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E-raamat: Algebraic Curves and Finite Fields: Cryptography and Other Applications

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Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples.

This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.
Introduction v
Generic Newton polygons for curves of given p-rank
1(22)
Jeffrey D. Achter
Rachel Pries
1 Introduction
1(2)
2 Structures in positive characteristic
3(6)
2.1 The p-rank
3(1)
2.2 Newton polygons
4(3)
2.3 Semicontinuity and purity
7(1)
2.4 Notation on stratifications and Newton polygons
8(1)
3 Stratifications on the moduli space of Abelian varieties
9(2)
3.1 The p-ranks of Abelian varieties
9(1)
3.2 Newton polygons of Abelian varieties
10(1)
4 The p-rank stratification of the moduli space of stable curves
11(3)
4.1 The moduli space of stable curves
11(1)
4.2 The p-rank stratification of Mg
12(1)
4.3 Connectedness of p-rank strata
13(1)
4.4 Open questions about the p-rank stratification
13(1)
5 Stratification by Newton polygon
14(2)
5.1 Newton polygons of curves of small genus
14(1)
5.2 Generic Newton polygons
15(1)
6 Hyperelliptic curves
16(2)
7 Some conjectures about Newton polygons of curves
18(5)
7.1 Nonexistence philosophy
19(1)
7.2 Supersingular curves
20(1)
7.3 Other nonexistence results
20(3)
Good towers of function fields
23(18)
Alp Bassa
Peter Beelen
Nhut Nguyen
1 Introduction
23(2)
2 The Drinfeld modular towers (X0(Pn))n≥0
25(7)
3 An example of a classical modular tower
32(1)
4 A tower obtained from Drinfeld modules over a different ring
33(8)
4.1 Explicit Drinfeld modules of rank 2
33(3)
4.2 Finding an isogeny
36(2)
4.3 Obtaining a tower
38(3)
Correlation-immune Boolean functions for easing counter measures to side-channel attacks
41(30)
Claude Carlet
Sylvain Guilley
1 Introduction
42(3)
2 Preliminaries
45(8)
2.1 The combiner model of pseudo-random generator in a stream cipher and correlation-immune functions
45(4)
2.2 Side-channel attacks
49(2)
2.3 Masking counter measure
51(2)
3 Methods for allowing masking to resist higher order side-channel attacks
53(5)
3.1 Leakage squeezing for first-order masking
53(2)
3.2 Leakage squeezing for second-order masking
55(1)
3.3 Rotating S-box masking
56(2)
4 New challenges for correlation-immune Boolean functions
58(13)
4.1 Basic facts on CI functions, orthogonal arrays and dual distance of codes
58(3)
4.2 Known constructions of correlation-immune functions
61(4)
4.3 Synthesis of minimal weights of d-CI Boolean functions
65(6)
The discrete logarithm problem with auxiliary inputs
71(22)
Jung Hee Cheon
Taechan Kim
Yongsoo Song
1 Introduction
72(1)
2 Algorithms for the ordinary DLP
73(5)
2.1 Generic algorithms
73(3)
2.2 Nongeneric algorithms
76(2)
3 The DLPwAI and Cheon's algorithm
78(4)
3.1 p - 1 cases
79(1)
3.2 Generalized algorithms
80(2)
4 Polynomials with small value sets
82(2)
4.1 Fast multipoint evaluation in a blackbox manner
82(1)
4.2 An approach using polynomials of small value sets
83(1)
5 Approach using the rational polynomials: Embedding to elliptic curves
84(1)
6 Generalized DLPwAI
85(2)
6.1 Representation of a multiplicative subgroup of Zxp-1
85(1)
6.2 A group action on Z*p and polynomial construction
86(1)
6.3 Main result
86(1)
7 Applications and implications
87(2)
7.1 Strong Diffie--Hellman problem and its variants
87(1)
7.2 Attack on the existing schemes using Cheon's algorithm
88(1)
8 Open problems and further work
89(4)
Garden of curves with many automorphisms
93(28)
Massimo Giulietti
Gabor Korchmaros
1 Introduction
93(1)
2 Notation and background
94(1)
3 Upper bounds on the size of G depending on g
95(1)
4 Upper bounds on the size of the p-subgroups of G depending on the p-rank
96(1)
5 Examples of curves with large automorphism groups
97(8)
5.1 Curves with unitary automorphism group
97(1)
5.2 Curves with Suzuki automorphism group
98(1)
5.3 Curves with Ree automorphism group
99(1)
5.4 The Giulietti--Korchmaros curve
99(1)
5.5 The generalized GK curve
100(1)
5.6 A curve admitting SU(3, p) as an automorphism group
101(1)
5.7 General hyperelliptic curves with a K-automorphism 2-group of order 2g + 2
101(1)
5.8 A curve with genus g = (2h - 1)2 admitting a K-automorphism 2-group of order of order 2(q - 1) + 2h+1 - 2
101(1)
5.9 General bielliptic curves with a dihedral K-automorphism 2-group of order 4(g - 1)
102(2)
5.10 A curve of genus q with a semidihedral K-automorphism 2-group of order 2(g - 1)
104(1)
6 Characterizations
105(5)
6.1 Curves with many automorphisms with respect to their genus
105(1)
6.2 Curves with a large nontame automorphism group
106(1)
6.3 Theorem 6.2 and some generalizations of Deligne--Lusztig curves
107(2)
6.4 Group-theoretic characterizations
109(1)
7 The possibilities for g when the p-rank is 0
110(2)
8 Large automorphism p-groups in positive p-rank
112(9)
8.1 p = 2
112(4)
8.2 p = 3
116(1)
8.3 p > 3
117(4)
Nonlinear shift registers -- A survey and challenges
121(24)
Tor Helleseth
1 Introduction
121(2)
2 Nonlinear shift registers
123(6)
2.1 The binary de Bruijn graph
124(2)
2.2 The pure cycling register
126(1)
2.3 The complementary cycling register
126(1)
2.4 De Bruijn sequences
126(3)
3 Mykkeltveit's proof of Golomb's conjecture
129(3)
4 The D-morphism
132(2)
5 Conjugate pairs in PCR
134(1)
6 Finite fields and conjugate pairs
135(4)
6.1 Cycle joining and cyclotomy
137(2)
7 Periodic structure of NLFSRs
139(3)
8 Conclusions
142(3)
Permutations of finite fields and uniform distribution modulo 1
145(16)
Florian Pausinger
Alev Topuzoglu
1 Introduction
145(1)
2 Preliminaries
146(4)
3 Good and weak families of permutations
150(1)
4 Existence of good families
151(1)
5 Permutation polynomials of Carlitz rank 3
152(2)
6 Bounds for f(Sσp)
154(2)
7 Computational results
156(1)
8 Concluding remarks
157(4)
Semifields, relative difference sets, and bent functions
161(18)
Alexander Pott
Kai-Uwe Schmidt
Yue Zhou
1 Introduction
161(1)
2 Semifields
162(3)
3 Relative difference sets
165(2)
4 Relative difference sets and semifields
167(4)
5 Planar functions in odd characteristic
171(1)
6 Planar functions in characteristic 2
172(1)
7 Component functions of planar functions
173(2)
8 Concluding remarks and open problems
175(4)
NTRU cryptosystem: Recent developments and emerging mathematical problems in finite polynomial rings
179(34)
Ron Steinfeld
1 Introduction
179(2)
2 Notation and preliminaries
181(2)
2.1 Notation
181(1)
2.2 Probability and algorithms
181(1)
2.3 Rings
182(1)
2.4 Lattices
182(1)
3 Review of the NTRU cryptosystem
183(6)
3.1 The NTRU construction
183(2)
3.2 Security of NTRU: Computational/statistical problems and known attacks
185(4)
4 Recent developments in security analysis of NTRU
189(11)
4.1 Overview
189(3)
4.2 Gaussian distributions modulo lattices and Fourier analysis
192(3)
4.3 Statistical hardness of the NTRU decision key cracking problem
195(3)
4.4 Computational hardness of the ciphertext cracking problem
198(2)
5 Recent developments in applications of NTRU
200(7)
5.1 NTRU-based homomorphic encryption
200(4)
5.2 NTRU-based multilinear maps
204(3)
6 Conclusions
207(6)
Analog of the Kronecker--Weber theorem in positive characteristic
213(26)
Gabriel D. Villa-Salvador
1 Introduction
213(2)
2 The classical case
215(1)
3 A proof of the Kronecker--Weber theorem based on ramification groups
216(3)
4 Cyclotomic function fields
219(2)
5 The maximal Abelian extension of k
221(2)
6 Reciprocity law
223(1)
7 The proof of David Hayes
224(1)
8 Witt vectors and the conductor
225(4)
8.1 The conductor
228(1)
8.2 The conductor according to Schmid
228(1)
9 The Kronecker--Weber--Hayes theorem
229(6)
10 Final remarks
235(4)
Index 239
Harald Niederreiter, JKU Linz, Austria; Alina Ostafe, Macquarie University, NSW, Australia; Daniel Panario, Carleton University, Ottawa, Ontario, Canada; Arne Winterhof, JKU Linz, Austria.
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