NTRU is a public key cryptosystem based on hard problems over lattices. In this paper, we present efficient methods for convolution product computation which is a dominant operation of NTRU. The new methods are based on the observation that repeating patterns in coefficients of an NTRU polynomial can be used for the construction of look-up tables, which is a similar approach to the sliding window methods for exponentiation. We provide efficient convolution algorithms to implement this idea, and we make a comprehensive analysis of the complexity of the new algorithms. We also give software implementations over a Pentium IV CPU, a MICAz mote, and a CUDA-based GPGPU platform. According to our analyses and experimental results, the new algorithms speed up the NTRU encryption and decryption operations by up to 41%.

We introduce efficient algorithms for the τ-adic sliding window method, which is a scalar multiplication algorithm on Koblitz curves over F2m. The τ-adic sliding window method is divided into two parts: the precomputation part and the main computation part. Until now, there has been no efficient way to deal with the precomputation part; the required points of the elliptic curves were calculated one by one. We propose two fast algorithms for the precomputation part. One of the proposed methods decreases the cost of the precomputation part by approximately 30%. Since more points are calculated, the total cost of scalar multiplication is decreased by approximately 7.5%.