1 3D Integer Factorization
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An Amazing Prime Heuristic.Pdf
This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. -
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. -
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. -
New Formulas for Semi-Primes. Testing, Counting and Identification
New Formulas for Semi-Primes. Testing, Counting and Identification of the nth and next Semi-Primes Issam Kaddouraa, Samih Abdul-Nabib, Khadija Al-Akhrassa aDepartment of Mathematics, school of arts and sciences bDepartment of computers and communications engineering, Lebanese International University, Beirut, Lebanon Abstract In this paper we give a new semiprimality test and we construct a new formula for π(2)(N), the function that counts the number of semiprimes not exceeding a given number N. We also present new formulas to identify the nth semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots 3 of N : P , P ....P 3 √N. 1 2 π( √N) ≤ Keywords: prime, semiprime, nth semiprime, next semiprime 1. Introduction Securing data remains a concern for every individual and every organiza- tion on the globe. In telecommunication, cryptography is one of the studies that permits the secure transfer of information [1] over the Internet. Prime numbers have special properties that make them of fundamental importance in cryptography. The core of the Internet security is based on protocols, such arXiv:1608.05405v1 [math.NT] 17 Aug 2016 as SSL and TSL [2] released in 1994 and persist as the basis for securing dif- ferent aspects of today’s Internet [3]. The Rivest-Shamir-Adleman encryption method [4], released in 1978, uses asymmetric keys for exchanging data. A secret key Sk and a public key Pk are generated by the recipient with the following property: A message enciphered Email addresses: [email protected] (Issam Kaddoura), [email protected] (Samih Abdul-Nabi) 1 by Pk can only be deciphered by Sk and vice versa. -
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). -
Cullen Numbers with the Lehmer Property
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]). -
Some New Results on Odd Perfect Numbers
Pacific Journal of Mathematics SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT,JOHN L. HUNSUCKER AND CARL POMERANCE Vol. 57, No. 2 February 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 57, No. 2, 1975 SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT, J. L. HUNSUCKER AND CARL POMERANCE If ra is a multiply perfect number (σ(m) = tm for some integer ί), we ask if there is a prime p with m = pan, (pa, n) = 1, σ(n) = pα, and σ(pa) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers neces- sarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) ~ 2n. 1* Introduction* In this paper we answer a question on odd perfect numbers posed by Suryanarayana [17]. It is known that if m is an odd perfect number, then m = pak2 where p is a prime, p Jf k, and p = a z= 1 (mod 4). Suryanarayana asked if it necessarily followed that (1) σ(k2) = pa , σ(pa) = 2k2 . Here, σ is the sum of the divisors function. We answer this question in the negative by showing that no odd perfect number satisfies (1). We actually consider a more general question. If m is multiply perfect (σ(m) = tm for some integer t), we say m has property S if there is a prime p with m = pan, (pa, n) = 1, and the equations (2) σ(n) = pa , σ(pa) = tn hold. -
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. -
On Types of Elliptic Pseudoprimes
journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N. -
On Repdigits As Sums of Fibonacci and Tribonacci Numbers
S S symmetry Article On Repdigits as Sums of Fibonacci and Tribonacci Numbers Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic; [email protected]; Tel.: +42-049-333-2860 Received: 17 September 2020; Accepted: 21 October 2020; Published: 26 October 2020 Abstract: In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a “symmetrical” type of numbers) that can be written in the form Fn + Tn, for some n ≥ 1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively. Keywords: Diophantine equations; repdigits; Fibonacci; Tribonacci; Baker’s theory MSC: 11B39; 11J86 1. Introduction A palindromic number is a number that has the same form when written forwards or backwards, i.e., of the form c1c2c3 ... c3c2c1 (thus it can be said that they are “symmetrical” with respect to an axis of symmetry). The first 19th palindromic numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99 and clearly they are a repdigits type. A number n is called repdigit if it has only one repeated digit in its decimal expansion. More precisely, n has the form ! 10` − 1 n = a , 9 for some ` ≥ 1 and a 2 [1, 9] (as usual, we set [a, b] = fa, a + 1, .. -
On Distribution of Semiprime Numbers
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2014, Vol. 58, No. 8, pp. 43–48. c Allerton Press, Inc., 2014. Original Russian Text c Sh.T. Ishmukhametov, F.F. Sharifullina, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53–59. On Distribution of Semiprime Numbers Sh. T. Ishmukhametov* and F. F. Sharifullina** Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia Received January 31, 2013 Abstract—A semiprime is a natural number which is the product of two (possibly equal) prime numbers. Let y be a natural number and g(y) be the probability for a number y to be semiprime. In this paper we derive an asymptotic formula to count g(y) for large y and evaluate its correctness for different y. We also introduce strongly semiprimes, i.e., numbers each of which is a product of two primes of large dimension, and investigate distribution of strongly semiprimes. DOI: 10.3103/S1066369X14080052 Keywords: semiprime integer, strongly semiprime, distribution of semiprimes, factorization of integers, the RSA ciphering method. By smoothness of a natural number n we mean possibility of its representation as a product of a large number of prime factors. A B-smooth number is a number all prime divisors of which are bounded from above by B. The concept of smoothness plays an important role in number theory and cryptography. Possibility of using the concept in cryptography is based on the fact that the procedure of decom- position of an integer into prime divisors (factorization) is a laborious computational process requiring significant calculating resources [1, 2]. -
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].