We present a Polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Z(X).Expand

This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field. Application of XTR in cryptographic… Expand

We give guidelines for the determination of cryptographic key sizes, based on a set of explicitly formulated hypotheses, combined with existing data points.Expand

This paper reports on the factorization of the 768-bit RSA-768 by the number field sieve factoring method and discusses some implications for RSA.Expand

We offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptsystems both over finite fields and over groups of elliptic curves.Expand

We consider the following integer feasibility problem: Given positive integer numbersa0,a1, ... , a n , with gcd( a1,..., a n ) = 1 anda = ( a1,..., a n ), does there exist a vectorx?Z n… Expand

The number field sieve is an algorithm to factor integers of the form $r^e-s$ for small positive $r$ and $s$. The authors present a report on work in progress on this algorithm. They informally… Expand

The number field sieve is an algorithm to factor integers of the form re − s for small positive r and |s| using arithmetic in an algebraic number field.Expand

We show how sloth can be used for uncontestable random number generation (unicorn), and how unicorn is used for a new trustworthy random ellip tic curves service.Expand