Efficient multiplication in GF(p^k) for elliptic curve cryptography

We present a new multiplication algorithm for the implementation of elliptic curve cryptography (ECC) over the finite extension fields GF(p^k) where p is a prime number greater than 2k. In the context of ECC we can assume that p is a 7-to-10-bit number, and easily find values for k which satisfy: p>2k, and for security reasons log₂(p)×k≃160. All the computations are performed within an alternate polynomial representation of the field elements which is directly obtained from the inputs. No conversion step is needed. We describe our algorithm in terms of matrix operations and point out some properties of the matrices that can be used to improve the design. The proposed algorithm is highly parallelizable and seems well adapted to hardware implementation of elliptic curve cryptosystems.