Minimal weight digit set conversions

Copyright © 2004 IEEEWe consider the problem of recoding a number to minimize the number of nonzero digits in its representation, that is, to minimize the weight of the representation. A general sliding window scheme is described that extends minimal binary sliding window conversion to arbitrary radix and to encompass signed digit sets. This new conversion expresses a number of known recoding techniques as special cases. Proof that this scheme achieves minimal weight for a given digit set is provided and results concerning the theoretical average and worst-case weight are derived.Braden Phillips and Neil Burges

Complements and signed digit representations: Analysis of a multi-exponentiation-algorithm of Wu, Lou, Lai and Chang

Wu, Lou, Lai and Chang proposed a multi-exponentiation algorithm using binary complements and the non-adjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the complement operation does not have significant influence on the performance of the algorithm and can therefore be omitted

Zot-Binary: A New Number System And Its Application On Number Theory Based Public-Key Cryptography

Kriptosistem Kunci Awam telah digunakan secara meluas dalam protokol seperti pengurusan kekunci, pengesahan, penyulitan kekunci, dan lain-lain. Teori Nombor yang berasaskan Kriptosistem Kunci Awam adalah salah satu cabang utama dalam sistem Kriptografi Kunci Awam. Dua operasi utama dalam Teori Nombor berasaskan Kriptografi Kunci Awam adalah pendaraban dan nombor besar. Antara contoh kriptosistem yang terkenal yang mendapat manfaat daripada operasi ini ialah enkripsi dan dekripsi RSA, tandatangan digital EIGamal, dan pertukaran kunci Diffie-Hellman. Prestasi kriptografi primitif ini sangat bergantung pada kecekapan kedua-dua operasi tersebut. Adalah menjadi sesuatu kebiasaan untuk melakukan penambahbaikan terhadap kecekapan pendaraban dan pengeksponen melalui penggunaan kaedah pengekodan semula atau penggunaan sistem nombor bagi mengurangkan ukuran berat Hamming. ZOT adalah kaedah pengekodan semula yang terkini bagi mengurangkan ukuran berat Hamming. Tetapi, oleh kerana ZOT bukan berasaskan sistem kedudukan nombor, maka kos perlaksanaannya adalah tinggi. Public-key cryptosystems are widely used in protocols such as key agreement, authentication, encryption; etc. Number theory based Public-key cryptosystems are one of the main branches in public-key cryptosystems. The two main operations in number theory based public-key cryptography are large number multiplication and exponentiation. For RSA encryption and decryption, ElGamal digital signature and Diffie-Hellman key exchange are some of the well-known example of these cryptosystems which benefit from these operations. The performance of these cryptographic primitives is highly dependent on the efficiency of these operations. Improving the efficiency of multiplication and exponentiation through the use of a recoding method or utilizing a number system which can decrease the Hamming weight of numbers is very common. ZOT recoding method is one of the latest recoding methods used to decrease the Hamming weight of numbers. However, since it is not positional number systems its cost is high

Some Optimizations of Hardware Multiplication by Constant Matrices

International audienceThis paper presents some improvements on the optimization of hardware multiplication by constant matrices. We focus on the automatic generation of circuits that involve constant matrix multiplication, i.e. multiplication of a vector by a constant matrix. The proposed method, based on number recoding and dedicated common sub-expression factorization algorithms was implemented in a VHDL generator. Our algorithms and generator have been extended to the case of some digital filters based on multiplication by a constant matrix and delay operations. The obtained results on several applications have been implemented on FPGAs and compared to previous solutions. Up to 40% area and speed savings are achieved

Algorithms and VLSI architectures for parametric additive synthesis

A parametric additive synthesis approach to sound synthesis is advantageous as it can model sounds in a large scale manner, unlike the classical sinusoidal additive based synthesis paradigms. It is known that a large body of naturally occurring sounds are resonant in character and thus fit the concept well. This thesis is concerned with the computational optimisation of a super class of form ant synthesis which extends the sinusoidal parameters with a spread parameter known as band width. Here a modified formant algorithm is introduced which can be traced back to work done at IRCAM, Paris. When impulse driven, a filter based approach to modelling a formant limits the computational work-load. It is assumed that the filter's coefficients are fixed at initialisation, thus avoiding interpolation which can cause the filter to become chaotic. A filter which is more complex than a second order section is required. Temporal resolution of an impulse generator is achieved by using a two stage polyphase decimator which drives many filterbanks. Each filterbank describes one formant and is composed of sub-elements which allow variation of the formant’s parameters. A resource manager is discussed to overcome the possibility of all sub- banks operating in unison. All filterbanks for one voice are connected in series to the impulse generator and their outputs are summed and scaled accordingly. An explorative study of number systems for DSP algorithms and their architectures is investigated. I invented a new theoretical mechanism for multi-level logic based DSP. Its aims are to reduce the number of transistors and to increase their functionality. A review of synthesis algorithms and VLSI architectures are discussed in a case study between a filter based bit-serial and a CORDIC based sinusoidal generator. They are both of similar size, but the latter is always guaranteed to be stable

Key Randomization Countermeasures to Power Analysis Attacks on Elliptic Curve Cryptosystems

It is essential to secure the implementation of cryptosystems in embedded devices agains side-channel attacks. Namely, in order to resist differential (DPA) attacks, randomization techniques should be employed to decorrelate the data processed by the device from secret key parts resulting in the value of this data. Among the countermeasures that appeared in the literature were those that resulted in a random representation of the key known as the binary signed digit representation (BSD). We have discovered some interesting properties related to the number of possible BSD representations for an integer and we have proposed a different randomization algorithm. We have also carried our study to the $\tau$-adic representation of integers which is employed in elliptic curve cryptosystems (ECCs) using Koblitz curves. We have then dealt with another randomization countermeasure which is based on randomly splitting the key. We have investigated the secure employment of this countermeasure in the context of ECCs

Proceedings of the 7th Conference on Real Numbers and Computers (RNC'7)

These are the proceedings of RNC7