In 2017, Shirase proposed a variant of Elliptic Curve Method combined with Complex Multiplication method for generating certain special kinds of elliptic curves. His algorithm can efficiently factorize a given composite integer when it has a prime factor p of the form 4p=1+Dv2 for some integer v, where -D is an auxiliary input integer called a discriminant. However, there is a disadvantage that the previous method works only for restricted cases where the class polynomial associated to -D has degree at most two. In this paper, we propose a generalization of the previous algorithm to the cases of class polynomials having arbitrary degrees, which enlarges the class of composite integers factorizable by our algorithm. We also extend the algorithm to more various cases where we have 4p=t2+Dv2 and p+1-t is a smooth integer.

In this paper, we describe an improvement of integer factorization of k RSA moduli Ni=piqi (1≤i≤k) with implicit hints, namely all pi share their t least significant bits. May et al. reduced this problem to finding a shortest (or a relatively short) vector in the lattice of dimension k obtained from a given system of k RSA moduli, for which they applied Gaussian reduction or the LLL algorithm. In this paper, we improve their method by increasing the determinant of the lattice obtained from the k RSA moduli. We see that, after our improvement, May et al.'s method works smoothly with higher probability. We further verify the efficiency of our method by computer experiments for various parameters.

In this letter the complexity of factoring an α-LSBS modulus is analyzed. This gives an improvement on the lower bound of the previous results.

In 1995, Vanstone and Zuccherato proposed a novel method of generating RSA moduli having a predetermined set of bits which are the ASCII representation of user's identification information (i.e., name, email address, etc.). This could lead to a savings in bandwidth for data transmission and storage. In this paper, we apply this idea of Vanstone and Zuccherato for reducing the storage requirement of RSA public moduli to integer factoring based public-key schemes with their moduli of the form prq. More precisely, we explicitly propose two efficient methods for specifying high-order bits of prime factors of their public-keys. We also consider the security of the proposed methods.

We describe our implementation of the Hypercube variation of the Multiple Polynomial Quadratic Sieve (HMPQS) integer factorization algorithm on a Parsytec GC computer with 128 processors. HMPQS is a variation on the Quadratic Sieve (QS) algorithm which inspects many quadratic polynomials looking for quadratic residues with small prime factors. The polynomials are organized as the nodes of an n-dimensional cube. We report on the performance of our implementations on factoring several large numbers for the Cunningham Project.