Generic Abelian Groups

A generic abelian group is a set whose elements form an abelian group with respect to a given law of composition. The user specifies the set Atogether with functions for composing two elements of A, constructing the inverse of an element of A, and recognizing the identity element of A.

• Definition as a set with given operations
• Arithmetic
• Random elements
• Order of an element: Baby-step giant-step algorithm; Pollard-rho algorithm
• Discrete logarithm: Baby-step giant-step algorithm; Pohlig-Hellman algorithm; Pollard-rho algorithm
• Order of the group
• Generating set and presentation
• Torsion invariants
• Construction of subgroups from generators
• Sylow p-subgroup
• Homomorphisms and isomorphisms

The three major calculations supported are: find the order of an element, compute the discrete logarithm of an element relative to a given base and determine the structure of the group. The algorithms used are improvements of those described in J. Buchmann, M.J. Jacobson and E. Teske  [5].