Integer factoring and compositeness witnesses

• Main Authors: Pomykała Jacek, Radziejewski Maciej
• Format: Article
• Language: English
• Published: De Gruyter 2020-08-01
• Series: Journal of Mathematical Cryptology
• Subjects: large sieve; factoring algorithms; zn*-generating sets; dirichlet characters; smooth numbers; discrete logarithm problem for composite numbers; primality testing; euler’s totient function; rsa; 11a05; 11a15; 11a51; 11m06; 11z05; computational number theory: 11y05; 11y16
• Online Access: https://doi.org/10.1515/jmc-2019-0023

We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp −cM(loglog⁡x)3(logloglog⁡x)2$\begin{array}{} \displaystyle \left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right) \end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored.