Title: Polynomial equations over finite fields and algebraic cryptanalysis Author: Jan Seidl Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: The present work deals with the procedure of algebraic crypta- nalysis, in which the problem of breaking cipher is at first converted to the problem of finding solutions to polynomial systems of equations and then the problem of finding a solution to this equation is converted to the SAT problem. The work specifically describes the methods that allow you to con- vert the problem of breaking cipher RC4 to the SAT problem. The individual methods were programmed in Mathematica programming language and then applied to RC4 with a word length of 2, 3. For finding of satisfiable evaluation of the resulting logical formula was used SAT-solver CryptoMiniSAT. In case of RC4 with word length 2 the solution was reached in the range from 0.09 to 0.34 second. In case of RC4 with word length 3 the solution was reached in the range from 1.10 to 1.23 second. Keywords: RC4, SAT, CryptoMiniSAT 1