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

https://www.jstage.jst.go.jp/article/essfr/1/3/1_3_3_58/_pdf

Recent Topics on Integer Factorization

A matching method includes: generating a first numerical vector; generating a second numerical vector by squaring each component of the first numerical vector and a third numerical vector by cubing each component of the first numerical vector; generating first to third polynomials by executing polynomial transformation of the first to third numerical vectors; encrypting the first to third polynomials by a homomorphic encryption scheme; executing a predetermined operation while keeping data used in the predetermined operation encrypted, by using fourth to sixth polynomials obtained by the polynomial transformation and the homomorphic encryption of fourth to sixth numerical vectors, wherein the fourth numerical vector is generated by numerically vectorizing second text, the fifth numerical vector is generated by squaring each component of the fourth numerical vector, and the sixth numerical vector is generated by cubing each component of the fourth numerical vector; and decrypting a result of the predetermined operation.

Information processing technique for pattern matching

With the widespread development of biometrics, concerns about security and privacy are increasing. In biometrics, template protection technology aims to protect the confidentiality of biometric templates (i.e., enrolled biometric data) by certain conversion. The fuzzy commitment scheme gives a practical way to protect biometric templates using a conventional error-correcting code. The scheme has both concealing and binding of templates, but it has some privacy problems. Specifically, in case of successful matching, stored biometric templates can be revealed. To address such problems, we improve the scheme. Our improvement is to coat with two error-correcting codes. In particular, our scheme can conceal stored biometric templates even in successful matching. Our improved scheme requires just conventional error-correcting codes as in the original scheme, and hence it gives a practical solution for both template security and privacy of biometric templates.

Privacy-preserving fuzzy commitment for biometrics via layered error-correcting codes

A key generation device includes a generation circuit, a concealment processing unit, and a cryptography processing unit. The generation circuit generates a value dependent on hardware. When acquiring a concealed cryptographic key, the concealment processing unit generates first data by performing a mask process to the concealed cryptographic key by using the value generated by the generation circuit, generates second data by decoding the first data by a first error correction decoding method, and generates a cryptographic key by decoding the second data by a second error correction decoding method. When acquiring the concealed cryptographic key and a plain text or an encrypted text, the cryptography processing unit acquires the cryptographic key corresponding to the concealed cryptographic key from the concealment processing unit, and encrypts the plain text or decrypts the encrypted text by using the cryptographic key.

Takeshi Shimoyama

Papers

Recent Topics on Integer Factorization

Information processing technique for pattern matching

Privacy-preserving fuzzy commitment for biometrics via layered error-correcting codes

Key generation device and key generation method

Key Length Estimation of Pairing-Based Cryptosystems Using η T Pairing over GF(3 n )