| In this thesis, we study the Karatsuba-Ofman Algorithm (KOA), which is a recursive multi-precision multiplication method, and improve it for certain special applications. This thesis is in two parts.; In the first part, we derive an efficient algorithm from the KOA to multiply the operands having a precision of 2m computer words for some integer m. This new algorithm is less complex and three times less recursive than the KOA. However, the order of the complexity is the same as the KOA.; In the second part of the thesis, we introduce a novel method to perform fast multiplication in GF(2m), using the KOA. This method is intended for software implementations and has two phases. In the first phase, we treat the field elements in GF (2m) as polynomials over GF(2) and multiply them by a technique based on the KOA, which we call the LKOA (lean KOA). In the second phase, we reduce the product with an irreducible trinomial or pentanomial.; The LKOA is similar to the KOA. However, it stops the recursions early and switches to some nonrecursive algorithms which can efficiently multiply small polynomials over GF(2). We derive these nonrecursive algorithms from the KOA, by removing its recursions. Additionally, we optimize them, exploiting the arithmetic of the polynomials over GF(2). As a result, we obtain a decrease in complexity, as well as a reduction in the recursion overhead. |
