This paper proposes an extended variant of the method by Brickel et al. for multiple scalar multiplication over an elliptic curve. In smartcard environments, the proposed method is superior to conventional methods. In particular, when the typical number of bases t=2, the proposed method is four times faster than the simultaneous multiple exponentiation method, a well known fast method for multiple scalar multiplication. Furthermore, the proposed method can change the amount of time and memory to fit various platform environment (e.g., personal computers as rich ones or mobile devices such as smartcards as poor ones) by adjusting the division bit width (division unit).

The need for electronic sealed-bid auction services with quantitative competition is increasing. This paper proposes a new method that combines one-way functions and a bit commitment technique for quantitative competitive sealed-bid auctions. Since each modular exponentiation is replaced with a one-way function, the proposed method's computational time is one forty thousandth that of the former methods and the proposed method suits mass bidder systems.

This paper proposes an electronic soccer lottery protocol suitable for the Internet environment. Recently, protocols based on public-key schemes such as digital signature have been proposed for electronic voting systems or other similar systems. For a soccer lottery system in particular, it is important to reduce the computational complexity and the amount of communication data required, because we must expect that a large number of tickets will be purchased simultaneously. These problems can be solved by introducing hash functions as the core of protocol. This paper shows a practical soccer lottery system based on bit commitment and hash functions, in which the privacy of prize-winners is protected and illegal acts by the lottery promoter or lottery ticket shops can be revealed.