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

Since the proposal of differential cryptanalysis and linear cryptanalysis in 1991 and 1993, respectively, the resistance to these cryptanalyses have been studied for many cryptosystems. Moreover, some block ciphers with provable security against differential and linear cryptanalysis have been proposed. One of them is the KN cipher proposed by Knudsen and Nyberg. The KN cipher is a prototype cipher with provable security against ordinary differential cryptanalysis, and has been proved to be secure against linear cryptanalysis, too. Recently a new method of attacking block ciphers, the higher order differential attack, was proposed, and Jakobsen and Knudsen showed that the KN cipher can be attacked by this method in FSE4. In this paper, we improve this attack to reduce both of the required chosen plaintexts and running time, and apply it to the cryptanalysis of the KN cipher. We show that, for the attacking of the KN cipher with 6 rounds, the number of required chosen plaintexts can be reduced by half and running time reduced from 241 to 214, and that all round keys can be derived in only 0.02 seconds on a Sun Ultra 1 (UltraSPARC 170MHz).

Improving the Higher Order Differential Attack and Cryptanalysis of the KN Cipher

PROBLEM TO BE SOLVED: To authenticate a secret key without bringing the key out to prevent leakage of biological information.SOLUTION: An authentication apparatus 101, which can authenticate encryption information by homomorphism encryption while the encryption information remains encrypted and which can update the encryption information, receives biological information encrypted by homomorphism encryption using a public key of a user, identification information of the user, and a public key of the user, by a receiver 304. Subsequently, the authentication apparatus 101 searches for the public key and the encrypted biological information stored in an encrypted biological information table 108, from the encrypted biological information table 108 based on the identification information received at the receiver 304, by a searching part 305. After searching, the authentication apparatus 101 authenticates the user based on the retrieved public key and encrypted biological information and the received public key and encrypted biological information, by an executing part 306.

Authentication apparatus, encryption apparatus, token device, authentication method, and authentication program

In biometrics, template protection aims to protect the confidentiality of templates (i.e., enrolled biometric data) by certain conversion. At ACNS 2015, as a new approach of template protection, Takahashi et al. proposed a new concept of digital signature, called “fuzzy signature”, that uses biometric data as a private key for securely generating a signature. After that, at ACNS 2016, Matsuda et al. modified the original scheme with several relaxing requirements. A main ingredient of fuzzy signature is “linear sketch”, which incorporates a kind of linear encoding and error correction process to securely output only the difference of signing keys without revealing any biometric data. In this paper, we give recovering attacks against the linear sketch schemes proposed at ACNS 2015 and 2016. Specifically, given encoded data by linear sketch (called a “sketch”), our attacks can directly recover both the signing key and the biometric data embedded in the sketch. Our attacks make use of the special structure that a sketch has the form of a sum of an integral part and a decimal part, and biometric data is embedded in the decimal part. On the other hand, we give a simple countermeasure against our attacks and discuss the effect in both theory and practice.

Recovering Attacks Against Linear Sketch in Fuzzy Signature Schemes of ACNS 2015 and 2016

This paper improves the fast DES implementation in software proposed by Biham at the 4-th Fast Software Encryption Workshop. That is, we propose a new algorithm which reduces the number of instructions for computation of S-boxes, which is the factor dominating the performance of Biham's implementation. When we apply our algorithm to DES S-boxes, we need only 87.75 instructions in average. We can reduce the number of instructions for 1 round of DES to 942, and in total, in terms of the number of instructions, our implementation is expected to be about 8% faster than Biham's implementation.

Improved fast software implementation of block ciphers (Extended abstract)

Abstract In 2015, Fukase and Kashiwabara proposed an efficient method to find a very short lattice vector. Their method has been applied to solve Darmstadt shortest vector problems of dimensions 134 to 150. Their method is based on Schnorr’s random sampling, but their preprocessing is different from others. It aims to decrease the sum of the squared lengths of the Gram–Schmidt vectors of a lattice basis, before executing random sampling of short lattice vectors. The effect is substantiated from their statistical analysis, and it implies that the smaller the sum becomes, the shorter sampled vectors can be. However, no guarantee is known to strictly decrease the sum. In this paper, we study Fukase–Kashiwabara’s method in both theory and practice, and give a heuristic but practical condition that the sum is strictly decreased. We believe that our condition would enable one to monotonically decrease the sum and to find a very short lattice vector in fewer steps.

Takeshi Shimoyama

Papers

Improving the Higher Order Differential Attack and Cryptanalysis of the KN Cipher

Authentication apparatus, encryption apparatus, token device, authentication method, and authentication program

Recovering Attacks Against Linear Sketch in Fuzzy Signature Schemes of ACNS 2015 and 2016

Improved fast software implementation of block ciphers (Extended abstract)

Analysis of decreasing squared-sum of Gram-Schmidt lengths for short lattice vectors