(Suggestion for the first time through: use P=257 and N=3)
Fix a modulus P = _________ and a base N = _________, which are known to everybody (typically, P is prime for this system and N is a "primitive root" meaning the powers of N are all different modulo P, until the (P-1) power of N is 1, by Fermat's little theorem).
Pick your secret exponent E = ________ (E and (P-1) should have no common factors, besides 1.).
Find all of the powers of N modulo P listed below, reducing modulo P after every step:
|N (mod P)||=||________||N2 (mod P)||=||________||N4 (mod P)||=||________|
|N8 (mod P)||=||________||N16 (mod P)||=||________||N32 (mod P)||=||________|
|N64 (mod P)||=||________||N128 (mod P)||=||________|
Now, find the powers above that sum to E, and compute J = NE (mod P) by multiplying those powers of N together. [For example, if E = 113, E = 64 + 32 + 16 + 1. Then, J = NE (mod P) = N64 * N32 * N16 * N (mod P).]
J = _________
Trade your J with another group (call theirs K).
Now compute KE (mod P) by the same repeated squaring method:
|K (mod P)||=||________||K2 (mod P)||=||________||K4 (mod P)||=||________|
|K8 (mod P)||=||________||K16 (mod P)||=||________||K32 (mod P)||=||________|
|K64 (mod P)||=||________||K128 (mod P)||=||________|
KE (mod P) is the secret number you have with the other group!