Abstract. The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463... in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408... if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.

Abstract. The general subset sum problem is NP-complete. However, there are two algorithms, one due to Brickell and the other to Lagarias and Odlyzko, which in polynomial time solve almost all subset sum problems of sufficiently low density. Both methods rely on basis reduction algorithms to find short non-zero vectors in special lattices. The Lagarias-Odlyzko algorithm would solve almost all subset sum problems of density < 0.6463... in polynomial time if it could invoke a polynomial-time algorithm for finding the shortest non-zero vector in a lattice. This paper presents two modifications of that algorithm, either one of which would solve almost all problems of density < 0.9408... if it could find shortest non-zero vectors in lattices. These modifications also yield dramatic improvements in practice when they are combined with known lattice basis reduction algorithms.

Theoretical results and asymptotic bounds in this area are plentiful (see [Dow81], e.g.), but we are not aware of anyone applying heuristics, as we do, to make chains useable in practice. Definitions and notationAn addition chain for a given number is a list of numbers having the following properties: l the frost number is one; . every number is the sum of two earlier numbers; l the given number occurs in the chain (at the end, that is). In the case of an addition sequence, the last condition becomes: l the given numbers occur in the sequence. We view such a list as a series of exponents used to do an exponentiation.The length of an addition chain or sequence is the number of elements in the chain, apart from the initial one. G. Brassard (Ed.): Advances in

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In this paper we prove some generalisations of congruences of Atkin and Swinnerton-Dyer type. This is done in the form of congruences for numbers P n(A/j']), where P,,(I) are the orthogonal polynomials of Legendre. The proofs are based on complex multiplication of elliptic functions.