SBIM (Q)-a Multivariate Polynomial Trapdoor Function over the Field of Rational Numbers

Abstract

. In this paper we define a trapdoor function called SBIM(Q) by using multivariate polynomials over the field of rational numbers Q. The public key consists of 2n multivariate polynomials with 3n variables y1, . . . , yn, z1, . . . , z2n. The yi variables take care for the information content, while the zi variables are for redundant information. Thus, for encryption of a plaintext of n rational numbers, a ciphertext of 2n rational numbers is used. The security is based on the fact that there are infinitely many solutions of a system with 2n polynomial equations of 3n unknowns. The public key is designed by quasigroup transformations obtained from quasigroups presented in matrix form. The quasigroups presented in matrix form allow numerical as well as symbolic computations, and here we exploit that possibility. The private key consists of several 1×n and n×n matrices over Q, and one 2n × 2n matrix.