In this thesis we presented binary polynomials through definition and operations on them. We explained key points of algorithms, gave examples of evaluation, pseudocodes of algorithms itself and analysed time complexity with comments in form suitable for appropriate thesis. We also offered few versions of algorithm for multiplication and exponentiation, and gave appropriate care and explanation with pseudocodes for division and addition. Factorization was described in detail: all necessary key steps for general approach in getting factors. We concluded that hybrid Karatsuba has better time complexity than Karatsuba, and Karatsuba is, in Big-O notation, faster than simple multiplication algorithm. Testing times are in accordance with theoretical predictions for multiplication algorithms. Also, addition and subtraction may be extremely fast because of simplicity of algorithm itself and structure of GF(2). We provided results for the rest of operations, too. It is obvious that we are not able to get results very fast when input size grows: division in case of divident’s degree higher than one million and factorization of polynomials with degree higher than twenty.