Introduction -- Contents -- Generic Newton polygons for curves of given p-rank -- 1 Introduction -- 2 Structures in positive characteristic -- 2.1 The p-rank -- 2.2 Newton polygons -- 2.3 Semicontinuity and purity -- 2.4 Notation on stratifications and Newton polygons -- 3 Stratifications on the moduli space of Abelian varieties -- 3.1 The p-ranks of Abelian varieties -- 3.2 Newton polygons of Abelian varieties -- 4 The p-rank stratification of the moduli space of stable curves -- 4.1 The moduli space of stable curves -- 4.2 The p-rank stratification of Mg -- 4.3 Connectedness of p-rank strata -- 4.4 Open questions about the p-rank stratification -- 5 Stratification by Newton polygon -- 5.1 Newton polygons of curves of small genus -- 5.2 Generic Newton polygons -- 6 Hyperelliptic curves -- 7 Some conjectures about Newton polygons of curves -- 7.1 Nonexistence philosophy -- 7.2 Supersingular curves -- 7.3 Other nonexistence results -- Good towers of function fields -- 1 Introduction -- 2 The Drinfeld modular towers (X0(Pn))n≥0 -- 3 An example of a classical modular tower -- 4 A tower obtained from Drinfeldmodules over a different ring -- 4.1 Explicit Drinfeld modules of rank 2 -- 4.2 Finding an isogeny -- 4.3 Obtaining a tower -- Correlation-immune Boolean functions for easing counter measures to side-channel attacks -- 1 Introduction -- 2 Preliminaries -- 2.1 The combiner model of pseudo-random generator in a stream cipher and correlation-immune functions -- 2.2 Side-channel attacks -- 2.3 Masking counter measure -- 3 Methods for allowing masking to resist higher order side-channel attacks -- 3.1 Leakage squeezing for first-order masking -- 3.2 Leakage squeezing for second-order masking -- 3.3 Rotating S-box masking -- 4 New challenges for correlation-immune Boolean functions.

4.1 Basic facts on CI functions, orthogonal arrays and dual distance of codes -- 4.2 Known constructions of correlation-immune functions -- 4.3 Synthesis of minimal weights of d-CI Boolean functions -- The discrete logarithm problem with auxiliary inputs -- 1 Introduction -- 2 Algorithms for the ordinary DLP -- 2.1 Generic algorithms -- 2.2 Nongeneric algorithms -- 3 The DLPwAI and Cheon's algorithm -- 3.1 p - 1 cases -- 3.2 Generalized algorithms -- 4 Polynomials with small value sets -- 4.1 Fast multipoint evaluation in a blackbox manner -- 4.2 An approach using polynomials of small value sets -- 5 Approach using the rational polynomials: Embedding to elliptic curves -- 6 Generalized DLPwAI -- 6.1 Representation of a multiplicative subgroup of Z×p-1 -- 6.2 A group action on Z*p and polynomial construction -- 6.3 Main result -- 7 Applications and implications -- 7.1 Strong Diffie-Hellman problem and its variants -- 7.2 Attack on the existing schemes using Cheon's algorithm -- 8 Open problems and further work -- Garden of curves with many automorphisms -- 1 Introduction -- 2 Notation and background -- 3 Upper bounds on the size of G depending on g -- 4 Upper bounds on the size of the p-subgroups of G depending on the p-rank -- 5 Examples of curves with large automorphism groups -- 5.1 Curves with unitary automorphism group -- 5.2 Curves with Suzuki automorphism group -- 5.3 Curves with Ree automorphism group -- 5.4 The Giulietti-Korchmáros curve -- 5.5 The generalized GK curve -- 5.6 A curve admitting SU(3, p) as an automorphism group -- 5.7 General hyperelliptic curves with a K-automorphism 2-group of order 2g + 2 -- 5.8 A curve with genus g = (2h - 1)2 admitting a K-automorphism 2-group of order of order 2(g - 1) + 2h+1 - 2.

5.9 General bielliptic curves with a dihedral K-automorphism 2-group of order 4(g - 1) -- 5.10 A curve of genus g with a semidihedral K-automorphism 2-group of order 2(g - 1) -- 6 Characterizations -- 6.1 Curves with many automorphisms with respect to their genus -- 6.2 Curves with a large nontame automorphism group -- 6.3 Theorem 6.2 and some generalizations of Deligne-Lusztig curves -- 6.4 Group-theoretic characterizations -- 7 The possibilities for G when the p-rank is 0 -- 8 Large automorphism p-groups in positive p-rank -- 8.1 p = 2 -- 8.2 p = 3 -- 8.3 p > 3 -- Nonlinear shift registers - A survey and challenges -- 1 Introduction -- 2 Nonlinear shift registers -- 2.1 The binary de Bruijn graph -- 2.2 The pure cycling register -- 2.3 The complementary cycling register -- 2.4 De Bruijn sequences -- 3 Mykkeltveit's proof of Golomb's conjecture -- 4 The D-morphism -- 5 Conjugate pairs in PCR -- 6 Finite fields and conjugate pairs -- 6.1 Cycle joining and cyclotomy -- 7 Periodic structure of NLFSRs -- 8 Conclusions -- Permutations of finite fields and uniform distribution modulo 1 -- 1 Introduction -- 2 Preliminaries -- 3 Good and weak families of permutations -- 4 Existence of good families -- 5 Permutation polynomials of Carlitz rank 3 -- 6 Bounds for f(Sσp) -- 7 Computational results -- 8 Concluding remarks -- Semifields, relative difference sets, and bent functions -- 1 Introduction -- 2 Semifields -- 3 Relative difference sets -- 4 Relative difference sets and semifields -- 5 Planar functions in odd characteristic -- 6 Planar functions in characteristic 2 -- 7 Component functions of planar functions -- 8 Concluding remarks and open problems.

NTRU cryptosystem: Recent developments and emerging mathematical problems in finite polynomial rings -- 1 Introduction -- 2 Notation and preliminaries -- 2.1 Notation -- 2.2 Probability and algorithms -- 2.3 Rings -- 2.4 Lattices -- 3 Review of the NTRU cryptosystem -- 3.1 The NTRU construction -- 3.2 Security of NTRU: Computational/statistical problems and known attacks -- 4 Recent developments in security analysis of NTRU -- 4.1 Overview -- 4.2 Gaussian distributions modulo lattices and Fourier analysis -- 4.3 Statistical hardness of the NTRU decision key cracking problem -- 4.4 Computational hardness of the ciphertext cracking problem -- 5 Recent developments in applications of NTRU -- 5.1 NTRU-based homomorphic encryption -- 5.2 NTRU-based multilinearmaps -- 6 Conclusions -- Analog of the Kronecker-Weber theorem in positive characteristic -- 1 Introduction -- 2 The classical case -- 3 A proof of the Kronecker-Weber theorem based on ramification groups -- 4 Cyclotomic function fields -- 5 The maximal Abelian extension of k -- 6 Reciprocity law -- 7 The proof of David Hayes -- 8 Witt vectors and the conductor -- 8.1 The conductor -- 8.2 The conductor according to Schmid -- 9 The Kronecker-Weber-Hayes theorem -- 10 Final remarks -- Index.