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Binary extension field arithmetic is widely used in several important applications such as error-correcting codes, cryptography and digital signal processing. Multiplication is usually considered the most important finite field arithmetic operation. Therefore efficient hardware architectures for multiplication are highly desired. In this brief, a new architecture for multiplication over finite fields generated by irreducible trinomials    $f(x) = x^{m}+x^{t}+1$    is presented. The architecture here proposed is based on the use of a polynomial multiplier and a cyclic shift register that can perform the multiplication in    $t-1$    clock cycles. The general architecture is applied to the trinomials recommended by NIST (National Institute of Standards and Technology). Furthermore, a RISC-V instruction set for the proposed multiplier is implemented and validated using VeeR-EL2 on a Nexys A7 FPGA. To the best knowledge of the authors, this is the first work that integrates the multiplication based on NIST trinomials into a RISC-V SoC. Results show an improvement of several orders of magnitude in terms of latency at a cost of less than 50&#x0025; more of area.

Efficient Low-Latency Multiplication Architecture for NIST Trinomials With RISC-V Integration