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Introducing Quaternions to Integer Factorisation

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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. The method in Section 2.2 using break RSA, which is integral to many cryptographic Gaussian integers is brought into Section 4.1. Then we systems worldwide. We also assume that the reader is subsequently loosen one of its constrains in Section familiar with the state-of-the-art factoring methods [2], 4.2. We will also show the Euclidean algorithm for although its knowledge is not required to understand quaternions. Then we propose an alternative method our ideas in this paper, but is useful to judge the novelty in Section 4.3, the one which we feel has better of our work. potential to contribute to the research on the integer 1.1 Outline of the Paper factorisation problem. Finally we discuss the results in Section 5 and The Euler factorisation method is introduced in conclude in Section 6. Section 2, while our probably new pathway to the solution with the Gaussian integers is explained in 1.2 Contributions of the Paper Section 2.2, after the introduction to Gaussian integers The main contribution of this paper is to open up in Section 2.1. Then we state the limitations of the the concepts of sum of four squares in quaternions Euler factorisation method and address one of the into the attempts of factoring semiprimes. Sections 2.1, work previously done to make Euler factorisation 4.1 and 4.2 are adapted from standard mathematical method workable. texts. Expressing the Euler’s factoring method with Gaussian integers has not been specifically published Corresponding author: HuiKang Tong. E-mail: [email protected]. on, so content in Section 2.2 may be new. To the 102 Introducing Quaternions to Integer Factorisation author’s knowledge, quaternions has never been Gaussian integer is a sum of two squares, which is the employed in integer factorisation, so works from basis of the Euler factorisation method. The Section 4 onwards are original. explanation will be illustrated by a numerical example that is first factored by Euler: 2. The Euler Factorisation Method pqna b c d We will now look at the much neglected Euler 1000009 1000 3 972 235 factorisation method [3]. In essence, it instantly The sum of two squares can be factorised into obtains the factors of a semiprime when it is written as Gaussian integers: a sum of two square in two distinct ways. pq abiabi cdicdi pqna b c d 1000009 1000 3i1000 3i 972 235i972 235i 2.1 The Gaussian Integers The Gaussian integers can be further factorised. The The Gaussian integersare described in this section so two forms of the semiprime are actually that the reader can draw parallels with the algebra of re-arrangements of each other: quaternions, where the main ideas are built on. pq rsituituirsi Gaussian integers are complex numbers with rsituituirsi rational integers as coefficients for its real and where p rsirsi and q t uit ui imaginary components [4]: The norm of one of the terms is one of the factors. i α abia,b The term is obtained by the Euclidean algorithm, which Similar to complex numbers, Gaussian integers has works for complex numbers [6]. The only difference its conjugate: from the algorithm forrational integersis is that the αabi remainder can be negative, so that the size of the The norm of any Gaussian integeris defined as: remainder can be continually reduced. Nmα ααabiabi a b 1000 3i 1972 235i 28 232i For Gaussian integers, the product of the norm is 972 235i 1 4i28 232i equal to the norm of its product: 72 109i NmαNmβ ααββαββααβαβ 28 232i 2i72 109i 7 58i Nmαβ 72 109i 2 i7 58i 0 A Gaussian prime cannot be expressed in terms of The norm of the greatest common divisor Nm7 two Gaussian integers that each has a smaller norm [5]. 58i 3413 is a prime factor. Hence, any prime number p congruent to 1 mod 4 is 2.3 Evaluation of the Method not a Gaussian prime, as it is factorable into: pρρ r sir si The Euler factorisation method works only when both prime factors is congruent to 1 mod 4, because 2.2 A Complex Approach only such semiprimes have the two representations [7]. The Euler factorisation method can be done with More importantly, there has not been a feasible way to Gaussian integers. We have independently derived this find the two representations of sum of two squares for relationship, although we expect one to have made this large semiprimes. minor result. However, to the author’s knowledge, However, in our literature review, we came across a nothing is specifically published on this concept. work that made the Euler factorisation method An alert reader would have noticed that norm of a feasible [8]. The result was an algorithm that factorises Introducing Quaternions to Integer Factorisation 103 / in On time. Our proposal focuses on quaternions a bicjdka bicjdk and the sum of four squares which should be totally aa bb cc dd different from his ideas. ab ba cd dci a c b d c a d b j 3. Extending to Quaternions ad bc cb dak Extending the concept of the Euler factorisation The conjugate of a quaternion has its imaginary method beyond Gaussian integers, we subsequently components negated: explore the quaternions, and the sum of four squares. P abicjdk Here is our justification why this may be novel. Our focus is on quaternions with integral coefficients, called a Lipchitz quaternion: 3.1 Comment on Mathematical Development P abicjdka,b,c,d Much rigorous work has already been done on the The concept of the norm is applicable to separate topics. RSA was patented in 1977, and the quaternions: RSA factoring challenge was put up on 1991. Attention NmP a b c d on factorising large semiprimes with factors of similar 3.3 Sum of Four Squares length, only then, has been emphasized. Before that, factorisation is acknowledged as difficult, and it is only The norm of the product of the quaternions is equal interested in special probable prime numbers like the to the product of the norm of the quaternions: Fermat numbers and the Mersenne prime candidates. NmPQ PQPQ PQQP P·NmQ ·P On the other hand, the theory of integral quaternions PP ·NmQ NmP ·NmQ and octonions are developed before the 1930s by The result is the Euler’s four-square identity [11]: Lipchitz and Hurwitz [9]. Modern usage on pq NmP ·NmQ quaternions only centres on rotation manipulation. a b c da b c d This may well explain and support the lack of usage of aa bb cc dd quaternions on the integer factorisation problem. ab ba cd dc ac bd ca db 3.2 Quaternions ad bc cb da We begin by defining integral quaternions [10], so NmPQ n as to draw parallels between them and Gaussian After knowing this set of information, the author integers. feels that the algebra of quaternions can be applied Quaternions has three instead of one imaginary into the problem of integer factorisation. component: In this paper, we denote a quaternion by an Aabicjdka,b,c,d upper-case letter, and the norm of the quaternion the Multiplication of quaternions is non-commutative, corresponding lower case. So quaternion P has a with the following properties: norm of p . Different subscripts distinguishes each i j k ijk1 quaternion with the same norm. More specifically: The Jacobi’s Theorem of Four Squares [7] specifies ij ji k the number of distinct representations of a number as jkkji a sum of four squares, which is equivalent to the kiikj number of quaternions with the same norm - the The multiplication of two quaternions is thus: number of ways to represent n as the sum of four 104 Introducing Quaternions to Integer Factorisation squares is eight times the sum of the divisors of n if n algorithm (explained in the following section). is odd, and 24 times the sum of the odd divisors of n However, there does not seem to be an easy method if n is even.
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