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132
A New Representation for Exact Real Numbers
, 1997
"... We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the rea ..."
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Cited by 42 (8 self)
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We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with nonnegative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first socalled sign matrix determines an interval on which the real number lies. The subsequent socalled digit matrices have nonnegative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.
Optimal Lefttoright Binary SignedDigit Recoding
, 2000
"... This paper describes new methods for producing optimal binary signeddigit representations. This can be useful in the fast computation of exponentiations. Contrary to existing algorithms, the digits are scanned from left to right (i.e., from the most significant position to the least significant ..."
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Cited by 34 (3 self)
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This paper describes new methods for producing optimal binary signeddigit representations. This can be useful in the fast computation of exponentiations. Contrary to existing algorithms, the digits are scanned from left to right (i.e., from the most significant position to the least significant position). This may lead to better performances in both hardware and software.
Semantics of Exact Real Arithmetic
, 1997
"... In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the exten ..."
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Cited by 29 (8 self)
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In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.
FloatingPoint Arithmetic And Message Authentication
, 2000
"... There is a wellknown class of message authentication systems guaranteeing that attackers will have a negligible chance of successfully forging a message. This paper shows how one of these systems can hash messages at extremely high speed  much more quickly than previous systems at the same securi ..."
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Cited by 28 (8 self)
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There is a wellknown class of message authentication systems guaranteeing that attackers will have a negligible chance of successfully forging a message. This paper shows how one of these systems can hash messages at extremely high speed  much more quickly than previous systems at the same security level  using IEEE floatingpoint arithmetic. This paper also presents a survey of the literature in a unified mathematical framework.
Theory and applications of the doublebase number system
 IEEE Trans. on Computers
, 1999
"... In this paper we present 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 operation ..."
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Cited by 27 (10 self)
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In this paper we present 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 logarithmiclike arithmetic with considerable hardware reductions in lookup 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. 1.
DigitSet Conversions: Generalizations and Applications
 IEEE Transactions on Computers
, 1995
"... The problem of digit set conversion for fixed radix is investigated for the case of converting into a nonredundant, as well as into a redundant digit set. Conversion may be from very general digit sets, and covers as special cases multiplier recodings, additions and certain multiplications. We gene ..."
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Cited by 22 (5 self)
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The problem of digit set conversion for fixed radix is investigated for the case of converting into a nonredundant, as well as into a redundant digit set. Conversion may be from very general digit sets, and covers as special cases multiplier recodings, additions and certain multiplications. We generalize known algorithms for conversions into nonredundant digit sets, as well as apply conversion to generalize the O(log n) time algorithm for conditional sum addition using parallel prefix computation, and a comparison is made with standard carrylookahead techniques. Examples on multioperand addition are used to illustrate the generality of this approach. O(1) time algorithms for converting into redundant digit sets are generalized based on a very simple lemma, which provides a framework for all conversions into redundant digit sets. Applications in multiplier recoding and partial product accumulation are used here as exemplifications. Keywords: Computer arithmetic, digit set conversio...
Hybrid SignedDigit number systems: A unified framework for redundant number representations with bounded carry propagation chains
, 1993
"... Abstract A novel hybrid number representation is proposed in this paper. It includes the two's complement representation and the signeddigit mpresentation as special cases. The hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addi ..."
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Cited by 20 (6 self)
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Abstract A novel hybrid number representation is proposed in this paper. It includes the two's complement representation and the signeddigit mpresentation as special cases. The hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addition to any desired value between 1 and the entire word length. The framework reveals a continuum of number representations between the two extremes of two's complement and signeddigit number systems and allows a unified performance analysis of the entire spectrum of implementations of adders, multipliers and alike. We present several static CMOS implementations of a twooperand adder which employ the proposed representations. We then derive quantitative estimates of area (in terms of the required number of transistors) and the maximum carry propagation delay for such an adder. The analysis clearly itlustrates the tradeoffs between area and execution time assodated with each of the possible representathns. We also discuss adder trees for parallel multipliers and show that the proposed representations lead to compact adder trees with fast execution times. In practice, the area available to a designer is often Umited. In such cases, the designer can select the particular hybrid fepresentation that yields the most suitable implementation (fastest, lowest power consumption, etc.) while satisfying the area constraint. Similarly, if the worst case delay is predetermined, the designer can select a hybrid representation that minimizes area or power under the delay constraint. Index TermsBounded carry propagation, carryfree addition, hybrid signeddigit number system, redundant number representation, signeddigit numbers, static CMOS implementation. I.
Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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Cited by 19 (0 self)
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Some characterizations of functions computable in online arithmetic
 I.E.E.E. Trans. on Computers
, 1994
"... AbsfmctAfter a short introduction to online computing, we prove that the functions computable in online by a finite automaton are piecewise afthe functions whose coefikients are rational numbers (i.e., the fnnctions f(s) = UP + b, or f(z, y) = nx + by + c where a. b, and c are rational). A con ..."
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Cited by 16 (0 self)
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AbsfmctAfter a short introduction to online computing, we prove that the functions computable in online by a finite automaton are piecewise afthe functions whose coefikients are rational numbers (i.e., the fnnctions f(s) = UP + b, or f(z, y) = nx + by + c where a. b, and c are rational). A consequence of this study is that multiplication, division, and elementary functions of operands of arbitrarily long length cannot be performed using boundedshe operators. Index Zhts4omputer arithmetic, finite automata, online arithmetic. I.
Extended results for minimumadder constant integer multipliers
 in Circuits and Systems, IEEE International Symposium on. IEEE, May 2002
, 2002
"... By introducing simplifications to multiplier graphs we extend the previous work on minimum adder multipliers to five adders and show that this is enough to express all coefficients up to 19 bits. The average savings are more than 25 % for 19 bits compared with CSD multipliers. The simplifications in ..."
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Cited by 16 (5 self)
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By introducing simplifications to multiplier graphs we extend the previous work on minimum adder multipliers to five adders and show that this is enough to express all coefficients up to 19 bits. The average savings are more than 25 % for 19 bits compared with CSD multipliers. The simplifications include addition reordering and vertex reduction to see that different graphs can generate the same coefficient sets. Thus, fewer graphs need to be evaluated. A classification of the graphs reduces the effort to search the coefficient space further. 1.