There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

Es wird ein Algorithmus zur Berechnung des Produktes von zweiN-stelligen Dualzahlen angegeben. Zwei Arten der Realisierung werden betrachtet: Turingmaschinen mit mehreren Bandern und logische Netze (aus zweistelligen logischen Elementen aufgebaut).

Schnelle Multiplikation großer Zahlen

In 1979 considerable progress was made in estimating the complexity of matrix multiplication. Here the new techniques and recent results are presented, based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication (some of the entries of the matrices may be zero) can efficiently be utilized to perform multiplication of large total matrices. By combining Pan’s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of $N \times N$ matrices (over arbitrary fields) is possible in time $O(N^\beta )$, where $\beta $ is a bit smaller than $3\ln 52/\ln 110 \approx 2.522$.

Partial and Total Matrix Multiplication

Polynomial multiplication of degree N can be accomplished in time O (N · log N) provided the scalar field contains suitable roots of unity. Otherwise at least O (N · log N · log log N) is obtained by a modified version of the Schonhage-Strassen multiplication which employs computations modulo 1 + xn (where N = 2n), if the field contains 2?1, or modulo 1 + xn + x2N, if 3?1 exists. The latter method covering all fields of characteristic 2 is presented here in detail.

Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2

In 1970 the author introduced a new machine model [A. Schonhage, Universelle Turing Speicherung, Dorr, Hotz, eds., Automatentheorie and Formale Sprachen, Bibliogr. Institut, Mannheim, 1970, pp. 369–383] now called storage modification machine (SMM). This paper gives a comprehensive presentation of our present knowledge of SMMs. It contains a complete description of the SMM model and its real time equivalence to the so-called successor RAMS. The preliminary version [A. Schonhage, Real-time simulation of multidimensional Turing machines by storage modification machines, Technical Memorandum 37, M.I.T. Project MAC, Cambridge, MA, 1973] of our proof for the real time simulation of multi-dimensional Turing machines is now worked out in its details. Moreover we show the existence of an SMM that performs integer-multiplication in linear time. The final discussion contains a brief comment on the relationship between SMMs and Kolmogorov algorithms [A. N. Kolmogorov and V. A. Uspenskii, On the definition of an algo...

Arnold Schönhage

Papers

Schnelle Multiplikation großer Zahlen

Partial and Total Matrix Multiplication

Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2

Storage Modification Machines

Finding the Median