Enhancing the Securing RSA Algorithm from Attack
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Introducing Quaternions to Integer Factorisation
Journal of Physical Science and Application 5 (2) (2015) 101-107 doi: 10.17265/2159-5348/2015.02.003 D DAVID PUBLISHING Introducing Quaternions to Integer Factorisation HuiKang Tong 4500 Ang Mo Kio Avenue 6, 569843, Singapore Abstract: The key purpose of this paper is to open up the concepts of the sum of four squares and the algebra of quaternions into the attempts of factoring semiprimes, the product of two prime numbers. However, the application of these concepts here has been clumsy, and would be better explored by those with a more rigorous mathematical background. There may be real immediate implications on some RSA numbers that are slightly larger than a perfect square. Key words: Integer factorisation, RSA, quaternions, sum of four squares, euler factorisation method. Nomenclature In Section 3, we extend the Euler factoring method to one using the sum of four squares and the algebra p, q: prime factors n: semiprime pq, the product of two primes of quaternions. We comment on the development of P: quaternion with norm p the mathematics in Section 3.1, and introduce the a, b, c, d: components of a quaternion integral quaternions in Section 3.2, and its relationship 1. Introduction with the sum of four squares in Section 3.3. In Section 3.4, we mention an algorithm to generate the sum of We assume that the reader know the RSA four squares. cryptosystem [1]. Notably, the ability to factorise a In Section 4, we propose the usage of concepts of random and large semiprime n (the product of two the algebra of quaternions into the factorisation of prime numbers p and q) efficiently can completely semiprimes. -
Question 1.1. What Is the RSA Laboratories' Frequently Asked
Copyright © 1996, 1998 RSA Data Security, Inc. All rights reserved. RSA BSAFE Crypto-C, RSA BSAFE Crypto-J, PKCS, S/WAN, RC2, RC4, RC5, MD2, MD4, and MD5 are trade- marks or registered trademarks of RSA Data Security, Inc. Other products and names are trademarks or regis- tered trademarks of their respective owners. For permission to reprint or redistribute in part or in whole, send e-mail to [email protected] or contact your RSA representative. RSA Laboratories’ Frequently Asked Questions About Today’s Cryptography, v4.0 2 Table of Contents Table of Contents............................................................................................ 3 Foreword......................................................................................................... 8 Section 1: Introduction .................................................................................... 9 Question 1.1. What is the RSA Laboratories’ Frequently Asked Questions About Today’s Cryptography? ................................................................................................................ 9 Question 1.2. What is cryptography? ............................................................................................10 Question 1.3. What are some of the more popular techniques in cryptography? ................... 11 Question 1.4. How is cryptography applied? ............................................................................... 12 Question 1.5. What are cryptography standards? ...................................................................... -
Integer Factorization and RSA Encryption
Integer Factorization and RSA Encryption Noah Zemel February 2016 1 Introduction Cryptography has been used for thousands of years as a means for securing a communications channel. Throughout history, all encryption algorithms utilized a private key, essentially a cipher that would allow people to both encrypt and decrypt messages. While the complexity of these private key algorithms grew significantly dur- ing the early 20th century because of the World Wars, significant change in cryptography didn't come until the 1970s with the invention of public key cryp- tography. Up until this point, all encryption and decryption required that both parties involved knew a secret key. This led to a weakness in the communica- tions systems that could be infiltrated, sometimes by brute force, but usually by more complicated backtracking algorithms. The most popular example of breaking a cipher was when the researchers at Bletchley Park (most notably Alan Turing) were able to crack some of the private keys being transmitted each morning by the Nazi forces,. This enabled the Allied forces to decipher some important messages, revealing key enemy positions. With the advent of computers and the power to do calculations much faster than humans can, there was a clear need for a more sophisticated encryption system|one that couldn't be cracked by intercepting or solving the private key. In public key cryptography, there are two keys: a private key and a public key. The public key is used to encrypt the message, while the private key is needed to decrypt it. Only the decryptor knows the private key. -
Elementary Number Theory, a Computational Approach
This is page i Printer: Opaque this Elementary Number Theory, A Computational Approach William Stein March 2007 ii To my wife Clarita Lefthand. This is page iii Printer: Opaque this Contents Preface 3 1 Prime Numbers 5 1.1 Prime Factorization . 5 1.2 The Sequence of Prime Numbers . 14 1.3 Exercises ............................ 22 2 The Ring of Integers Modulo n 25 2.1 Congruences Modulo n ..................... 25 2.2 The Chinese Remainder Theorem . 32 2.3 Quickly Computing Inverses and Huge Powers . 35 2.4 PrimalityTesting........................ 40 2.5 The Structure of (Z/pZ)∗ ................... 43 2.6 Exercises ............................ 48 3 Public-Key Cryptography 51 3.1 The Diffie-Hellman Key Exchange . 54 3.2 TheRSACryptosystem .................... 60 3.3 AttackingRSA ......................... 64 3.4 Exercises ............................ 70 4 Quadratic Reciprocity 73 4.1 Statement of the Quadratic Reciprocity Law . 74 4.2 Euler’sCriterion ........................ 77 Contents 1 4.3 First Proof of Quadratic Reciprocity . 79 4.4 A Proof of Quadratic Reciprocity Using Gauss Sums . 85 4.5 Finding Square Roots . 90 4.6 Exercises ............................ 92 5 Continued Fractions 95 5.1 Finite Continued Fractions . 96 5.2 Infinite Continued Fractions . 103 5.3 The Continued Fraction of e .................. 109 5.4 Quadratic Irrationals . 112 5.5 Recognizing Rational Numbers From Their Decimal Expansion116 5.6 Sums of Two Squares . 118 5.7 Exercises ............................ 121 6 Elliptic Curves 125 6.1 TheDefinition ......................... 125 6.2 The Group Structure on an Elliptic Curve . 126 6.3 Integer Factorization Using Elliptic Curves . 129 6.4 Elliptic Curve Cryptography . 135 6.5 Elliptic Curves Over the Rational Numbers . -
An Investigation on Integer Factorization Applied to Public Key Cryptography
Universit`adegli Studi di Trento DIPARTIMENTO DI MATEMATICA Corso di Dottorato di Ricerca in Matematica XXXI CICLO A thesis submitted for the degree of Doctor of Philosophy An investigation on Integer Factorization applied to Public Key Cryptography Supervisor: Ph.D. Candidate: Prof. Massimiliano Sala Giordano Santilli “I was just guessing at numbers and figures, pulling your puzzles apart...” (The Scientist - Coldplay) To all the people that told me their secrets and asked me their questions. Contents Introduction VII 1 Preliminaries 1 1.1 Historical Overview . .1 1.1.1 RSA . .2 1.1.2 A quick review on Factorization Methods . .2 1.1.3 Factorization Records . .5 1.2 Preliminaries on Elementary Number Theory . .6 1.2.1 Basics on Floor and Ceiling . .6 1.2.2 Definitions and basic properties . .6 1.2.3 Integer solutions of a General Quadratic Diophantine Equation having as discriminant a square . 11 1.3 Algebraic Number Theory Background . 13 1.3.1 Number Fields and Ring of Integers . 13 1.3.2 Norm of an element . 15 1.3.3 Ideals in the Ring of Integers . 16 1.4 Groebner Bases Theory . 20 1.4.1 Multivariate polynomials . 20 1.4.2 Monomial Orderings . 22 1.4.3 Groebner Bases . 24 1.4.4 Elimination Theory . 27 2 An elementary approach to factorization 29 2.1 Successive moduli . 29 2.2 A formula for successive moduli . 36 2.3 Successive moduli in factorization . 39 2.4 Interpolation . 42 2.4.1 Successive Remainders and Interpolation . 42 III CONTENTS 2.4.2 A conjecture on interpolating polynomials . -
Public-Key Cryptography and the RSA Algorithm
Public-Key Cryptography Public-key Cryptography ■ Breakthrough idea due to Diffie, Hellman and Merkle in their and the RSA Algorithm 1976 works ■ “Is it possible to exchange information confidentially without Ozalp Babaoglu having to first agree on a key?” ■ Yes, as long as we can implement “one-way trap-door” concept mathematically ALMA MATER STUDIORUM – UNIVERSITA’ DI BOLOGNA © Babaoglu 2001-2021 Cybersecurity 2 RSA Algorithm Notation ■ One of the first practical responses to the challenge posed by Diffie-Hellman was developed by Ron Rivest, Adi Shamir, and Len Adleman of MIT in 1977 ■ Resulting algorithm is known as RSA Let ■ Based on properties of prime numbers and results from ℤ = { . , − 3, − 2, − 1, 0, 1, 2, 3, . } denote the set of integers number theory ℤn = {0, 1, 2, 3, . , n − 1} denote the set of integers modulo n GCD(m,n) denote the greatest common divisor of m and n ℤn* denote the integers relatively prime with n φ(n) = |ℤn*| denote Euler’s totient function © Babaoglu 2001-2021 Cybersecurity 3 © Babaoglu 2001-2021 Cybersecurity 4 Some Facts Example ■ Let n=15 ■ What is !(15)=? If GCD(n, m) = 1 (n and m are relatively prime or coprime) then ■ Integers relatively prime with 15: {1, 2, 4, 7, 8, 11, 13, 14} φ(nm) = φ(n)φ(m) ■ Therefore, !(15)=8 ■ Observe that 15=3×5 If p and q are two primes, then ■ Therefore, !(n)=!(3×5) φ(p) = (p − 1) =!(3)×!(5) φ(pq) = (p − 1)(q − 1) =(3−1)×(5−1) =2×4 =8 © Babaoglu 2001-2021 Cybersecurity 5 © Babaoglu 2001-2021 Cybersecurity 6 RSA RSA: Generation of the keys ■ Choose two very large primes p, q ■ -
Factorization of a 768-Bit RSA Modulus Version 1.4, February 18, 2010
Factorization of a 768-bit RSA modulus version 1.4, February 18, 2010 Thorsten Kleinjung1, Kazumaro Aoki2, Jens Franke3, Arjen K. Lenstra1, Emmanuel Thomé4, Joppe W. Bos1, Pierrick Gaudry4, Alexander Kruppa4, Peter L. Montgomery5;6, Dag Arne Osvik1, Herman te Riele6, Andrey Timofeev6, and Paul Zimmermann4 1 EPFL IC LACAL, Station 14, CH-1015 Lausanne, Switzerland 2 NTT, 3-9-11 Midori-cho, Musashino-shi, Tokyo, 180-8585 Japan 3 University of Bonn, Department of Mathematics, Beringstraße 1, D-53115 Bonn, Germany 4 INRIA CNRS LORIA, Équipe CARAMEL - bâtiment A, 615 rue du jardin botanique, F-54602 Villers-lès-Nancy Cedex, France 5 Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA 6 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Abstract. This paper reports on the factorization of the 768-bit number RSA-768 by the num- ber field sieve factoring method and discusses some implications for RSA. Keywords: RSA, number field sieve 1 Introduction On December 12, 2009, we factored the 768-bit, 232-digit number RSA-768 by the number field sieve (NFS, [20]). The number RSA-768 was taken from the now obsolete RSA Challenge list [38] as a representative 768-bit RSA modulus (cf. [37]). This result is a record for factoring general integers. Factoring a 1024-bit RSA modulus would be about a thousand times harder, and a 768-bit RSA modulus is several thousands times harder to factor than a 512-bit one. Because the first factorization of a 512-bit RSA modulus was reported only a decade ago (cf. [7]) it is not unreasonable to expect that 1024-bit RSA moduli can be factored well within the next decade by an academic effort such as ours or the one in [7]. -
Strategy for Algorithm Design in Factoring RSA Numbers
IOSR Journal of Computer Engineering (IOSR-JCE) e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 19, Issue 3, Ver. II (May.-June. 2017), PP 01-07 www.iosrjournals.org Strategy For Algorithm Design in Factoring RSA Numbers Xingbo WANG (Department of Mechatronics, Foshan University, Foshan City, Guangdong Province, PRC, 528000) Abstract: The article first makes an analysis on characteristic of the RSA numbers based on the factorized RSA numbers and the Digital Signature Standard; then it investigates the affection of the divisor-ratio of a RSA number to the efficiency of searching the number’s divisors in term of valuated binary tree and puts forward a framework for designing algorithms of fast factoring the RSA numbers. Thoughts of the framework are presented with detail mathematical deductions. Keywords - RSA number, Factorization, Algorithm Design, Binary Tree I. INTRODUCTION Ever since the RSA Laboratories declared the RSA Factoring Challenge in March 1991, factorization of the RSA numbers has become a practical criterion to test an algorithm of factoring integers and thousands of people have still kept doing the factorization in spite that the RSA Laboratories have retracted their rewards. Literatures show that, nowadays, the Number Field Sieve (NFS) is regarded as the most efficient method to factorize the RSA numbers, as introduced and surveyed in articles [1] to [7]. But unfortunately the NFS will take a vast mount of memory in the computation and it has no chance to apply the method on conventional personal computers. Hence new approaches are still under development. For example, article [8] proposed MPFA, article [9] proposed Factorization Using Multiplication Table, articles [10] proposed an formula for approximately testing divisors of a semiprime, articles [11] and [12] proposed constructing sieves of odd composite numbers. -
Directions on Factoring Utility Demo
TCSV Whitenoise Demonstration of Factoring bi-prime public key equivalents TCSV Deep Dive on Security in the Snowden Era at Microsoft How secure are we? The security of RSA style public key asymmetric frameworks is predicated on the premise that the creation of a public key is a one-way-function. This means that it is “easy” to do the math in one direction but impossible to go backwards and unwind it. It is easy for anyone to multiply two prime numbers together with a calculator but it was considered impossible or infeasible to go backwards and determine the two original prime number values. This process is called factoring. For a long time RSA ran factoring challenges. Most persons were left with the sense that this validates the security of those PKI frameworks. It had once been very difficult to factor co-primes. It was supposed to take thousands of years to do so. But there are few if any public key networks using keys of the strength found in their contests. That is an academic distraction. The majority of networks globally rely on 128-bit keys i.e. SSL, TSL etc. Just trying to move up to the use of 256-bit and 1024-bit keys makes processing problematic and additional resources like accelerators become necessary. The overhead required in public key technology excludes that technology from deployment in the majority of inexpensive Internet of Everything endpoints. “The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography. -
Number Theory Resolves a Problem in a Divorce
Number theory resolves a problem in a divorce Sunil K. Chebolu Illinois State University 1/23 Sunil Chebolu Number theory resolves a problem in a divorce . Beer is proof that God loves us and wants us to have a good time Benjamin Franklin Number Theory is proof that God loves us and wants us to have a good time Sunil Chebolu 2/23 Sunil Chebolu Number theory resolves a problem in a divorce A divorce problem Alice and Bob are getting a divorce and have to discuss who gets what. They are already separated and they live in different cities and can't stand facing each other. They don't seem to agree on one thing: 3/23 Sunil Chebolu Number theory resolves a problem in a divorce Who gets the car? 4/23 Sunil Chebolu Number theory resolves a problem in a divorce After much deliberation on the matter, they decide to flip a coin 5/23 Sunil Chebolu Number theory resolves a problem in a divorce Coin flipping over the telephone is tricky Problem: If they don't trust each other how can they flip a coin over the telephone without bringing in a 3rd party (a referee)? \Flipping a coin" really means performing some random experiment akin to coin tossing which has two equally likely outcomes and in which no party can cheat. 6/23 Sunil Chebolu Number theory resolves a problem in a divorce Assumptions Before we go further we will make a couple of mild assumptions on the couple. I Bob and Alice have math degrees and they love number theory (more than their partner). -
The Role of Prime Numbers in RSA Cryptosystems
The Role Of Prime Numbers in RSA Cryptosystems Henry Rowland December 5, 2016 Abstract Prime numbers play an essential role in the security of many cryptosystems that are cur- rently being implemented. One such cryptosystem, the RSA cryptosystem, is today’s most popular public-key cryptosystem used for securing small amounts of sensitive information. The security of the RSA cryptosystem lies in the difficulty of factoring an integer that is the product of two large prime numbers. Several businesses rely on the RSA cryptosystem for making sure that sensitive information does not end up in the wrong hands. This paper high- lights the importance of prime numbers in modern day implementations of RSA cryptosys- tems. We will discuss how an RSA cryptosystem can be successfully implemented, some of the methods used to find primes considered appropriate for RSA cryptosystems, as well as cryptanalysists techniques for factoring a product consisting of such primes. 1 Introduction Throughout history, there has been a need for secure, or private, communication. In war, it would be devastating for the opposing side to intercept information during communication over an inse- cure line about the plan of attack. With all of the information that is stored on the internet today, it is important that sensitive information, such as a person’s credit card number, is stored securely. Cryptosystems, or ciphers, use a key, or keys, for encrypting and decrypting information being communicated with the intention of keeping this information out of the hands of unwanted recip- ients. Prior to the 1970s, the key(s) used in cryptosystems had to be agreed upon and kept private between the originator, or sender, of a message and the intended recipient. -
Dynamic Distributed PKI
Rapid Factorization of Semiprimes [email protected] RSA Conference 2016 Dynamic Distributed PKI Whitenoise is a trade mission member for the Canadian Ontario delegation to RSA. Title Two new techniques for the rapid factorization of semiprimes (modulus): bisection then volatility test and prime number dictionary attacks. Abstract This document aims to stimulate experimentation and research with new techniques for rapid factorization of semiprimes. These techniques start by taking the square root of a semiprime (modulus) to radically reduce attack work space. Thereafter, Newton’s bisection theory is applied and a simple volatility test tells us which direction one of the semiprime factors is located relative to subsequent bisections. The semiprime square-root and the volatility test for prime number location direction might be incorporated into traditional sieve methods like the general number field sieve and the quadratic sieve by creative mathematicians who can make their techniques work in the search for the single factor to find the other. Taking the square root of the semiprime reduces the attack work space to such a degree that it is possible to construct a simple “prime number dictionary” attack just with prime numbers already calculated and available on the internet. It is possible (maybe probable) that starting with the square root of the semiprime for factorization was not considered previously because of the difficulty of manipulating such large numbers. Most Rapid Factorization of Semiprimes [email protected] sieves are based on Fermat’s Factorization. Fermat died in 1665 long before computers and calculators. The Whitenoise factorial utility has a large number math library capable of handling numbers many hundreds of digits long on a basic computer that can be easily expanded upon for ever increasing number sizes.