A modular multiplication method implemented in an electronic digital processing system takes advantage of the case where one of the operands W is known in advance or used multiple times with different second operands V to speed calculation. The operands V and W and the modulus M may be integers or polynomials over a variable X. A possible choice for the type of polynomials can be polynomials of the binary finite field GF (2N). Once operand W is loaded (30; 60) into a data storage (12) location, a value P = Lw-Xn+δ/M J is pre- computed (32; 62) by the processing system (10). Then when a second operand V is loaded (34; 64), the quotient qΛ for the product V.W being reduced modulo M is quickly estimated (36; 66), qΛ = Lv-P/Xn+δJ, optionally randomized (40; 70), q' = qΛ - E, and can be used to obtain (44; 74) the remainder r' = V.W - q'-M, which is congruent to (V.W) mod M. A final reduction (46; 76) can be carried out, and the later steps repeated (52; 82) with other second operands V.

Modular multiplication method with precomputation using one known operand

A method, device and cipher for performing an optimized multiplicative inverse on received data in substantially real-time and without use of a look-up table. The received data represented as a Galois field GF(2 N ) values.

Optimized multiplicative inverse

A table matching method for multiplication of elements in Galois Field. First, a table of the byte value in Galois Field and the corresponding exponent is formed in the hardware. To perform the multiplication between two elements in the Galois Field, the corresponding exponents of the two elements are found out in advance. The two exponents are then added up to obtain a sum. Then, by using the table, a corresponding byte value of the sum can be obtained. The byte value is the product of the two elements in the Galois Field.

Lai Yi-Lin

Papers

Table matching for multiplication of elements in Galois Field