Is anyone able to point me towards a known-to-be-good diffie-hellman group? I currently use group 24 (RFC5114, 2048-bit MODP Group with 256-bit Prime Order Subgroup). It came to me that this group is not adequate even for AES-128 (this is according to www.cryptopp.com/wiki/Diffie-Hellman)...so far it seems my only options are 1) generating a diffie-hellman group myself, but I don't want to do this when I don't know whether the group I will generate has the ideal properties or 2) Have the application generate a new, adequate diffie-hellman group on the first session start... I would like to avoid this, because it introduces a lot of unnecessary complication on the already complicated system.
In short, my need is to include a pre-generated diffie-hellman group on my application, but I don't want to generate this group myself and include it (and neither do I want to generate a diffie-hellman group on the first session start).
I have also looked into RFC3526 but even the biggest group there (group 16, 8192-bit MODP Group) is not strong enough for my needs according to NIST guidelines (I need at least AES-256) the RFC also doesn't specify the strength of the prime-order subgroup, which further annoys me, it also doesn't specify how, where, when, or under what criteria those groups were generated (RFC5114 got its groups from NIST).
The only groups that satisfy are the elliptic curve diffie-hellman groups, but I do not have the necessary support for elliptic curve diffie-hellman, and already got diffie-hellman implemented.
I seem to have hit a dead end here, can anyone chime in?