Testwiki:Reference desk/Archives/Mathematics/2024 November 15

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November 15

Are there morphisms when enlarging a prime field sharing a common suborder/subgroup ?

Simple question : I have a prime field having modulus $p$ where p−1 contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as it’s suborder/subgroup. Are there special cases where it’s possible to lift 2 $p$’s elements to modulus $q$ while keeping their discrete logarithm if those 2 elements lies only within the $O$’s subgroup ? Without solving the discrete logarithm of course ! 82.66.26.199 (talk) 11:36, 15 November 2024 (UTC)

Clearly it is possible, since any two groups of order o are isomorphic. Existence of a general algorithm, however, is equivalent to solving the discrete log problem (consider the problem of determining a non-trivial character). Tito Omburo (talk) 11:40, 15 November 2024 (UTC)

So how to do it without solving the discrete logarithm ? Because of course, I was meaning without solving the discrete logarithm. 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 12:51, 15 November 2024 (UTC)

It can't. You're basically asking if there is some canonical isomorphism between two groups of order O, and there just isn't one. Tito Omburo (talk) 15:00, 15 November 2024 (UTC)

Even if it’s about enlarging instead of shrinking ? Is in theory impossible to build a relation/map or is that no such relation exists yet ? 2A01:E0A:401:A7C0:9CB:33F3:E8EB:8A5D (talk) 08:48, 16 November 2024 (UTC)

At least into the group of complex roots of unity, where a logarithm is known, it is easily seen to be equivalent to discrete logarithm. In general, there is no relation between the groups of units in GF(p) and GF(q) for p and q distinct primes. Any accidental isomorphisms between subgroups are not canonical. Tito Omburo (talk) 15:02, 16 November 2024 (UTC)