The method of obtaining the set of noncanonical hypercomplex number systems by conversion of infinite hypercomplex number system to finite hypercomplex number system depending on multiplication rules and factorization method is described. Systems obtained by this method starting from the 3dimension are noncanonical. The obtained systems of even dimension can be re-factorized. As a result of it ...

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 2 − 1. Although this factorization is orders of magnitude ‘easier’ than a factorization of a 1024-bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of t...

We describe the complete factorization of the tenth Fermat number F10 by the elliptic curve method (ECM). F10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations

We describe the main ideas underlying integer factorization using the number field sieve.

I provide the details of the factorization of the Mersenne number 21061 − 1 by the Special Number Field Sieve. Although this factorization is easier than the completed factorization of RSA-768, it represents a new milestone for factorization using publicly available software.

It was shown in 2] that under reasonable assumptions the general number eld sieve (GNFS) is the asymptotically fastest known factoring algorithm. It is, however, not known how this algorithm behaves in practice. In this report we describe practical experience with our implementation of the GNFS whose rst version was completed in January 1993 at the

A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2 > ... ^ aM. We establish the asymptotic distribution, as «-»• oo, of the vector A(«) = (loga,/logiV: i: > 1) in a transparent manner. By randomly re-ordering the components of A(«), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is ...

The integer q is called the quotient and r is the remainder. Proof. Consider the rational number b a . Since R = ⋃ k∈Z[k, k + 1) (disjoint), there exists a unique integer q such that b a ∈ [q, q + 1), i.e., q ≤ b a < q + 1. Multiplying through by the positive integer a, we obtain qa ≤ b < (q + 1)a. Let r = b− qa. Then we have b = qa + r and 0 ≤ r < a, as required. Proposition 3. Let a, b, d ∈ Z...