We investigate image and video convolutions based on Fermat number transform (FNT) modulo q=2/sup M/+1 where M is an integer power of two. These transforms are found to be ideal for image convolutions, except that the choices for the word length, restricted by the transform modulus, are rather limited. We discuss two methods to overcome this limitation. First, we allow M to be an arbitrary integer. This gives much wider variety in possible moduli, at the cost of decreased transform length of 16 or 32 points for M<32. Nevertheless, the transform length appears still to be useful especially with block-based image and video filtering applications. We call these transforms the generalized FNT (GFNT). The second solution is to use a residue number system (RNS) to enlarge the effective modulus, while performing actual number theoretic transforms with smaller moduli. This approach appears to be particularly useful with moduli q/sub 1/=2/sup 16/+1 and q/sub 2/=2/sup 8/+1, which allow transforms up to 256 points with a dynamic range of about 24 bits. We design an efficient reconstruction circuit based on mixed radix conversion for converting the result from diminished-1 RNS into normal binary code. The circuit is implemented in VHDL and found to be very small in area. We also discuss the necessary steps in performing convolutions with the GFNT and evaluate the integrated circuit implementation cost for various elementary operations.

Video filtering with Fermat number theoretic transforms using residue number system

This paper describes the theory, simulation, and construction of a two-dimensional number theoretic transform (NTT) convolver. The convolver performs indirect convolution by using the cyclic convolution property of a class of generalized discrete Fourier transforms (DFT's) defined over rings isomorphic to direct sums of Galois fields. The paper first presents the theoretical development of the computational element required for computing the generalized discrete Fourier transform (GDFIT) and its inverse. The theory extends the use of base fields to second degree extension fields and provides efficient choices for transform parameters to minimize hardware. The paper next presents results of recent work in multidimensional transform memory structures, and extends this work to the complete convolution process. The two theories are then "married" to produce efficient, very high speed convolution architectures. Simulation results are presented for a second degree extension field image convolver and constructional details are presented for a fast image convolver using 2 base fields and designed to operate as a peripheral to a fast 32 bit minicomputer.

Processor Architectures for Two-Dimensional Convolvers Using a Single Multiplexed Computational Element with Finite Field Arithmetic

Processor architectures for two-dimensional convolvers using a single multiplexed computational element with finite field arithmetic

The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m log m log log m), for multiplication of m-bit integers or (m− 1) degree polynomials. Unfortunately, this method bears significant overhead due to the conversions between the time and the frequency domains. This makes the original DFT based method impractical for small operands, e.g. less than 1000 bits in length as used in many applications. In this work, we investigate the application of the number theoretical transform (NTT), which found many applications in digital signal processing, to finite field multiplication with an emphasis on elliptic curve cryptography (ECC). We introduce an efficient algorithm for computing Montgomery products of polynomials in the frequency domain. Our algorithm performs the entire modular multiplication (including the reduction step) in the frequency domain, and thus eliminates costly back and forth conversions. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to elliptic curve cryptography (ECC). This paper is an expanded version of the earlier paper [19] on the same topic which, for the first time, proposes the use of frequency domain arithmetic for ECC and shows that it can be efficient.

Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography

A residue arithmetic implementation of the fft

Using the Residue Number System (RNS) as a basis for the hardware construction, two different hard-ware structures are discussed for implementing NTTs over a direct sum of Galois Fields GF(m i 2). The first structure uses arrays of read only memories and the second uses arrays of microprocessors; in particular, single chip microprocessors are proposed. The two techniques offer trade offs between speed of operation, and cost. Using the RNS, rather than conventional binary arithmetic, allows more flexibility in the choice of the generator, α, and consequently more flexibility in allowable transform parameters. A selection of parameters, for the two realizations, are discussed when the Galois Field of m i 2elements is a finite field of Gaussian or quadratic integers.

G. Jullien

Papers

Hardware implementation of convolution using number theoretic transforms