Addition Chain Heuristics

Abstract

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

A Survey of Fast Exponentiation Methods

“…, r t }. Bos and Coster [5] give some heuristics for constructing good addition sequences. A vector addition chain is a sequence of elements v i in N t such that v i = e i for 1 ≤ i ≤ t, and v i = v j + v k for j ≤ k < i.…”

“…A vector addition chain is a sequence of elements v i in N t such that v i = e i for 1 ≤ i ≤ t, and v i = v j + v k for j ≤ k < i. For example, a vector addition chain for [7,15,23] is: [1,3,5], [2,4,6], [3,7,11], [4,8,12], [7,15,23].…”

Minimal weight digit set conversions

“…This method does not take advantage of strings of zeros that do not appear on m-bit boundaries. A more flexible window method is demonstrated in [27]. Similar conversions are presented in [4], [7], [20], [28].…”

Securing RSA against Fault Analysis by Double Addition Chain Exponentiation

“…What we call here double exponentiation shall not be confused with multi-exponentiations (also known as simultaneous exponentiations) that compute a product of powers i m ai i (see for instance [32]). What we call double addition chain is also called addition sequence in the general case where possibly more than two powers must be computed [11,19]. Addition sequences have not been so much investigated.…”

“…Addition sequences have not been so much investigated. In [11], the authors propose some heuristics but these are not suitable for implementations constrained in memory.…”