High-Speed Function Approximation using a Minimax Quadratic Interpolator

A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent p), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs)

RSA Power Analysis Obfuscation: A Dynamic FPGA Architecture

The modular exponentiation operation used in popular public key encryption schemes, such as RSA, has been the focus of many side channel analysis (SCA) attacks in recent years. Current SCA attack countermeasures are largely static. Given sufficient signal-to-noise ratio and a number of power traces, static countermeasures can be defeated, as they merely attempt to hide the power consumption of the system under attack. This research develops a dynamic countermeasure which constantly varies the timing and power consumption of each operation, making correlation between traces more difficult than for static countermeasures. By randomizing the radix of encoding for Booth multiplication and randomizing the window size in exponentiation, this research produces a SCA countermeasure capable of increasing RSA SCA attack protection

Optimized linear, quadratic and cubic interpolators for elementary function hardware implementations

This paper presents a method for designing linear, quadratic and cubic interpolators that compute elementary functions using truncated multipliers, squarers and cubers. Initial coefficient values are obtained using a Chebyshev series approximation. A direct search algorithm is then used to optimize the quantized coefficient values to meet a user-specified error constraint. The algorithm minimizes coefficient lengths to reduce lookup table requirements, maximizes the number of truncated columns to reduce the area, delay and power of the arithmetic units, and minimizes the maximum absolute error of the interpolator output. The method can be used to design interpolators to approximate any function to a user-specified accuracy, up to and beyond 53-bits of precision (e.g., IEEE double precision significand). Linear, quadratic and cubic interpolator designs that approximate reciprocal, square root, reciprocal square root and sine are presented and analyzed. Area, delay and power estimates are given for 16, 24 and 32-bit interpolators that compute the reciprocal function, targeting a 65 nm CMOS technology from IBM. Results indicate the proposed method uses smaller arithmetic units and has reduced lookup table sizes compared to previously proposed methods. The method can be used to optimize coefficients in other systems while accounting for coefficient quantization as well as truncation and rounding effects of multiple arithmetic units.Peer reviewedElectrical and Computer Engineerin