Integer Factoring Algorithms

Most cryptographic systems are based on an underlying difficult problem. The RSA cryptosystem and many other cryptosystems rely on the fact that factoring a large composite number into two prime numbers is a hard problem. The are many algorithms for factoring integers. This chapter presents some of the basic algorithms for integer factorization like the Trial Division, Fermat's Algorithm. Pollard's Rho Method, Pollard's p-1 method and the Elliptic Curve Method. The Number Field Sieve algorithm along with Special Number field Sieve and the General Number Field Sieve are also used in factoring large numbers. Other factoring algorithms discussed in this chapter are the Continued Fractions Algorithms and the Quadratic Sieve Algorithm.

JKL-ECM: an implementation of ECM using Hessian curves

We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over$\mathbb{Q}$and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$of Bernsteinet al. are completely recovered by the$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.

COMPARISON OF ALGORITHMS FOR FACTORIZATION OF LARGE NUMBERS HAVING SEVERAL DISTINCT PRIME FACTORS

We present analysis of security of the most known assymetric algorythm RSA and its modern version MultiPrime RSA. We focused on more precisious estimations of time complexity of two factorization algorithms: Elliptic Curve Method and General Number Field Sieve. Additionally for the MultiPrime RSA algorithm we computed the maximal number of prime factors for given modulus length which does not decrease the security level.

Implementation of Embedded Mobile Device Elliptic Curve Algorithm Fast Generating Algorithm

Elliptic curve cryptosystem is used in the process of embedded systems, the selection and generation algorithm of the elliptic curve will directly affect the efficiency of systems. From Elliptic Curve's selection, Elliptic Curve's structure, Elliptic Curve's generation, this paper discussed the realization of a random elliptic curve method of Embedded Mobile Device, the SEA algorithm and its improved algorithm. The results show that this method can achieve a quick implementation of the elliptic curve method to improve the operating efficiency of embedded systems in the same security guarantees.