From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." J.B. Little, MAA, March 2004 The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.

A Singular Introduction to Commutative Algebra

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

To appear as a chapter of the volume " Boolean Methods and Models " ,

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Vectorial Boolean Functions for Cryptography

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Computational methods in commutative algebra and algebraic geometry

Twofish is a 128-bit block cipher that accepts a variable-length key up to 256 bits. The cipher is a 16-round Feistel network with a bijective F function made up of four key-dependent 8-by-8-bit S-boxes, a fixed 4-by-4 maximum distance separable matrix over GF(2), a pseudo-Hadamard transform, bitwise rotations, and a carefully designed key schedule. A fully optimized implementation of Twofish encrypts on a Pentium Pro at 17.8 clock cycles per byte, and an 8-bit smart card implementation encrypts at 1660 clock cycles per byte. Twofish can be implemented in hardware in 14000 gates. The design of both the round function and the key schedule permits a wide variety of tradeoffs between speed, software size, key setup time, gate count, and memory. We have extensively cryptanalyzed Twofish; our best attack breaks 5 rounds with 2 chosen plaintexts and 2 effort.

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Twofish : A 128-bit block cipher

We analyze the security of the SEED block cipher against differential attacks. SEED is a 16-round Feistel cipher developed by the Korea Information Security Agency. The SEED proposers estimated their cipher against differential cryptanalysis in a self-estimation document and found a six-round differential characteristic with probability 2 -130 . We present an improved method of examining the differential characteristics of SEED and show three six-round differential characteristics with probability 2 -124 . These characteristics allow us to attack seven-round SEED, which surpasses the proposers estimation. Our differential attack needs 2 126 chosen-plaintext pairs and 2 126 computations of the F function to deduce the subkey used in the last round of seven-round SEED.

Differential cryptanalysis of a reduced-round SEED

In this paper, we give the theoretical analysis of χ attack proposed by Knudsen and Meier on the RC6 block cipher. To this end, we propose the novel method of security evaluation against χ attack precisely including key dependency by introducing a technique “Transition Matrix Computing.” On the other hand, the way of security evaluation against χ attack has not been known except the computer experiment. We should note that it is the first results the way of security evaluation against χ attack is shown theoretically. Using this method, we can obtain the “weakest keys” against the attack. Keyword: RC6, χ attack, Transition Matrix,

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Theoretical Analysis of "Correlations in RC6".

A proximity determination method includes: collecting a first concealed vector based on a linear combination of a row vector of a key matrix, first numerical information, and a first random number, the key matrix being generated by adding a random number vector as a last column to a matrix in which a threshold for determining a proximity range is a diagonal element; collecting a second concealed vector based on the linear combination, second numerical information, and a second random number; calculating a remainder vector, the remainder vector being a remainder obtained from a difference between the collected first concealed vector and the second concealed vector when the key matrix is used as a modulus; and determining whether the first numerical information and the second numerical information belong to the proximity range, based on the calculated remainder vector.

Proximity determination method, proximity determination device, and proximity determination system

An information processing method according to an embodiment causes a computer(300) to execute a process of receiving an input of matching information encrypted with an encryption algorithm allowing a Hamming distance to be calculated with the matching information encrypted. The information processing method also causes the computer to execute a process of calculating(S3) a first Hamming distance between the received matching information and registered information that is different from encrypted registered information of a user, the registered information being encrypted with the encryption algorithm. The information processing method also causes the computer to execute a process of determining(S7) legitimacy of the matching information based on whether the calculated first Hamming distance falls into a distance distribution representing matches with another person that is different from the user.

Information processing method, information processing program, and information processing apparatus

PROBLEM TO BE SOLVED: To reduce computational complexity and also improve data spreading performance in a common key block ciphering system. SOLUTION: One or more means 2 for performing data conversion using a Feistel structure and one or more means 3 for performing data conversion using an SPN structure are cascaded between a data input and a data output. In an SPN structure, a non-linear conversion means is used, by which the linear relational expression related only to a half number of fixed input bits and a half number of output bits at the same position satisfies all the probabilities as 1/2 holding between all the input and output data.

Takeshi Shimoyama

Papers

Differential cryptanalysis of a reduced-round SEED

Theoretical Analysis of "Correlations in RC6".

Proximity determination method, proximity determination device, and proximity determination system

Information processing method, information processing program, and information processing apparatus

Arithmetic unit combining feistel structure and spn structure and arithmetic method