Abstract: The cryptographic system is comprised of two stations A and B in which a respective string S.sub.A and S.sub.B of digits is stored. In station A an error-control information string C is generated from string S.sub.A and transmitted to station B over an error-free public channel CH1. In station B a particular string D and a decision bit F are generated. String D maximizes a predetermined reliability function of a string combination (S.sub.B, C, D). Decision bit F is assigned the value 1 if and only if a maximum value taken on by the reliability function is greater than a predetermined threshold. Decision bit F is transmitted to station A over an error-free public channel CH2. In both stations A and B, the respective strings S.sub.A and S.sub.B are tagged as accepted when said decision bit F has the value 1. The above sequence is repeated, resulting in a plurality of tagged strings which are concatenated at stations A and B to result in a random cipher key shared by the stations A and B.

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.

Abstract: The cryptographic system comprises at least one encryption station, one decryption station and a trapdoor generator. The trapdoor generator comprises means for selecting r distinct prime numbers pi, generating a modulus m that is a product of the prime numbers pi, selecting a pair of integers (a,b) satisfying 0.ltoreq.a<m and 0.ltoreq.b<m, computing for each pi a number N(pi) of distinct pairs of integers (x,y) satisfying 0.ltoreq.x<pi and 0.ltoreq.y<pi and y.sup.2 .ident.x.sup.3 +a.multidot.x+b (mod pi) and a sum N(pi)+1 representative of an order of an elliptic curve defined as the set of pairs of integers (x,y), computing a least common multiple .mu. of the values N(pi)+1, selecting a public multiplier e relatively prime to .mu., computing a secret multiplier d according to d.ident.1/e (mod .mu.), and transferring data comprising at least m, a, b and e to a storage means provided in the cryptographic system.