DOCSLIB.ORG

The Improvement of Elliptic Curve Factorization Method to Recover RSA’S Prime Factors

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Improvement of Elliptic Curve Factorization Method to Recover RSA’S Prime Factors S S symmetry Article The Improvement of Elliptic Curve Factorization Method to Recover RSA’s Prime Factors Kritsanapong Somsuk Department of Computer and Communication Engineering, Faculty of Technology, Udon Thani Rajabhat University, UDRU, Udon Thani 41000, Thailand; [email protected] Abstract: Elliptic Curve Factorization Method (ECM) is the general-purpose factoring method used in the digital computer era. It is based on the medium length of the modulus; ECM is an efficient algorithm when the length of modulus is between 40 and 50 digits. In fact, the main costs for each iteration are modular inverse, modular multiplication, modular square and greatest common divisor. However, when compared to modular multiplication and modular square, the costs of modular inverse and greatest common divisor are very high. The aim of this paper is to improve ECM in order to reduce the costs to compute both of modular inverse and greatest common divisor. The proposed method is called Fast Elliptic Curve Factorization Method (F-ECM). For every two adjacent points on the curve, only one modular inverse and one greatest common divisor will be computed. That means it implies that the costs in both of them can be split in half. Furthermore, the length of modulus in the experiment spans from 30 to 65 bits. The experimental results show that F-ECM can finish the task faster than ECM for all cases of the modulus. Furthermore, the computation time is reduced by 30 to 38 percent. Keywords: ECM; F-ECM; computation time; RSA; modulus Citation: Somsuk, K. The Improvement of Elliptic Curve Factorization Method to Recover 1. Introduction RSA’s Prime Factors. Symmetry 2021, 13, 1314. https://doi.org/10.3390/ RSA [1] is one of the most well-known cryptography algorithms in the digital com- sym13081314 puter era. It is classified as asymmetric key cryptography or public key cryptography [2]. In fact, this algorithm which was proposed in 1977 by Ron Rivest, Adi Shamir and Leonard Academic Editor: Debiao He Adleman, can be used to secure secret information transferred via an unsecure channel as well as sign digital signatures. RSA algorithm, on the other hand, is based on the integer Received: 26 June 2021 factorization problem [3,4]. As a result, when one of the two prime factors is revealed, RSA Accepted: 20 July 2021 is broken. Furthermore, there are two types of integer factorization algorithms [5]. The Published: 21 July 2021 first category is known as special-purpose factoring. The efficiency of each algorithm in this group is determined by the properties of prime factors. The other group is known Publisher’s Note: MDPI stays neutral as general-purpose factoring, and the algorithms are based on the length of the modulus. with regard to jurisdictional claims in Assuming that two strong prime factors and 1024 bits of the modulus are selected, none of published maps and institutional affil- proposed integer factorization algorithms can break RSA in polynomial time by using the iations. digital computer. Elliptic Curve Cryptography (ECC) [6,7] is another public key cryptography that can be used for data security as well as digital signature. In addition, Neal Koblitz and Victor S. Miller proposed ECC in 1985. Despite the fact that the bit-length of ECC is shorter, the Copyright: © 2021 by the author. security level is the same as RSA. ECC, on the other hand, differs from RSA in which it Licensee MDPI, Basel, Switzerland. is based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) [8–10]. Moreover, This article is an open access article Elliptic Curve can be used to factor the composite number as well. That is, it is possible to distributed under the terms and break RSA. The Lenstra Elliptic Curve Factorization or Elliptic Curve Factorization Method conditions of the Creative Commons (ECM) is used for this task. However, ECM requires a large number of modular inverses Attribution (CC BY) license (https:// and greatest common divisors, which are known as the expensive costs. That is, it takes a creativecommons.org/licenses/by/ significant amount of time to finish the task. 4.0/). Symmetry 2021, 13, 1314. https://doi.org/10.3390/sym13081314 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1314 2 of 15 The modified method for speeding up ECM is proposed in this paper. In fact, the proposed method is known as the Fast Elliptic Curve Factorization Method (F-ECM). The main idea is to reduce the number of modular inverses and greatest common divisors in order to shorten the computation time. Even though the number of modular multiplications and modular squares are increased, these costs are very low when compared to modular inverse and greatest common divisor. In fact, every classical algorithm linked to the RSA and ECC algorithms is the digital computer-based. If the digital computer era is not disrupted, they are all still effective. Therefore, it implies that F-ECM is still an efficient method for the medium length of the modulus in the digital computer era. However, if the quantum computer that is being developed becomes practical, all algorithms pertaining to digital technology, including RSA, ECC and the block chain that is controlled by cryptography, will be insecure. For example, RSA may be easily broken by using Shor’s quantum algorithm [11] when the quantum computer is completely used. As a result, if such a scenario occurs, new security algorithms based on quantum computers may be devised in order to control all technology at the time. The rest of the paper is organized as follows. In Section2, RSA and ECC which are public key cryptography are reviewed. Some integer factorization algorithms will be mentioned in Section3. The overviews of quantum computing and Shor’s Factoring algorithm will be mentioned in Section4. The proposed method is presented in Section5. Loop analysis will be discussed in Section6. In Section7, the experimental results about the comparison between ECM and F-ECM are shown. The conclusion is discussed in the last section. 2. Overviews of RSA and ECC The idea behind public key cryptography is that the encryption and decryption processes necessitate the use of a pair of keys that are mathematically related to one another. The first key, named the public key, is made available to all group members. The owner keeps the other key, known as the private key, hidden. In fact, if one of the keys is chosen for encryption, the other must be chosen for decryption. Furthermore, for data security, the public key is used in the encryption process. In the case of digital signatures, however, the private key is used in this process. This section examines two types of public key cryptography algorithms, RSA and ECC. 2.1. RSA RSA algorithm is a type of public key cryptography. Its security is based on the integer factorization problem. As a result, it should be assigned at least 1024 bits to avoid intruders attacking rapidly. Furthermore, RSA can be used for a variety of tasks, including data security and digital signatures. For RSA, there are three processes: the key generation algorithm, the encryption and the decryption. However, the processes for data security are as follows: Process 1 (Key generation Algorithm): it is the algorithm to generate a pair of keys and the modulus. There are four steps as follows: Step 1: Choose two prime numbers randomly, p and q Step 2: Calculate the modulus, n = p ∗ q, and calculate the Euler totient function, F(n) = (p − 1) ∗ (q − 1), using the Extended Euclidean Algorithms [12–14] Step 3: Choose the public key (e) from the following conditions, 1 < e < F(n) and gcd (e, F(n)) = 1 Step 4: Compute the private key (d) from the following equation, e ∗ d mod F(n) = 1 Where, the public key is {e, n} and the private key is {d, n} Process 2 (Encryption): senders must convert the original plaintext (m) into the unreadable message or the ciphertext (c). There are three steps as follows: Step 1: Receive the receiver’s public key {e, n} Step 2: Represent m as the positive integer, M, where 1 < M < n Symmetry 2021, 13, 1314 3 of 15 Step 3: Encrypt M by using the Equation (1): c = Memod n (1) Step 4: Send c to receivers Process 3 (Decryption): After receiving c, receivers can recover m by using the following steps: Step 1: Use the private key {d, n} to decrypt c by using the Equation (2): M = cdmod n (2) Step 2: Transform M as the original plaintext, m For this system, p and q are the attackers’ target, because F(n) can be computed rapidly to recover d. 2.2. Elliptic Curve Cryptography Another type of public key cryptography is Elliptic Curve Cryptography (ECC). This method comes in a variety of forms, including ECC over Field of Characteristic Two, ECC over Field of Characteristic Three and ECC over Finite Field (Zp), where p is a prime number. However, only ECC over Finite Field (Zp) is focused in this paper. Assuming that p is a prime number, the ECC’s equation over Finite Field (Zp) is as follows: y2 = x3 + ax + b mod p (3) where, 4a3 + 27b2 mod p 6= 0. Assuming that P = (xp, yp) and Q = (xq, yq) are the points on the curve, the main processes for finding the new point on the curve are point addition and point doubling. In fact, there are two cases to find R = (xr, yr) = P + Q as follows: Case 1 (P = Q): Point doubling is required: 2 3xp + a m = mod p (4) 2yp 2 xr = m − 2xp mod p (5) yr = m(xp − xr) − yp mod p (6) Case 2 (P 6= Q): Point addition is required: yq − yp m = mod p (7) xq − xp 2 xr = m − xq − xp mod p (8) yr = m(xp − xr) − yp mod p (9) Assuming that I, M, S and G are represented as the number of modular inverses, number of modular multiplications, number of modular squares and number of greatest common divisor computations, point doubling requires 1I, 2M and 2S.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Computing Prime Divisors in an Interval
    MATHEMATICS OF COMPUTATION Volume 84, Number 291, January 2015, Pages 339–354 S 0025-5718(2014)02840-8 Article electronically published on May 28, 2014 COMPUTING PRIME DIVISORS IN AN INTERVAL MINKYU KIM AND JUNG HEE CHEON Abstract. We address the problem of finding a nontrivial divisor of a com- posite integer when it has a prime divisor in an interval. We show that this problem can be solved in time of the square root of the interval length with a similar amount of storage, by presenting two algorithms; one is probabilistic and the other is its derandomized version. 1. Introduction Integer factorization is one of the most challenging problems in computational number theory. It has been studied for centuries, and it has been intensively in- vestigated after introducing the RSA cryptosystem [18]. The difficulty of integer factorization has been examined not only in pure factoring situations but also in various modified situations. One such approach is to find a nontrivial divisor of a composite integer when it has prime divisors of special form. These include Pol- lard’s p − 1 algorithm [15], Williams’ p + 1 algorithm [20], and others. On the other hand, owing to the importance and the usage of integer factorization in cryptog- raphy, one needs to examine this problem when some partial information about divisors is revealed. This side information might be exposed through the proce- dures of generating prime numbers or through some attacks against crypto devices or protocols. This paper deals with integer factorization given the approximation of divisors, and it is motivated by the above mentioned research.
    [Show full text]
  • Integer Factorization and Computing Discrete Logarithms in Maple
    Integer Factorization and Computing Discrete Logarithms in Maple Aaron Bradford∗, Michael Monagan∗, Colin Percival∗ [email protected], [email protected], [email protected] Department of Mathematics, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada. 1 Introduction As part of our MITACS research project at Simon Fraser University, we have investigated algorithms for integer factorization and computing discrete logarithms. We have implemented a quadratic sieve algorithm for integer factorization in Maple to replace Maple's implementation of the Morrison- Brillhart continued fraction algorithm which was done by Gaston Gonnet in the early 1980's. We have also implemented an indexed calculus algorithm for discrete logarithms in GF(q) to replace Maple's implementation of Shanks' baby-step giant-step algorithm, also done by Gaston Gonnet in the early 1980's. In this paper we describe the algorithms and our optimizations made to them. We give some details of our Maple implementations and present some initial timings. Since Maple is an interpreted language, see [7], there is room for improvement of both implementations by coding critical parts of the algorithms in C. For example, one of the bottle-necks of the indexed calculus algorithm is finding and integers which are B-smooth. Let B be a set of primes. A positive integer y is said to be B-smooth if its prime divisors are all in B. Typically B might be the first 200 primes and y might be a 50 bit integer. ∗This work was supported by the MITACS NCE of Canada. 1 2 Integer Factorization Starting from some very simple instructions | \make integer factorization faster in Maple" | we have implemented the Quadratic Sieve factoring al- gorithm in a combination of Maple and C (which is accessed via Maple's capabilities for external linking).
    [Show full text]
  • Integer Factorization with a Neuromorphic Sieve
    Integer Factorization with a Neuromorphic Sieve John V. Monaco and Manuel M. Vindiola U.S. Army Research Laboratory Aberdeen Proving Ground, MD 21005 Email: [email protected], [email protected] Abstract—The bound to factor large integers is dominated by to represent n, the computational effort to factor n by trial the computational effort to discover numbers that are smooth, division grows exponentially. typically performed by sieving a polynomial sequence. On a Dixon’s factorization method attempts to construct a con- von Neumann architecture, sieving has log-log amortized time 2 2 complexity to check each value for smoothness. This work gruence of squares, x ≡ y mod n. If such a congruence presents a neuromorphic sieve that achieves a constant time is found, and x 6≡ ±y mod n, then gcd (x − y; n) must be check for smoothness by exploiting two characteristic properties a nontrivial factor of n. A class of subexponential factoring of neuromorphic architectures: constant time synaptic integration algorithms, including the quadratic sieve, build on Dixon’s and massively parallel computation. The approach is validated method by specifying how to construct the congruence of by modifying msieve, one of the fastest publicly available integer factorization implementations, to use the IBM Neurosynaptic squares through a linear combination of smooth numbers [5]. System (NS1e) as a coprocessor for the sieving stage. Given smoothness bound B, a number is B-smooth if it does not contain any prime factors greater than B. Additionally, let I. INTRODUCTION v = e1; e2; : : : ; eπ(B) be the exponents vector of a smooth number s, where s = Q pvi , p is the ith prime, and A number is said to be smooth if it is an integer composed 1≤i≤π(B) i i π (B) is the number of primes not greater than B.
    [Show full text]
  • Cracking RSA with Various Factoring Algorithms Brian Holt
    Cracking RSA with Various Factoring Algorithms Brian Holt 1 Abstract For centuries, factoring products of large prime numbers has been recognized as a computationally difficult task by mathematicians. The modern encryption scheme RSA (short for Rivest, Shamir, and Adleman) uses products of large primes for secure communication protocols. In this paper, we analyze and compare four factorization algorithms which use elementary number theory to assess the safety of various RSA moduli. 2 Introduction The origins of prime factorization can be traced back to around 300 B.C. when Euclid's theorem and the unique factorization theorem were proved in his work El- ements [3]. For nearly two millenia, the simple trial division algorithm was used to factor numbers into primes. More efficient means were studied by mathemeticians during the seventeenth and eighteenth centuries. In the 1640s, Pierre de Fermat devised what is known as Fermat's factorization method. Over one century later, Leonhard Euler improved upon Fermat's factorization method for numbers that take specific forms. Around this time, Adrien-Marie Legendre and Carl Gauss also con- tributed crucial ideas used in modern factorization schemes [3]. Prime factorization's potential for secure communication was not recognized until the twentieth century. However, economist William Jevons antipicated the "one-way" or ”difficult-to-reverse" property of products of large primes in 1874. This crucial con- cept is the basis of modern public key cryptography. Decrypting information without access to the private key must be computationally complex to prevent malicious at- tacks. In his book The Principles of Science: A Treatise on Logic and Scientific Method, Jevons challenged the reader to factor a ten digit semiprime (now called Jevon's Number)[4].
    [Show full text]
  • Primality Testing and Integer Factorization in Public-Key Cryptography Pdf, Epub, Ebook
    PRIMALITY TESTING AND INTEGER FACTORIZATION IN PUBLIC-KEY CRYPTOGRAPHY PDF, EPUB, EBOOK Song Y. Yan | 371 pages | 29 Nov 2010 | Springer-Verlag New York Inc. | 9781441945860 | English | New York, NY, United States Primality Testing and Integer Factorization in Public-Key Cryptography PDF Book RA 39 math. Blog at WordPress. The two equations give. Book Description Condition: New. Stock Image. The square and multiply algorithm is equivalent to the Python one-liner pow x, k, p. One can use a crude version of the prime number theorem to get the upper bound on. Since has order greater than in , we see that the number of residue classes of the form is at least. Factoring integers with elliptic curves. HO 12 math. They matter because of an important class of cryptosystems, in a multibillion dollar industry. You have to search a smaller space until you brute force the key. View 2 excerpts, references background and methods. By Terence Tao. So to establish the proposition it suffices to show that all these products are distinct. View all copies of this ISBN edition:. Hello I think that binomial test is also suitable for factoring. View 9 excerpts, references background. Each shard is capable of processing transactions in parallel, yielding a high throughput for the network. Sorry, your blog cannot share posts by email. Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Note that can be computed in time for any fixed by expressing in binary, and repeatedly squaring. I implemented those steps in the Python code below.
    [Show full text]
  • 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.
    [Show full text]
  • Implementing and Comparing Integer Factorization Algorithms
    Implementing and Comparing Integer Factorization Algorithms Jacqueline Speiser jspeiser p Abstract by choosing B = exp( logN loglogN)) and let the factor base be the set of all primes smaller than B. Next, Integer factorization is an important problem in modern search for positive integers x such that x2 mod N is B- cryptography as it is the basis of RSA encryption. I have smooth, meaning that all the factors of x2 are in the factor implemented two integer factorization algorithms: Pol- 2 e1 e2 ek base. For all B-smooth numbers xi = p p ::: p , lard’s rho algorithm and Dixon’s factorization method. 2 record (xi ;~ei). After we have enough of these relations, While the results are not revolutionary, they illustrate we can solve a system of linear equations to find some the software design difficulties inherent to integer fac- subset of the relations such that ∑~ei =~0 mod 2. (See the torization. The code for this project is available at Implementation section for details on how this is done.) https://github.com/jspeiser/factoring. Note that if k is the size of our factor base, then we only need k + 1 relations to guarantee that such a solution 1 Introduction exists. We have now found a congruence of squares, 2 2 2 ∑i ei1 ∑i eik a = xi and b = p1 ::: pk . This implies that The integer factorization problem is defined as follows: (a + b)(a − b) = 0 mod N, which means that there is a given a composite number N, find two integers x and y 50% chance that gcd(a−b;N) factorspN.
    [Show full text]
  • The Elliptic Curve Method and Other Integer Factorization Algorithms
    The Elliptic Curve Method and Other Integer Factorization Algorithms John Wright April 12, 2012 Contents 1 Introduction 2 2 Preliminaries 3 2.1 Greatest common divisors and modular arithmetic . 3 2.2 Basic definitions and theorems . 6 2.3 RSACryptosystem........................ 9 3 Factorization Algorithms 11 3.1 TheSieveofEratosthenes . 11 3.2 TrialDivision ........................... 12 3.3 Fermat’sLittleTheorem . 13 3.4 PseudoprimeTest......................... 14 3.5 StrongPseudoprimeTest . 16 3.6 AmethodofFermat ....................... 17 3.7 TheQuadraticSieve . 20 3.8 Pollard Rho Factorization Method . 22 4 Elliptic curves 24 4.1 Additiononanellipticcurve . 28 4.2 Reduction of elliptic curves defined modulo N ......... 31 4.3 Reduction of curves modulo p .................. 33 4.4 Lenstra’s Elliptic Curve Integer Factorization Method . 34 4.5 TheECMintheprojectiveplane . 36 5 Improving the Elliptic Curve Method 38 5.1 MontgomeryCurves . 38 5.2 Addition for elliptic curves in Montgomery form . 42 5.3 Montgomery multiplication . 47 5.4 Recentdevelopments . 49 5.5 Conclusion ............................ 50 1 Chapter 1 Introduction The Fundamental Theorem of Arithmetic, first proved by Gauss [2], states that every positive integer has a unique factorization into primes. That is, for every positive integer N, a1 a2 ak N = p1 p2 ...pk where the pi’s are distinct primes and each ai is a positive integer. This paper is motivated by a computational question: given an arbitrary integer N, how might we find a non-trivial factor of N? That is, a factor of N other than N and 1. ± ± While it is computationally easy to multiply numbers together, factor- ing a general integer is “generally believed to be hard” [4].
    [Show full text]
  • Integer Factoring
    Designs, Codes and Cryptography, 19, 101–128 (2000) c 2000 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Integer Factoring ARJEN K. LENSTRA [email protected] Citibank, N.A., 1 North Gate Road, Mendham, NJ 07945-3104, USA Abstract. Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. Keywords: Integer factorization, quadratic sieve, number field sieve, elliptic curve method, Morrison–Brillhart Approach 1. Introduction Factoring a positive integer n means finding positive integers u and v such that the product of u and v equals n, and such that both u and v are greater than 1. Such u and v are called factors (or divisors)ofn, and n = u v is called a factorization of n. Positive integers that can be factored are called composites. Positive integers greater than 1 that cannot be factored are called primes. For example, n = 15 can be factored as the product of the primes u = 3 and v = 5, and n = 105 can be factored as the product of the prime u = 7 and the composite v = 15. A factorization of a composite number is not necessarily unique: n = 105 can also be factored as the product of the prime u = 5 and the composite v = 21. But the prime factorization of a number—writing it as a product of prime numbers—is unique, up to the order of the factors: n = 3 5 7isthe prime factorization of n = 105, and n = 5 is the prime factorization of n = 5.
    [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]
  • ”Primality Testing and Integer Factorization in Public-Key Cryptography” by Song Y
    Review of the book "Primality Testing and Integer Factorization in Public-Key Cryptography" by Song Y. Yan Springer 2009 (2nd Edition) ISBN-13: 978-0-387-77267-7 Dr. Joerg Gerschuetz 2010-01-01 1 What the book is about This book is a successful compilation of the actual state of research regarding Primality Testing and Integer Factorization, both fundamental problems in number theory having far reaching implications to factoring-based public-key cryptography. The book itself is divided into four chapters: 1 Number-Theoretic Preliminaries This chapter collects the basics of number theory necessary for understanding the following chapters and algorithms. Some of the covered topics are Euclid's algorithm, continued fractions, the Chinese remainder theorem (CRT), quadratic and power residues and some of the essential arithmetic functions. There is also a short introduction into the arithmetic of Elliptic Curves. 2 Primality Testing and Prime Generation For a security practitioner this might be the most interesting chapter as it covers the most crucial operations for public-key cryptography. The reader finds a detailed introduction into the most widely used as well as the most recent algorithms for primality testing and reliable prime number generation. One exciting section deals with primality tests for special numbers (Fermat numbers and Mersenne primes). 3 Integer Factorization and Discrete Logarithms Many public-key cryptography schemes rely on the intractability of the integer factorization prob- lem (IFP) or the discrete logarithm problem (DLP). There are still no efficient algorithms for these problems. During the course of the subsections the reader will gain insight into past and recent ef- forts to solve these, e.g.
    [Show full text]