Abstract: A method, system and apparatus for performing user identification, digital signatures and other secure communication functions in which keys are chosen essentially at random from a large set of vectors and key lengths are comparable to the key lengths in other common identification and digital signature schemes at comparable security levels. The signing technique of an embodiment of the identification/digital signature scheme hereof uses a mixing system based on multiplication in a ring and reduction modulo an ideal q in that ring; while the verification technique uses special properties of products of elements whose validity depends on elementary probability theory. The security of the identification/digital signature scheme comes from the interaction of reduction modulo q and the difficulty of forming products with special properties.

Abstract: A method for performing a cryptographic operation involving transforming digital information is described. A digital operator is provided that has a component selected from a large set of elements. The component is expanded into a plurality of factors, each factor having a low Hamming weight. Digital information is transformed using the digital operator. Computer readable medium embody instructions for the method.

Abstract: A method for authenticating, by a second user, the identity of a first user, that includes a challenge communication from the second user to the first user, a response communication from the first user to the second user, and a verification by the second user, includes the steps: selection by the first user of a private key f in a ring R and a public key that includes ?(f) in a ring B that is mapped from f using the ring homomorphism ?: R?B, and publication by the first user of the public key; generation of the challenge communication by the second user that includes selection of a challenge c in the ring R; generation of the response communication by the first user that includes computation of a response comprising h in the ring R, where h is a function of c and f; and performing of a verification by the second user that includes determination of ?(c) from c, ?(h) from h, and an evaluation that depends on ?(h), ?(c) and ?(f).

Abstract: A method of communicating information between users of a communications system includes the following steps: generating a ring R, ideals P and Q in R, a set of coset representatives CQ for the ring R modulo the ideal Q, and a set of coset representatives Cp for the ring R modulo the ideal P; generating at least one public key element h1, . . . , hk in the ring R as a function of at least two private key elements ƒ1, . . . ƒn in R and the ideal Q of the first user; and transmitting from a first user to a second user a description of the ring R, the ideal Q, the ideal P, and the elements h1, . . . , hk in R; generating an element e in R as a function of the ideals P and Q, the public key elements h1, . . . , hk, a private message element m in R, and at least one private random element ø1, . . .

Abstract: The public key encryption system of the present invention has short and easily created encryption keys and wherein the encoding and decoding processes are performed extremely rapidly, and has low memory requirements. The encoding and decoding processes use both the addition and multiplication operations in a ring modulo with two different ideals. The cryptosystem of the present invention allows encryption keys to be chosen essentially at random from a large set of binary vectors, for which key lengths are comparable to the key lengths of the most widely used prior art cryptosystems. The present invention features an appropriate security level (.about.2.sup.80), with encoding and decoding processes ranging from approximately one to two orders of magnitude faster than the prior art, particularly the exponentiation cryptosystems.