful p a = a(mod p) This form does not require that a be relatively prime to p useful in public key and primality testing Cf. Fermat Last Theorem (Fermat’s Conjecture): n n n states that no three positive integers a, b, and c satisfy the equation a + b = c for any integer value of n strictly greater than two. The cases n = 1 and n = 2 have been known to have infinitely many solutions. Andrew Wiles 교수가 증명함 13 Fermat's Theorem mod p로 만들어진 set에 a를 곱해서(p와 서로소인)만들어진 set도 결국 원래의 set과 동일하다는 것을 증명함. p-1 Proof of a = 1 (mod p), gcd(a,p)=1 Consider the set of positive integers less than p. that is {1,2,…,p-1} and multiply each element by a, and then apply modulo p. We can get X={a mod p, 2a mod p, …,(p-1)a mod p}. Here, none of the elements of X is equal to zero because p does not divide a. Furthermore no two of the integers in X are equal. To see this, assume that ja=ka(mod p) where 1<=j<k<=p-1. Because a is relatively prime to p, gcd(a,p)=1, we can eliminate a from both sides of ja=ka(mod p). We now get j=k(mod p). This is impossible because j and k are both positive integers less than p. Therefore, we know that the (p-1) elements of X are all positive integers, with no two elements equal. We can conclude the X consists of the set of integers {1,2,…,p-1} in some order. Multiplying the numbers in both sets and taking the result mod p yields: a x 2a x … x (p-1)a = [ (1x2x…x(p-1)](mod p) p-1 a (p-1)! = (p-1)! (mod p) p-1 That is, a = 1 (mod p) 14 Fermat's Theorem In some parts of proof step, (In the case of p is 5 and a is 3) Gcd(a,p) = gcd(3,5) = 1 Consider the set of positive integers less than p: that is, {1,2,3,4} Also the set of X, X = a mod p, 2a mod p, …,(p-1)a mod p} = {3 mod p, 2*3 mod p, 3*3 mod p, 4*3 mod p} = {3,6,9,12} mod 5 = {3,1,4,2} mod 5 We can conclude the X consists of the set of integers {1,2,…,p-1} in some order. Multiplying the numbers in both sets and taking the result mod p yields: a x 2a x … x (p-1)a = [ (1x2x…x(p-1)](mod p) p-1 a (p-1)! = (p-1)! (mod p) p-1 That is, a = 1 (mod p) 15 Fermat's Theorem Example a = 7, p = 19 72 = 49 = 11(mod 19) 74 = 121 = 7(mod 19) 78 = 49 = 11(mod 19) 716 = 121 = 7(mod 19) ap-1 = 718 = 716 x 72 = 7 x 11 = 1(mod 19) 16 Euler Totient Function ø(n) Before presenting the Euler’s theorem. We need to introduce the important quantity in number theory complete set of residues is: 0..n-1 reduced set of residues, in which those numbers (residues) are relatively prime to n eg for n=10, complete set of residues is {0,1,2,3,4,5,6,7,8,9} reduced set of residues is {1,3,7,9} number of elements in reduced set of residues is called the Euler Totient Function ø(n) ø(10)=4 and the set is {1,3,7,9} 17 Euler Totient Function ø(n) Some values of Euler Totient Function ø(n) 18 Euler Totient Function ø(n) In general, we need prime factorization to compute ø(n) But, we have some easy ways to get ø(n) The order of for p (p prime) ø(p) = p-1 GF(p) is p-1 for p.q (p,q prime) ø(pq)= ø(p) x ø(q) =(p-1)x(q-1) ø(pqr)!= ø(p) x ø(q) x ø(r) eg. ø(37) = 36 ø(21) = ø(3)x ø(7)= (3–1)x(7–1) = 2x6 = 12 19 Euler Totient Function ø(n) Proof of ø(n)=ø(p) x ø(q): To see ø(n)=ø(p) x ø(q), consider that the set of positive integers less than n is the set {1,…,(pq- 1)}. The integers in this set that are not relatively prime to n are the set {p,2p,…,(q-1)p} and the set {q,2q, …, (p-1)q}. (since n=pxq) Accordingly, ø(n)=(pq-1)-[ (q-1) + (p-1)] p의 배수와 q의 배수를 빼야 즉, n=p x q이므로 n과 relatively = pq – (p+q)+1 prime하지 않은걸 뺀다는 의미 = (p-1) x (q-1) = ø(p)x ø(q) 20 Euler's Theorem A generalisation of Fermat's Theorem ø(n) a = 1 (mod n) for any a,n where gcd(a,n)=1 if n is prime and gcd(a,n)=1, this is Fermat’s theorem. That is, aø(n) = an-1 = 1 (mod n) eg. a=3;n=10; ø(10)=4; hence 34 = 81 = 1 mod 10 a=2;n=11; ø(11)=10; hence 210 = 1024 = 1 mod 11 Alternative form of the Euler’s theorem also useful. ø(n)+1 a = a (mod n), where gcd(a,n)=1 21 Agenda Prime Numbers Fermat’s & Euler’s Theorems Carmichael Number Testing for Primality The Chinese Remainder Theorem Discrete Logarithms 22 Recap: Fermat's Little Theorem Formulation 1: If p is prime, then for every number a with 1 ≤ a <p, ap-1 1 (mod p) Formulation 2: If p is prime, then for every number a with 1 ≤ a <p, ap a (mod p) Q4-5 Easy Primality Test? Is N prime? "composite" Pick some a with 1 < a < N means "not prime" N-1 Is a 1 (mod N)? If so, N is prime; if not, N is composite Nice try, but… Fermat's Little Theorem is not an "if and only if" condition.
Load more

Recommended publications
  • Lecture 9: Arithmetics II 1 Greatest Common Divisor
    DD2458, Problem Solving and Programming Under Pressure Lecture 9: Arithmetics II Date: 2008-11-10 Scribe(s): Marcus Forsell Stahre and David Schlyter Lecturer: Douglas Wikström This lecture is a continuation of the previous one, and covers modular arithmetic and topics from a branch of number theory known as elementary number theory. Also, some abstract algebra will be discussed. 1 Greatest Common Divisor Definition 1.1 If an integer d divides another integer n with no remainder, d is said to be a divisor of n. That is, there exists an integer a such that a · d = n. The notation for this is d | n. Definition 1.2 A common divisor of two non-zero integers m and n is a positive integer d, such that d | m and d | n. Definition 1.3 The Greatest Common Divisor (GCD) of two positive integers m and n is a common divisor d such that every other common divisor d0 | d. The notation for this is GCD(m, n) = d. That is, the GCD of two numbers is the greatest number that is a divisor of both of them. To get an intuition of what GCD is, let’s have a look at this example. Example Calculate GCD(9, 6). Say we have 9 black and 6 white blocks. We want to put the blocks in boxes, but every box has to be the same size, and can only hold blocks of the same color. Also, all the boxes must be full and as large as possible . Let’s for example say we choose a box of size 2: As we can see, the last box of black bricks is not full.
    [Show full text]
  • 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.
    [Show full text]
  • Enclave Security and Address-Based Side Channels
    Graz University of Technology Faculty of Computer Science Institute of Applied Information Processing and Communications IAIK Enclave Security and Address-based Side Channels Assessors: A PhD Thesis Presented to the Prof. Stefan Mangard Faculty of Computer Science in Prof. Thomas Eisenbarth Fulfillment of the Requirements for the PhD Degree by June 2020 Samuel Weiser Samuel Weiser Enclave Security and Address-based Side Channels DOCTORAL THESIS to achieve the university degree of Doctor of Technical Sciences; Dr. techn. submitted to Graz University of Technology Assessors Prof. Stefan Mangard Institute of Applied Information Processing and Communications Graz University of Technology Prof. Thomas Eisenbarth Institute for IT Security Universit¨atzu L¨ubeck Graz, June 2020 SSS AFFIDAVIT I declare that I have authored this thesis independently, that I have not used other than the declared sources/resources, and that I have explicitly indicated all material which has been quoted either literally or by content from the sources used. The text document uploaded to TUGRAZonline is identical to the present doctoral thesis. Date, Signature SSS Prologue Everyone has the right to life, liberty and security of person. Universal Declaration of Human Rights, Article 3 Our life turned digital, and so did we. Not long ago, the globalized commu- nication that we enjoy today on an everyday basis was the privilege of a few. Nowadays, artificial intelligence in the cloud, smartified handhelds, low-power Internet-of-Things gadgets, and self-maneuvering objects in the physical world are promising us unthinkable freedom in shaping our personal lives as well as society as a whole. Sadly, our collective excitement about the \new", the \better", the \more", the \instant", has overruled our sense of security and privacy.
    [Show full text]
  • Independence of the Miller-Rabin and Lucas Probable Prime Tests
    Independence of the Miller-Rabin and Lucas Probable Prime Tests Alec Leng Mentor: David Corwin March 30, 2017 1 Abstract In the modern age, public-key cryptography has become a vital component for se- cure online communication. To implement these cryptosystems, rapid primality test- ing is necessary in order to generate keys. In particular, probabilistic tests are used for their speed, despite the potential for pseudoprimes. So, we examine the commonly used Miller-Rabin and Lucas tests, showing that numbers with many nonwitnesses are usually Carmichael or Lucas-Carmichael numbers in a specific form. We then use these categorizations, through a generalization of Korselt’s criterion, to prove that there are no numbers with many nonwitnesses for both tests, affirming the two tests’ relative independence. As Carmichael and Lucas-Carmichael numbers are in general more difficult for the two tests to deal with, we next search for numbers which are both Carmichael and Lucas-Carmichael numbers, experimentally finding none less than 1016. We thus conjecture that there are no such composites and, using multi- variate calculus with symmetric polynomials, begin developing techniques to prove this. 2 1 Introduction In the current information age, cryptographic systems to protect data have become a funda- mental necessity. With the quantity of data distributed over the internet, the importance of encryption for protecting individual privacy has greatly risen. Indeed, according to [EMC16], cryptography is allows for authentication and protection in online commerce, even when working with vital financial information (e.g. in online banking and shopping). Now that more and more transactions are done through the internet, rather than in person, the importance of secure encryption schemes is only growing.
    [Show full text]
  • Improving the Speed and Accuracy of the Miller-Rabin Primality Test
    Improving the Speed and Accuracy of the Miller-Rabin Primality Test Shyam Narayanan Mentor: David Corwin MIT PRIMES-USA Abstract Currently, even the fastest deterministic primality tests run slowly, with the Agrawal- Kayal-Saxena (AKS) Primality Test runtime O~(log6(n)), and probabilistic primality tests such as the Fermat and Miller-Rabin Primality Tests are still prone to false re- sults. In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer n. We also extend the Miller- '(n) Rabin Theorem by determining when the number of nonwitnesses N(n) equals 4 5 and by proving that for all n, if N(n) > · '(n) then n must be of one of these 3 32 forms: n = (2x + 1)(4x + 1), where x is an integer, n = (2x + 1)(6x + 1), where x is an integer, n is a Carmichael number of the form pqr, where p, q, r are distinct primes congruent to 3 (mod 4). We then find witnesses to certain forms of composite numbers with high rates of nonwitnesses and find that Jacobi nonresidues and 2 are both valuable bases for the Miller-Rabin test. Finally, we investigate the frequency of strong pseudoprimes further and analyze common patterns using MATLAB. This work is expected to result in a faster and better primality test for large integers. 1 1 Introduction Data is growing at an astoundingly rapid rate, and better information security is re- quired to protect increasing quantities of data. Improved data protection requires more sophisticated cryptographic methods.
    [Show full text]
  • Factoring & Primality
    Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount of research over the years. These prob- lems started receiving attention in the mathematics community far before the appearance of computer science, however the emergence of the latter has given them additional importance. Gauss himself wrote in 1801 \The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." 1 Factorization Problems The following result is known as the Fundamental Theorem of Arithmetic: Theorem 1 Every n 2 Z with n 6= 0 can be written as: r Y ei n = ± pi i=1 where pi are distinct prime numbers and ei are positive integers. Moreover, this representa- tion is unique (except for term re-ordering). This celebrated result (already known at Euclid's time) unfortunately does not tell us any- thing about computing the primes pi or the powers ei except for their existence. This naturally leads to the following problem statement known as the integer factorization problem: Problem 1 Given n 2 Z with n 6= 0, compute primes pi and positive integers ei for i = Qr ei 1; :::; r s.t. i=1 pi = n. Clearly, if n is a prime number then the factorization described above is trivial.
    [Show full text]
  • On El-Gamal and Fermat's Primality Test
    On El-Gamal and Fermat’s Primality Test IRVAN JAHJA / 13509099 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia [email protected] May 6, 2012 Abstract Gamal reacts with non prime p, and then discuss the case with Carmichael Numbers. The paper then closes We discuss the feasibility of using the simple Fermat’s with a possible way to take advantage of the behavior Primality Test to generate prime required as a key to the of El-Gamal with composite p. public key El-Gamal cryptosystem. This option is ap- Throughout this paper, pseudocodes in Python-like pealing due to the simplicity of Fermat’s Primality Test, languages will be presented as to make things less ab- but is marred with the existance of Carmichael Num- stract. bers. 2 Background Theory 1 Introduction 2.1 Fermat Primality Testing 1.1 Background Fermat Primality Testing is one of the fastest primality El-Gamal is one of the most used public key cryptosys- testing algorithm. Interested readers are referred to the tem. This accounts to its simplicity of implementation book by Cormen et. al [20], and we will only give a as well for being one of the earliest relatively secure brief explanation in this section. public key cryptosystem to be published. For instance, The Fermat’s Primality Testing is based on the Fer- GNU Privacy Guard software and recent versions of mat’s Little Theorem: PGP are major user of this system. If p is prime and 1 ≤ p < p, then ap−1 ≡ 1 (mod p).
    [Show full text]
  • RSA and Public Key Cryptography
    RSA and Public Key Cryptography Chester Rebeiro IIT Madras CR STINSON : chapter 5, 6 Ciphers • Symmetric Algorithms – EncrypAon and DecrypAon use the same key – i.e. KE = KD – Examples: • Block Ciphers : DES, AES, PRESENT, etc. • Stream Ciphers : A5, Grain, etc. • Asymmetric Algorithms – EncrypAon and DecrypAon keys are different – KE ≠ KD – Examples: • RSA • ECC CR 2 Asymmetric Key Algorithms KE KD untrusted communicaon link Alice E D Bob #%AR3Xf34^$ “APack at Dawn!!” Plaintext encrypAon (ciphertext) decrypon “APack at Dawn!!” The Key K is a secret Encrypon Key KE not same as decryp<on key KD Advantage : No need of secure KE known as Bob’s public key; K is Bob’s private key key exchange between Alice and D Bob Asymmetric key algorithms based on trapdoor one-way func<ons CR 3 One Way Func<ons • Easy to compute in one direcAon • Once done, it is difficult to inverse Press to lock Once locked it is (can be easily done) difficult to unlock without a key CR 4 Trapdoor One Way Func<on • One way funcAon with a trapdoor • Trapdoor is a special funcAon that if possessed can be used to easily invert the one way trapdoor Locked (difficult to unlock) Easily Unlocked CR 5 Public Key Cryptography (An Anology) • Alice puts message into box and locks it • Only Bob, who has the key to the lock can open it and read the message CR 6 Mathema<cal Trapdoor One way funcons • Examples – Integer Factorizaon (in NP, maybe NP-complete) • Given P, Q are two primes • and N = P * Q – It is easy to compute N – However given N it is difficult to factorize into P and Q •
    [Show full text]
  • The Miller-Rabin Randomized Primality Test
    Introduction to Algorithms (CS 482) Cornell University Instructor: Bobby Kleinberg Lecture Notes, 5 May 2010 The Miller-Rabin Randomized Primality Test 1 Introduction Primality testing is an important algorithmic problem. In addition to being a fun- damental mathematical question, the problem of how to determine whether a given number is prime has tremendous practical importance. Every time someone uses the RSA public-key cryptosystem, they need to generate a private key consisting of two large prime numbers and a public key consisting of their product. To do this, one needs to be able to check rapidly whether a number is prime. The simplestp algorithm to test whether n is prime is trial division:p for k = 2; 3;:::; b nc test whether n ≡ 0 (mod k). This runs in time O( n log2(n)), but this running time is exponential in the input size since the input represents n as a binary number with dlog2(n)e digits. (A good public key these days relies on using prime numbers with at least 2250 binary digits; testing whether such a number is prime using trial division would require at least 2125 operations.) In 1980, Michael Rabin discovered a randomized polynomial-time algorithm to test whether a number is prime. It is called the Miller-Rabin primality test because it is closely related to a deterministic algorithm studied by Gary Miller in 1976. This is still the most practical known primality testing algorithm, and is widely used in software libraries that rely on RSA encryption, e.g. OpenSSL. 2 Randomized algorithms What does it mean to say that there is a randomized polynomial-time algorithm to solve a problem? Here are some definitions to make this notion precise.
    [Show full text]
  • Improved Error Bounds for the Fermat Primality Test on Random Inputs
    IMPROVED ERROR BOUNDS FOR THE FERMAT PRIMALITY TEST ON RANDOM INPUTS JARED D. LICHTMAN AND CARL POMERANCE Abstract. We investigate the probability that a random odd composite num- ber passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to 2200, our results improve on prior estimates by close to 3 orders of magnitude. 1. Introduction Part of the basic landscape in elementary number theory is the Fermat congru- ence: If n is a prime and 1 ≤ b ≤ n − 1, then (1.1) bn−1 ≡ 1 (mod n): It is attractive in its simplicity and ease of verification: using fast arithmetic sub- routines, (1.1) may be checked in (log n)2+o(1) bit operations. Further, its converse (apparently) seldom lies. In practice, if one has a large random number n that sat- isfies (1.1) for a random choice for b, then almost certainly n is prime. To be sure, there are infinitely many composites (the Carmichael numbers) that satisfy (1.1) for all b coprime to n, see [1]. And in [2] it is shown that there are infinitely many Carmichael numbers n such that (1.1) holds for (1−o(1))n choices for b in [1; n−1]. (Specifically, for each fixed k there are infinitely many Carmichael numbers n such that the probability a random b in [1; n − 1] has (b; n) > 1 is less than 1= logk n.) However, Carmichael numbers are rare, and if a number n is chosen at random, it is unlikely to be one.
    [Show full text]
  • Average Case Error Estimates of the Strong Lucas Probable Prime Test
    Average Case Error Estimates of the Strong Lucas Probable Prime Test Master Thesis Semira Einsele Monday 27th July, 2020 Advisors: Prof. Dr. Kenneth Paterson Prof. Dr. Ozlem¨ Imamoglu Department of Mathematics, ETH Z¨urich Abstract Generating and testing large prime numbers is crucial in many public- key cryptography algorithms. A common choice of a probabilistic pri- mality test is the strong Lucas probable prime test which is based on the Lucas sequences. Throughout this work, we estimate bounds for average error behaviour of this test. To do so, let us consider a procedure that draws k-bit odd integers in- dependently from the uniform distribution, subjects each number to t independent iterations of the strong Lucas probable prime test with randomly chosen bases, and outputs the first number that passes all t tests. Let qk,t denote the probability that this procedurep returns a com- 2 2.3− k posite number. We show that qk,1 < log(k)k 4 for k ≥ 2. We see that slightly modifying the procedure, using trial division by the first l odd primes, gives remarkable improvements in this error analysis. Let qk,l,t denote the probability that the now modified procedurep returns a 2 1.87727− k composite number. We show that qk,128,1 < k 4 for k ≥ 2. We also give general bounds for both qk,t and qk,l,t when t ≥ 2, k ≥ 21 and l 2 N. In addition, we treat the numbers, that add the most to our probability estimate differently in the analysis, and give improved bounds for large t.
    [Show full text]
  • PRIMALITY TESTING a Journey from Fermat to AKS
    PRIMALITY TESTING A Journey from Fermat to AKS Shreejit Bandyopadhyay Mathematics and Computer Science Chennai Mathematical Institute Abstract We discuss the Fermat, Miller-Rabin, Solovay-Strassen and the AKS tests for primality testing. We also speak about the probabilistic or de- terministic nature of these tests and their time-complexities. 1 Introduction We all know what a prime number is { it's a natural number p 6= 1 for which the only divisors are 1 and p. This essentially means that for a prime number p, gcd(a; p)=1 8a 2 N and1 ≤ a ≤ (p − 1). So the value of Euler's totient function for a prime p, φ(p) equals (p − 1) as all the (p − 1) values of a in (1,2,...p − 1) satisfy gcd(a; p)=1. A primality test, the topic of this paper, is simply an algo- rithm that tests, either probabilistically or deterministically, whether or not a given input number is prime. A general primality test does not provide us with a prime factorisation of a number not found to be prime, but simply labels it as composite. In cryptography, for example, we often need the generation of large primes and one technique for this is to pick a random number of requisite size and detrmine if it's prime. The larger the number, the greater will be the time required to test this and this is what prompts us to search for efficient primality tests that are polynomial in complexity. Note that the desired complexity is logarithmic in the number itself and hence polynomial in its bit{size as a num- ber n requires O(log n) bits for its binary representation.
    [Show full text]