Çetin Kaya Koç

Bit-level systolic arrays for modular multiplication

This paper presents bit-level cellular arrays implementing Blakleys algorithm for multiplication of twon-bit integers modulo anothern-bit integer. The semi-systolic version uses 3n(n+3) single-bit carry save adders and 2n copies of 3-bit carry look-ahead logic, and computes a pair of binary numbers (C, S) in 3n clock cycles such thatC+Sε[0, 2N). The carry look-ahead logic is used to estimate the sign of the partial product, which is needed during the reduction process. The final result in the correct range [0,N) can easily be obtained by computingC+S andC+S−N, and selecting the latter if it is positive; otherwise, the former is selected. We construct a localized process dependence graph of this algorithm, and introduce a systolic array containing 3nw simple adder cells. The latency of the systolic array is 6n+w−2, wherew=⌈n/2⌉. The systolic version does not require broadcast and can be used to efficiently compute several modular multiplications in a pipelined fashion, producing a result in every clock cycle.

High-radix and bit recoding techniques for modular exponentiation

Algorithms that make use of high-radix and bit recoding techniques to perform modular exponentiation are proposed. It is shown that the high-radix methods with optimal choice of the radix provide significant reductions in the number of multiplications required for modular exponentiation. It is then shown that bit recoding techniques similar to those used in binary multiplication algorithms can be used to further reduce the total number of multiplications. The algorithms presented are analyzed by counting the maximum and the average number of multiplications required.

Systolic computation of interpolating polynomials

Several time-optimal and spacetime-optimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating polynomial. Multivariate generalized divided differences are shown to be efficiently computed on a 2-dimensional systolic array. The techniques also are applied to the Neville algorithm, producing similar results.ZusammenfassungZur Berechnung eines zum Aitken-Algorithmus gehörigen Prozeßabhängigkeitsgraphen werden einige Zeit-optimale und Raum-Zeit-optimale systolische Felder vorgestellt. Es wird gezeigt, daß man diese Felder auch zur Berechnung verallgemeinerter dividierter Differenzen verwenden kann, wie sie als Koeffizienten des Hermiteschen Interpolationspolynoms auftreten. Die effiziente Berechnung multivariater verallgemeinerter dividierter Differenzen auf einem zweidimensionalen systolischen Feld wird gezeigt. Für den Neville-Algorithmus ergeben die Techniken ähnliche Ergebnisse.

Parallel prefix computation with few processors

Abstract We present a parallel prefix algorithm which uses (2(p + 1) p (p + 1) + 2 )n − 1 arithmetic and (p (p − 1) p (p + 1) + 2 )n + ( 1 2 ) p (p − 1) routing steps to compute the prefixes of n elements on a distributed-memory multiprocessor with p n nodes. The algorithm is compared with the distributed-memory implementation of the parallel prefix algorithm proposed by Kruskal, Rudolph, and Snir. We show that there is a trade-off between the two algorithms in terms of the number of processors, and the parameter τ = τ R τ A , which is the ratio of the time required to transfer an operand to the time required to perform the operation of the prefix problem. The new algorithm is shown to be more efficient when n is large and p 2 (p − 1) ≤ 4 τ .

Adaptive m-ary segmentation and canonical recoding algorithms for multiplication of large binary numbers

We propose a variable-length segmentation strategy which significantly reduces the average number of additions required by the m-ary segmentation and the canonical recodiing algorithms for multiplication of large binary numbers. This strategy produces two new algorithms: the adaptive m-ary segmentation algorithm utilizes both the speedup inherent in high-radix multiplication and the ability to skip zero bits; the adaptive m-ary segmentation canonical recoding algorithm gains additional benefit from the increased probability of zero after the canonical recoding. The average number of additions required is computed using Markov chains.

Fast computation of divided differences and parallel hermite interpolation

Abstract We present parallel algorithms for fast polynomial interpolation. These algorithms can be used for constructing and evaluating polynomials interpolating the function values and its derivatives of arbitrary order (Hermite interpolation). For interpolation, the parallel arithmetic complexity is O(log2 M + log N) for large M and N, where M − 1 is the order of the highest derivative information and N is the number of distinct points used. Unlike alternate approaches which use the Lag-range representation, the algorithms described in this paper are based on the fast parallel evaluation of a closed formula for the generalized divided differences. Applications to the solution of dual Vandermonde and confluent Vandermonde systems are described. This work extends previous results in polynomial interpolation and improves the parallel time complexity of existing algorithms. 1989 Academic Press, Inc.

A parallel method for fast and practical high-order Newton interpolation

We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n2), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.

Parallel hermite interpolation: an algebraic approach

Givenn+1 distinct points and arbitrary order derivative information at these points, a parallel algorithm to compute the coefficients of the corresponding Hermite interpolating polynomial inO (logn) parallel arithmetic operations usingO (n2) processors is presented. The algorithm relies on a novel closed formula that yields the expansion of the generalized divided differences in terms of the given function and derivative values. We show that each one of the coefficients in this expansion and the required linear combinations can be evaluated efficiently.The particular cases where up to first and second order derivative information is available are treated in detail. The proof of the general case, where arbitrarily high order derivative information is available, involves algebraic arguments that make use of the theory of symmetric, functions.ZusammenfassungGegeben seienn+1 verschiedene Punkte sowie die Werte von Ableitungen beliebiger Ordnung in diesen Punkten. Für die Berechnung der Koeffizienten des zugehörigen Hermiteschen Interpolationspolynoms wird ein paralleler Algorithmus vorgestellt, derO (logn) parallele arithmetische Operationen aufO (n2) Prozessoren benötigt. Der Algorithmus basiert auf einer neuartigen geschlossenen Darstellung der verallgemeinerten Differenzenquotienten durch die gegebenen Funktions- und Ableitungswerte. Wir zeigen, daß sowohl die Koeffizienten in dieser Darstellung als auch die benötigten Linearkombinationen effizient berechnet werden können.Detailliert behandelt werden die Spezialfälle, daß die Ableitungen bis zur ersten bzw. zweiten Ordnung bekannt sind. Für den Beweis des allgemeinen Falles, wo Ableitungswerte beliebiger höherer Ordnung verfügbar sind, wird ein algebraischer Zugang gewählt, bei dem die Theorie symmetrischer Funktionen herangezogen wird.

A parallel algorithm for exact solution of linear equations via congruence technique

Abstract We present a parallel algorithm for computing the exact solution of a system of linear equations via the congruence technique. The basic idea of the technique is to convert the original system of equations into a system of congruences modulo various primes, and combine the solutions by the application of the Chinese remainder theorem. The parallel congruence algorithm proposed in this paper requires only local communication among the processors and is particularly suitable for implementation on distributed-memory multiprocessors and systolic computing systems. We have implemented the parallel congruence algorithm on an Intel cube and obtained an efficiency of greater than 90%.

Schwarz-Christoffel transformation for the simulation of two-dimensional capacitance (VLSI circuits)

An inherent problem in the use of simulators for the determination of capacitance in VLSI circuits is the verification of the reliability of the simulation. The problem is due to the numerical approximations made in order to achieve a versatile simulation. The Schwarz-Christoffel transformation provides theoretically exact simulation of a limited class of problems consisting of two odd shaped conductors embedded in a uniform dielectric. It is proposed that the Schwarz-Christoffel technique can be used to calibrate simulators designed for more general problems. >