Electronic signature with ordered and divided private key

Abstract

The encryption systems with public keys are related to algebraic structures, which have to ensure by the means of their computational properties or their dimensions, a high level of security for the generated keys, which are private. The ElGamal algorithm is defined over the group of remainder classes modulo n, which is a suitable structure for an encryption algorithm. This algorithm can be defined over another mathematical structure: symmetrical group of degree p. The properties which ensure the opportunity of choosing this structure are: the cardinal of the group is sufficiently great, the operation of composition of the permutations is simple from the computational point of view and every permutation can be decomposed uniquely into a product of disjunctive functions. The main objective of this paper is to create a signature scheme with ordered and divided private key starting from the ElGamal algorithm with public keys over the symmetrical group of degree p. The auxiliary objectives of the paper are: the outline of the obtained results in this field from the corresponding literature and putting forward a comparative study. The paper is structured according to these objectives and includes a study of the literature of the field, the algorithms for the created schemes and conclusions. The scheme is included in the category of the ones with divided or shared private key and it is not part of the category of group signatures or of the ring signature. In this situation the private key is divided and from a number of entities, each owns a division. In order to create the signature, it is necessary for each entity to use, in a default order its own division from the private key, while the checking of the signature is realized using a unique public key. The conclusions and the contributions of the scheme are addressed in the last part of the paper.