Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates

Exact computer arithmetic has a variety of uses including, but not limited to, the robust implementation of geometric algorithms. This report has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive-precision arithmetic that can often speed these algorithms when one wishes to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to provide a practical demonstration of these techniques, in the form of implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small. These algorithms work on computers whose floating-point arithmetic uses radix two and exact rounding, including machines complying with the IEEE 754 standard. The inputs to the predicates may be arbitrary single or double precision floating-point numbers. C code is publicly available for the 2D and 3D orientation and incircle tests, an

An Implementation of the Chor-Rivest Knapsack Type Public Key Cryptosystem

The Chor-Rivest cryptosystem is a public key cryptosystem first proposed by MIT cryptographers Ben Zion Chor and Ronald Rivest [Chor84]. More recently Chor has imple mented the cryptosystem as part of his doctoral thesis [Chor85]. Derived from the knapsack problem, this cryptosystem differs from earlier knapsack public key systems in that computa tions to create the knapsack are done over finite algebraic fields. An interesting result of Bose and Chowla supplies a method of constructing higher densities than previously attain able [Bose62]. Not only does an increased information rate arise, but the new system so far is immune to the low density attacks levied against its predecessors, notably those of Lagarias- Odlyzko and Radziszowski-Kreher [Laga85, Radz86]. An implementation of this cryptosystem is really an instance of the general scheme, dis tinguished by fixing a pair of parameters, p and h , at the outset. These parameters then remain constant throughout the life of the implementation (which supports a community of users). Chor has implemented one such instance of his cryptosystem, where p =197 and h =24. This thesis aspires to extend Chor\u27s work by admitting p and h as variable inputs at run time. In so doing, a cryptanalyst is afforded the means to mimic the action of arbitrary implementations. A high degree of success has been achieved with respect to this goal. There are only a few restrictions on the choice of parameters that may be selected. Unfortunately this general ity incurs a high cost in efficiency; up to thirty hours of (VAX1 1-780) processor time are needed to generate a single key pair in the desired range (p = 243 and h =18)

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

These are the proceedings of RNC7

Optimising Code Generation with haggies

This article describes haggies, a program for the generation of optimised programs for the efficient numerical evaluation of mathematical expressions. It uses a multivariate Horner-scheme and Common Subexpression Elimination to reduce the overall number of operations. The package can serve as a back-end for virtually any general purpose computer algebra program. Built-in type inference that allows to deal with non-standard data types in strongly typed languages and a very flexible, pattern-based output specification ensure that haggies can produce code for a large variety of programming languages. We currently use haggies as part of an automated package for the calculation of one-loop scattering amplitudes in quantum field theories. The examples in this articles, however, demonstrate that its use is not restricted to the field of high energy physics.Comment: 66 pages, 5 figures, program files for download at http://www.nikhef.nl/~thomasr

An alternative architecture for performing basic computer arithmetical operations

The arithmetic portions of almost all modern processor architectures are of very similar design. We use the term "traditional" to describe this design, the primary characteristics of which are native support for integer and floating-point number types and special disjoint instructions and hardware for each supported type. Decades of refinement have endowed this traditional arithmetic architecture with high performance, but also certain inherent limitations. The highly-specific instruction sets and circuitry that provide optimized performance for supported number types, also make it difficult to synthesize unsupported number types and manipulate them in an efficient manner. This trait also applies when using supported number types for arbitrary ranges greater than those directly implemented by the processor. In this thesis we present an alternative to the traditional computer arithmetic architecture, designed to address the limitations of the traditional approach while preserving most of its benefits. Instead of the specific number representation support provided by the instructions, hardware and native data types in a traditional ALU/FPU pair, we define a single data type, the XLU digit that forms a base from which other number types may be easily derived, along with a set of instruction primitives from which basic arithmetic operations may be efficiently realized. Our data type has a signed-digit representation, which allows algorithms for addition, subtraction and multiplication to achieve a high degree of parallelism at the primitive instruction level. The instruction primitives and algorithms are designed to hide or eliminate as much branching as possible, further increasing instruction-level independence. We provide details of the data type, an overview of the set of instruction primitives, and a discussion of how to use those instruction primitives to perform basic arithmetic algorithms for addition, subtraction and multiplication. We also give examples for three derived number representations; integer, fixed-point and floating-point numbers. We believe that our approach of building from a unified base provides flexibility and scalability beyond that of the traditional arithmetic architecture. Our data type, the XLU digit, and the primitive operations to manipulate it may be implemented with modest amounts of circuitry, and this, together with the highly parallel nature of the entire design means that many XLU circuit blocks can be realized in the same silicon area as one traditional ALU/FPV pair. An ALU or FPU may only work when it has the correct type to work on, whereas we believe any and all XLUs available to the processor can be kept busy almost all of the time, achieving greater utilization of the available silicon