Vassil S. Dimitrov

An algorithm for modular exponentiation

Abstract A practical technique for improving the performance of modular exponentiations (ME) is described. The complexity of the ME algorithm is O ( n log n ) modular multiplications (MMs), where n is the length of the exponent, requiring an O(n2) precomputed look-up table size with very small constant of proportionality. The algorithm uses a double-based number system which we introduce in this paper.

A new DCT algorithm based on encoding algebraic integers

In this paper we introduce an algebraic integer encoding scheme for the basis matrix elements of 8/spl times/8 DCTs and IDCTs. In particular, we encode the function cos(/spl pi//16) and generate the other matrix elements using standard trigonometric identities. This encoding technique eliminates the requirement to approximate the matrix elements; rather we use algebraic placeholders for them. Using this encoding scheme we are able to produce a multiplication free implementation of the Feig-Winograd algorithm.

Theory and applications for a double-base number system

Presents a rigorous theoretical analysis of the main properties of a double-base number system, using bases 2 and 3. In particular, we emphasize the sparseness of the representation. A simple geometric interpretation allows an efficient implementation of the basic arithmetic operations, and we introduce an index calculus for logarithmic-like arithmetic with considerable hardware reductions in look-up table size. Two potential areas of applications are discussed: applications in digital signal processing for computation of inner products and in cryptography for computation of modular exponentiations.

The use of the multi-dimensional logarithmic number system in DSP applications

A recently introduced double-base number representation has proved to be successful in improving the performance of several algorithms in cryptography and digital signal processing. The index-calculus version of this number system can be regarded as a two-dimensional extension of the classical logarithmic number system. This paper builds on previous special results by generalizing the number system both in multiple dimensions (multiple bases) and by the use of multiple digits. Adopting both generalizations the paper shows that large reductions in hardware complexity are achievable compared to an equivalent precision logarithmic number system.

A hybrid DBNS processor for DSP computation

This paper introduces a modification to an index calculus representation for the double-base number system (DBNS). The DBNS uses the bases 2 and 3; it is redundant (very sparse) and has a simple two-dimensional representation. An extremely sparse form of the DBNS uses a single non-zero digit to represent any real number with arbitrary precision. In this case the single digit can be identified by its coordinates (indices) in the two-dimensional representation space. The modification proposed in this paper, targeted to DSP inner product computations, uses a single digit representation for the coefficient vector and a 2-digit representation for the data vector. We show that a reduction of over 80% in hardware cost is possible using this hybrid representation compared to the original single-digit technique.

An efficient technique for error-free algebraic-integer encoding for high performance implementation of the DCT and IDCT

A recently introduced algebraic integer encoding scheme allows low complexity, virtually error free computation of the DCT and IDCT. Efficiencies can be introduced into this method, but at the expense of some increase in error. In this paper, a modification to the encoding scheme is introduced for specific architectures which provides increased implementation efficiency, but with no sacrifice in accuracy. We provide a theoretical study of this new approach and illustrate the technique using selected DCT and IDCT algorithms.

A 2-digit DBNS filter architecture

We have previously reported on a novel number representation using 2 bases which we refer to as the double-base number system (DBNS). Our preferred implementation uses the relatively prime bases {2,3}. If we allow the exponents of the bases to be arbitrarily large signed integers, then we can represent any real number to any arbitrary precision by a single digit DBNS representation. By representing the digit position by the exponent values, we generate a logarithmic-like representation which we can manipulate using an index calculus. A multiplier accumulator architecture for a FIR filter application has been reported which uses a half-index domain to remove the problem of addition within the index calculus. In this paper we show that using a 2-digit DBNS representation for both the input data and the filter coefficients can result in substantial hardware savings compared to both the single-digit a DBNS approach and an equivalent binary implementation of a general multiplier accumulator. In the paper we discuss the filter architecture, techniques for converting between binary and the 2-digit DBNS representations, and also the design technique used to generate the 2-digit DBNS FIR filter coefficients.

A 2-digit multidimensional logarithmic number system filterbank for a digital hearing aid architecture

This paper addresses the design and implementation of a filterbank for digital hearing aids using a multi-dimensional logarithmic number system (MDLNS). The logarithmic properties of the MDLNS allow for reduced complexity multiplication, and large dynamic range, and a multiple-digit MDLNS provides a considerable reduction in hardware complexity compared to a conventional logarithmic number system (LNS) approach. In this paper we discuss the design and implementation of both a 1-digit and 2-digit 2-D MDLNS filterbank and provide initial simulation results.

On efficient techniques for difficult operations in one and two-digit DBNS index calculus

The Double Base Number System (DBNS), using orthogonal bases of 2 and 3, has similar properties to the logarithmic number system (LNS) if an index calculus is used. The DBNS provides more degrees of freedom than the LNS by virtue of both the orthogonal bases and the ability to use multiple digits. As with the LNS, multiplication and division are easy but addition and subtraction are difficult. This paper introduces a technique that uses a map to a DBNS index sequence domain, removing the need for the relatively large look-up table solution of the LNS.

Computing Haar transform using algebraic integers

A novel algorithm for computing the Haar transform is presented that is based on the algebraic integer-encoding scheme. For the implementation of the algorithm a fully pipelined systolic architecture with O(N) throughput is proposed.