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Number Theory and Cryptographic Hardness Assumptions
Published in Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography, 2020
A fundamental theorem of arithmetic is that every integer greater than 1 can be expressed uniquely (up to ordering) as a product of primes. That is, any positive integer N > 1 can be written as N=∏ipiei, where the {pi} are distinct primes and ei > 1 for all i; furthermore, the {pi} (and {ei}) are uniquely determined up to ordering.
Integer Factorization Problem
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
Pinkimani Goswami, Madan Mohan Singh
Integer factorization of a positive integer n is the decomposition of n into a product of two small positive integers a,b (say), i.e., n=a⋅b. Here, a and b are called factors of n. If a,b≠1,n, then they are the nontrivial factors of n and n is called a composite number; otherwise, n is a prime number. If a,b are primes (or prime power), then the factorization is called prime factorization of n. Note that integer factorization of a number is not unique, but the prime factorization of n is unique. The fundamental theorem of arithmetic states that for any positive integer n>1 can be expressed as the product of prime numbers (not necessarily distinct). Therefore, it is trivial to get n from the given primes, but it is difficult to factor a large n. It is believed that the factoring a large composite number is difficult as there is no efficient algorithm to find the factors. Note that the problem of finding integer factors of a composite number is called integer factorization problem. The concept of integer factorization has been around since Euclid. He defined the idea of unique factorization in around 300 bc. The practical importance of the integer factorization problem arises with the development of public key cryptography. For example, the security of RSA cryptosystem (Rivest, Shamir & Adleman, 1978) is based on integer factorization problem, i.e., RSA cryptosystem will break if integer factorization problem is solved. Consequently, finding an efficient algorithm for integer factorization has practical importance.
Meaning making in a sixth-grade mathematics classroom through touch screen technology
Published in International Journal of Mathematical Education in Science and Technology, 2022
This study builds on socio-cultural learning theories that argue for the individual being an active participant in his or her own education (Cobb & Bowers, 1999). Mathematics is often perceived as an abstract knowledge area, and the learning of mathematical concepts occurs through abstraction processes. In the natural world, learning the properties of a tangible object can often help us develop an understanding of that object. In mathematics, one can consider a ‘mathematical object’ (Sfard, 2008) as an abstract concept in which its content is, in a sense, hidden (Steinbring, 2005). To represent how mathematical concepts are abstract constructs (Thompson & Sfard, 1994), we can consider the case of prime numbers, which are one set of natural numbers. One can identify whether a natural number is prime by verifying whether the number satisfies the definition of a prime number. The fundamental theorem of arithmetic states that all natural numbers can be written uniquely as products of prime factors (Rosen, 1988). This fact expresses a multiplicative relationship that is initially hidden. Therefore, the presentation of mathematical objects through character signs and symbols is initially not directly related to its contents (Duval, 2017)