The standard prime groups have their highest and lowest 64 bits set, and as many as possible of the rest taken from the binary expansion of pi. The group #14 prime for example is
The value 124476 is the smallest non-negative value which results in p and (p-1)/2 both being prime and 2 being a quadratic residue mod p; this ensures that the subgroup {2^0, 2^1, 2^2, 2^3, ... } is cyclic.
We assume that the binary expansion of Pi is not selected maliciously. ;-)
> We assume that the binary expansion of Pi is not selected maliciously. ;-)
Well, I suppose that we must assume that, as assuming otherwise is self-defeating: anyone running the simulation we're living in already has direct access to the plaintexts anyway.
p = 2^2048 - 2^1984 - 1 + 2^64 * { [2^1918 pi] + 124476 }
The value 124476 is the smallest non-negative value which results in p and (p-1)/2 both being prime and 2 being a quadratic residue mod p; this ensures that the subgroup {2^0, 2^1, 2^2, 2^3, ... } is cyclic.
We assume that the binary expansion of Pi is not selected maliciously. ;-)