Abstract: A secret key generator for a trusted authority means in a cryptographic system comprising means for receiving a publicly-known user identity ID.sub.A for a user A; and further comprising secret key generator means for transforming the publicly-known user identity ID.sub.A into secret key s.sub.A for the user A with the inverse of the exponentiation functiona.sup.s.sbsp.A .ident.(ID.sub.A).sup.2 (mod m),that is solved for s.sub.A, where a is a publicly-known base parameter, where m is a publicly-known modulus parameter and m=p.sub.1 *p.sub.2 * . . . *p.sub.r and where p.sub.1, p.sub.2, . . . , p.sub.r are secret system prime numbers. The inverse of the exponentiation function is infeasible to compute in order to calculate the secret key s.sub.A without knowing the prime numbers p.sub.1, p.sub.2, . . . , p.sub.r. The secret key s.sub.A enables user A to communicate securely to a user B having publicly-known user identity ID.sub.B with a mutual secure cipher key K.sub.AB, defined by the equation:K.sub.AB .ident.