Systems of Linear Restrictions Over a Finite Field

Abstract


 
 
 This paper considers the problem of recovering an unknown vector based on partial information presented in the form of certain linear dependencies. Such problem is an alternative to the standard one of solving a system of polynomial equations over a finite field, which arises in the context of algebraic cryptanalysis of stream ciphers, and it models a situation when it’s not possible to formulate specific equations with an unknown vector, but certain restrictions on linear dependencies with this vector. To formalize such linear dependencies, the notation of the system of linear restrictions over a finite field is introduced, and the problem of recovering the unknown vector is replaced by the problem of solving the system of linear restrictions over a finite field. In this paper, we researched some properties of this problem using its equivalent forms and important partial cases.