Viktor Bunimov

Area and time efficient modular multiplication of large integers

A new modular multiplication algorithm and its corresponding architecture is presented. It is optimised with respect to hardware complexity and latency. Based on the dataflow of the well known interleaved modular multiplication the product of two n-bit-integers X and Y modulo M is computed by n iterations of a simple loop. The loop consists of one single carry save addition, a comparison of constant complexity, and a table lookup, where the table contains 6 precomputed values and two constants. By this construction the arithmetical complexity of the modular multiplication is reduced to n additions without carry propagation in total which leads to a speedup of at least two in comparison to all methods previously known. It consists of a first algorithm A2 implementing the new idea of combining carry save addition and constant time comparison. A2 is not optimal with respect to area and time. Its correctness is proven. By use of a small amount of precomputing the loop of A2 can be modified such that the effort within the loop is minimised. This leads to the algorithm A3 and it is verified.

Fast modular multiplication by operand changing

A new algorithm for modular multiplication for public key cryptography is presented. The algorithm is optimised with respect to area and time by use of a combination of adders and fast lookup tables. This leads to a multiplication method that can significantly speed up exponentiation, because the values of the lookup table do not depend on the operands of the individual multiplication. The speedup is achieved by continuous modification of one operand.

Efficient Parallel Multiplication Algorithm for Large Integers

A new algorithm for calculating A*B is presented. It is based on precomputation of a small number of values that allow for an efficient parallel implementation of multiplication. The algorithm requires O(logn) time and O(n 2 /logn 2 ) area with small constant factors which makes it feasible for practical implementations.