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

An encryption server receives encrypted data which is encrypted by using a public key of a first user and which is added with auxiliary information calculated using a combination of a secret key and the public key of the first user and data to be encrypted. Then, the encryption server generates re-encrypted data obtained by re-encrypting the encrypted data using the auxiliary information added to the encrypted data, a public key of a second user, and encrypted information related to the secret key of the first user which is encrypted with the public key of the second user.

Encryption device, encryption method, information processing device, and encryption system

In this paper, we discuss solving the DLP over GF(36·97) by using the function field sieve (FFS) for breaking paring-based cryptosystems using the ηT pairing over GF(397). The extension degree 97 has been intensively used in benchmarking tests for the implementation of the ηT pairing, and the order (923-bit) of GF(36·97) is substantially larger than the previous world record (676-bit) of solving the DLP by using the FFS. We implemented the FFS for the medium prime case, and proposed several improvements of the FFS. Finally, we succeeded in solving the DLP over GF(36·97). The entire computational time requires about 148.2 days using 252 CPU cores.

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Breaking pairing-based cryptosystems using η T pairing over GF(3 97 ).

PROBLEM TO BE SOLVED: To perform an arithmetic operation using a linear conversion part in an SPN structure excellent in a data spreading performance, when the numbers of input and output bits are not the same among plural S-boxes in the SPN structure in an F-function SOLUTION: This arithmetic unit is provided with a means 1 for receiving an input of a set of bit numbers unequally dividing the total number of bits of input data to be given to the arithmetic unit 1, and a means 3 for outputting a value AT presenting a possibility of existence of a suitable linear conversion part corresponding to the plural S-boxes having the divided bit numbers as their input-output bit numbers, respectively Further, the arithmetic unit is provided with a means 4 for judging that the proper linear conversion part exists when the value of AT is positive, and a means 5 for generating a pseudo MDS matrix as such a linear conversion part

Arithmetic unit and method using spn structure in f- function

A ciphertext processing device acquires a first polynomial into which first text data is converted with a first order and which is encrypted with a first public key and a first square-value polynomial into which square-value vector data of each component of the first text data is converted with the first order and which is encrypted with the first public key. The ciphertext processing device acquires a second polynomial into which second text data is converted with a second order and which is encrypted with the first public key and a second square-value polynomial into which square-value vector data of each component of the second text data is converted with the second order and which is encrypted with the first public key. The ciphertext processing device determines whether the second text data is included in the first text data, using the each polynomial.

Ciphertext processing device, ciphertext processing method, computer-readable recording medium, and information processing device

Pollard's rho method is the asymptotically fastest known attack for the elliptic curve discrete logarithm problem (ECDLP) except special cases It works by giving a pseudo-random sequence defined by an iteration function and then detecting a collision in the sequence We note that the number of iterations before obtaining a collision is significant for the running time of the rho method and depends on the choice of an iteration function For many iteration functions suitable for the ECDLP on elliptic curves except Koblitz curves, the number of iterations before obtaining a collision had been investigated In this paper, we propose a new iteration function on Koblitz curves which is an extension of the iteration function proposed by Gallant et al and analyze the performance on our iteration function experimentally

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Takeshi Shimoyama

Papers

Encryption device, encryption method, information processing device, and encryption system

Breaking pairing-based cryptosystems using η T pairing over GF(3 97 ).

Arithmetic unit and method using spn structure in f- function

Ciphertext processing device, ciphertext processing method, computer-readable recording medium, and information processing device

On random walks of Pollard's rho method for the ECDLP on Koblitz curves