On some computational problems in finite abelian groups

Abstract

Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the O-constants and Ω-constants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results.Our algorithms are based on a modification of Shanks' baby-step giantstep strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order.

A modification of Shanks' baby-step giant-step algorithm

“…The numbers in brackets indicate which of the five giant-step sizes used was optimal for the given ideal class and algorithm. Note that in the case of ∆ = −4(10 10 + 1), and v = 2, |∆| 1 4 /2, and |∆| 1 4 (middle four rows of Tables 1, 4, and 5) my results for the BJT algorithm agree with those in [1].…”

“…every positive integer n can be expressed as the difference t j − i, where t j is the least number in the sequence (t j ) not exceeding n and 0 ≤ i < j + v. The algorithm proceeds as follows. (My notation is the same as in [1].) (17) Proof.…”

“…Shanks' baby-step giant-step algorithm [1,2] is a well-known procedure for finding the order n of an element g of a finite group G. Running it involves 2 √ K + O(1) group multiplications (GM), and √ K + O(1) table lookups (TL), where K is an upper bound on n (for instance, one often uses K = |G|). Often, however, K is unknown or much larger than n. In this case, a faster algorithm is desired.…”

A space efficient algorithm for group structure computation

“…There is another generic algorithm for group structure computation [BJT97], which is based on Shanks' Baby-Step Giant-Step method [Sha71]. It has runtime complexity O( |G|), but it has the disadvantage that it has high storage requirements.…”

Determining the small solutions to 𝑆-unit equations

“…Now, as ord p (µ i ) = 0 for all i, we can consider the group M/p n . Using an algorithm like the ones in [1] or [6], one can determine the group structure of the M/p n as a product of cyclic groups C 1 ×. .…”