We propose a parallel precomputation method for real-time model predictive control. The key idea is to use predicted input values produced by model predictive control to solve an optimal control problem in advance. It is well known that control systems are not suitable for multi- or many-core processors because feedback-loop control systems are inherently based on sequential operations. However, since the proposed method does not rely on conventional thread-/data-level parallelism, it can be easily applied to such control systems without changing the algorithm in applications. A practical evaluation using three real-world model predictive control system simulation programs demonstrates drastic performance improvement without degrading control quality offered by the proposed method.

Given an integer n-dimensional lattice basis, the random sampling reduction was proven to find a short vector in arithmetic steps with an integer k, which is freely chosen by users. This paper introduces new random sampling reduction using precomputation techniques. The computation cost is almost independent of the lattice dimension number. The new method is therefore especially advantageous to find a short lattice vector in higher dimensions. The arithmetic operation number of our new method is about 20% of the random sampling reduction with 200 dimensions, and with 1000 dimensions it is less than 1% ( 1/130) of that of the random sampling reduction with representative parameter settings under reasonable assumptions.

This paper presents a new approach to precompute points [3]P, [5]P,..., [2k-1]P, for some k ≥ 2 on an elliptic curve over Fp. Those points are required for the efficient evaluation of a scalar multiplication, the most important operation in elliptic curve cryptography. The proposed method precomputes the points in affine coordinates and needs only one single field inversion for the computation. The new method is superior to all known methods that also use one field inversion, if the required memory is taken into consideration. Compared to methods that require several field inversions for the precomputation, the proposed method is faster for a broad range of ratios of field inversions and field multiplications. The proposed method benefits especially from ratios as they occur on smart cards.

We develop efficient precomputation methods of multi-scalar multiplication on ECC. We should recall that multi-scalar multiplication is required in some elliptic curve cryptosystems including the signature verification of ECDSA signature scheme. One of the known fast computation methods of multi-scalar multiplication is a simultaneous method. A simultaneous method consists of two stages; precomputation stage and evaluation stage. Precomputation stage computes points of precomputation, which are used at evaluation stage. Evaluation stage computes multi-scalar multiplication using precomputed points. In the evaluation stage of simultaneous methods, we can compute the multi-scalar multiplied point quickly because the number of additions is small. However, if we take a large window width, we have to compute an enormous number of points in precomputation stage. Hence, we have to compute an abundance of inversions, which have large computational amount. As a result, precomputation stage requires much time, as well known. Our proposed method reduces from O(22w) inversions to O(w) inversions for a window width w, using Montgomery trick. In addition, our proposed method computes uP and vQ first, then compute uP+vQ, where P,Q are elliptic points. This procedure enables us to remove unused points of precomputation. Compared with the method without Montgomery trick, our proposed method is 3.6 times faster in the case of the precomputation stage for simultaneous sliding window NAF method with window width w=3 and 160-bit scalars under the assumption that I/M=30, S/M=0.8, where I,M,S respectively denote computational amounts of inversion, multiplication and squaring on a finite field.

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%.