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In this paper, we propose fast finite field and elliptic curve (EC) algorithms useful for embedding cryptographic functions on high performance device such that most instructions take just one cycle. In such case, the integer multiplications and additions have the same computational cost so that the computational cost analyses that were previously done in traditional manner may be invalid and in some cases the new algorithms should be introduced for fast computation. In our implementation, column major method for field multiplication and BP inversion algorithm are used for fast field arithmetic, and mixed coordinates method is used for efficient EC exponentiation. We give here analyses on various algorithms that are useful for implementing EC exponentiation on CalmRISC microcontroller with MAC2424 coprocessor, as well as new exact analyses on BP (Bailey-Paar) inversion algorithm and EC exponentiation. Using techniques shown in this paper, we implemented EC exponentiation for various coordinate systems and the best result took 122ms, assuming 50ns clock cycle.

Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor