MaplePrimes - answers and comments on Question, mapping function between Z(2^m) and GF(2^m)

Mapping

alec — Sat, 31 Jan 2009 01:45:10 Z

First, you don't have to use that mapping. It is much easier to use the Groebner:-Basis without it, defining in your example x as RootOf(z^2+z+1).

Second, any other mapping can be used (preferably with 0 being mapped to 0). This one has an advantage of being easy to calculate - substitute x=2 going in one direction, and using base 2 representation of a number, going in another direction.

Alec

Thank you for your

gepo — Sat, 31 Jan 2009 03:00:09 Z

Thank you for your explain.

After further study, I do not think there is such a mapping exist. If such a mapping exist, I mean, if there is a polynomial function to map between Z(2^m) and GF(2^m) to satisfy the addition and multiplication ,then, Z(2^m) will be a field which is impossible.

polynomial function

alec — Sat, 31 Jan 2009 21:20:14 Z

There is no an isomorpism between them even considered only as abelian groups (with addition) - one of them is cyclic of order 2^n (and 1 in it has order 2^n), and another one is a direct sum of n cyclic groups of order 2 (with all non-identity elements having order 2). Any mapping can be considered as a polynomial function though (in some weird sense :) It just doesn't have much sense outside of cryptography.

Alec