Paper 2014/739

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

Smile Markovski, Aleksandra Mileva, and Vesna Dimitrova

Abstract

In this paper we define a trapdoor function called SBIM(Q) by using multivariate polynomials over the field of rational numbers $\mathbb Q.$ The public key consists of $2n$ multivariate polynomials with $3n$ variables $y_1,\dots,y_n,$ $z_1,\dots,z_{2n}$. The $y_i$ variables take care for the information content, while the $z_i$ 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 $1\times n$ and $n\times n$ matrices over $\mathbb Q$, and one $2n\times 2n$ matrix.