Arithmetic of Finite Fields

[1]  Peter Schwabe,et al.  Solving Binary MQ with Grover's Algorithm , 2016, SPACE.

[2]  Elinor Velasquez,et al.  Is there life on finite upper half planes , 1993 .

[3]  Jianqin Zhou,et al.  Generalized Partially Bent Functions , 2007, Future Generation Communication and Networking (FGCN 2007).

[4]  C. Dong,et al.  Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine , 1997, q-alg/9703016.

[5]  Michele Mosca,et al.  Estimating the Cost of Generic Quantum Pre-image Attacks on SHA-2 and SHA-3 , 2016, SAC.

[6]  S. Vanstone,et al.  OPTIMAL NORMAL BASES IN GF(p”)* , 2002 .

[7]  J. Littlewood.  Some problems in real and complex analysis , 1968 .

[8]  Giacomo Micheli,et al.  Irreducible polynomials over finite fields produced by composition of quadratics , 2019, Revista Matemática Iberoamericana.

[9]  Carlos J. Moreno,et al.  Algebraic curves over finite fields: Frontmatter , 1991 .

[10]  Donald L. Kreher,et al.  Combinatorial algorithms: generation, enumeration, and search , 1998, SIGA.

[11]  Xiang-dong Hou.  q-ary Bent Functions Constructed from Chain Rings , 1998 .

[12]  Patrick Solé,et al.  Connections between Quaternary and Binary Bent Functions , 2009, IACR Cryptol. ePrint Arch..

[13]  Daniel J. Katz,et al.  Low Correlation Sequences From Linear Combinations of Characters , 2017, IEEE Transactions on Information Theory.

[14]  Wiman’s and Edge’s sextics attaining Serre’s bound , 2018 .

[15]  Kaisa Nyberg,et al.  Constructions of Bent Functions and Difference Sets , 1991, EUROCRYPT.

[16]  Vincent Rijmen,et al.  The Cipher SHARK , 1996, FSE.

[17]  Yoonjin Lee,et al.  Explicit Criteria for Construction of Plateaued Functions , 2016, IEEE Transactions on Information Theory.

[18]  Dilip V. Sarwate,et al.  Bounds on aperiodic cross-correlation for binary sequences , 1976 .

[19]  Xuejia Lai,et al.  Additive and Linear Structures of Cryptographic Functions , 1994, FSE.

[20]  Lucia Moura,et al.  Low Complexity Normal Elements over Finite Fields of Characteristic Two , 2008, IEEE Transactions on Computers.

[21]  Zhou Dexin,et al.  Hardware Implementation of Finite-field Arithmetic , 2003 .

[22]  Robert J. Lechner.  HARMONIC ANALYSIS OF SWITCHING FUNCTIONS , 1971 .

[23]  Claude Carlet,et al.  Vectorial Boolean Functions for Cryptography , 2006 .

[24]  Qingju Wang,et al.  Design of Lightweight Linear Diffusion Layers from Near-MDS Matrices , 2017, IACR Trans. Symmetric Cryptol..

[25]  Arne Winterhof,et al.  Recent Results on Recursive Nonlinear Pseudorandom Number Generators - (Invited Paper) , 2010, SETA.

[26]  Igor E. Shparlinski,et al.  On the length of critical orbits of stable quadratic polynomials , 2009, 0909.3972.

[27]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[28]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[29]  Sihem Mesnager.  Characterizations of Plateaued and Bent Functions in Characteristic p , 2014, SETA.

[30]  Xiang-Dong Hou,et al.  p-Ary and q-ary versions of certain results about bent functions and resilient functions , 2004, Finite Fields Their Appl..

[31]  Joachim von zur Gathen,et al.  Polynomials with two values , 1997, Comb..

[32]  Eric Filiol,et al.  Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity , 1998, EUROCRYPT.

[33]  D. Jungnickel.  Finite fields : structure and arithmetics , 1993 .

[34]  Mahdi Sajadieh,et al.  On construction of involutory MDS matrices from Vandermonde Matrices in GF(2q) , 2011, Designs, Codes and Cryptography.

[35]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[36]  Matthew G. Parker.  Even Length Binary Sequence Families with Low Negaperiodic Autocorrelation , 2001, AAECC.

[37]  G. Mason,et al.  On vector-valued modular forms and their Fourier coefficients , 2003 .

[38]  Anthony Shaheen,et al.  Fourier expansions of complex-valued Eisenstein series on finite upper half planes , 2006, Int. J. Math. Math. Sci..

[39]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[40]  Bo-Yin Yang,et al.  Design Principles for HFEv- Based Multivariate Signature Schemes , 2015, ASIACRYPT.

[41]  Marcel J. E. Golay.  Hybrid low autocorrelation sequences (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[42]  Wilfried Meidl,et al.  Partially bent functions and their properties , 2014, Applied Algebra and Number Theory.

[43]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[44]  Sylvie Dubuc,et al.  Characterization of Linear Structures , 2001, Des. Codes Cryptogr..

[45]  Caroline Fontaine,et al.  On Some Cosets of the First-Order Reed-Muller Code with High Minimum Weight , 1999, IEEE Trans. Inf. Theory.

[46]  Yuan Li,et al.  k-th order symmetric SAC boolean functions and bisecting binomial coefficients , 2005, Discret. Appl. Math..

[47]  Jacques Patarin,et al.  About the XL Algorithm over GF(2) , 2003, CT-RSA.

[48]  Thor Martinsen.  Correlation Immunity, Avalanche Features, and Other Cryptographic Properties of Generalized Boolean Functions , 2017 .

[49]  Sumanta Sarkar,et al.  Lightweight Diffusion Layer: Importance of Toeplitz Matrices , 2016, IACR Trans. Symmetric Cryptol..

[50]  Anxiao Jiang,et al.  Rank modulation for flash memories , 2008, 2008 IEEE International Symposium on Information Theory.

[51]  Walter Rudin,et al.  Some theorems on Fourier coefficients , 1959 .

[52]  Don Zagier,et al.  The Theory of Jacobi Forms , 2013 .

[53]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[54]  Michael Rosen,et al.  Idempotent relations and factors of Jacobians , 1989 .

[55]  François Arnault,et al.  Construction of self-dual normal bases and their complexity , 2010, Finite Fields Their Appl..

[56]  A. Sebbar,et al.  Equivariant functions and integrals of elliptic functions , 2012 .

[57]  Henning Stichtenoth,et al.  A generalization of the Giulietti-Korchmaros maximal curve , 2010 .

[58]  Gregory A. Freiman,et al.  On Solvability of a System of Two Boolean Linear Equations , 1996 .

[59]  Selçuk Kavut,et al.  Modifying Maiorana-McFarland Type Bent Functions for Good Cryptographic Properties and Efficient Implementation , 2019, SIAM J. Discret. Math..

[60]  Natalia Tokareva.  Bent Functions: Results and Applications to Cryptography , 2015 .

[61]  A. Sebbar,et al.  Equivariant functions and vector-valued modular forms , 2013, 1312.2991.

[62]  A. Terras.  Harmonic Analysis on Symmetric Spaces―Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane , 2013 .

[63]  Robert L. McFarland,et al.  A Family of Difference Sets in Non-cyclic Groups , 1973, J. Comb. Theory, Ser. A.

[64]  Giovanni Zini,et al.  On maximal curves that are not quotients of the Hermitian curve , 2015, Finite Fields Their Appl..

[65]  Howard M. Heys,et al.  Hardware Design and Analysisof Block Cipher Components , 2002, ICISC.

[66]  Thomas Peyrin,et al.  Looting the LUTs: FPGA Optimization of AES and AES-like Ciphers for Authenticated Encryption , 2017, INDOCRYPT.

[67]  Tommaso Toffoli,et al.  Reversible Computing , 1980, ICALP.

[68]  V. V. Yashchenko,et al.  Bent functions on a finite Abelian group , 1997 .

[69]  Martin Rötteler,et al.  Post-Quantum Cryptography , 2015, Lecture Notes in Computer Science.

[70]  DOMINGO GÓMEZ-PÉREZ,et al.  Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators , 2014, Math. Comput..

[71]  Sihem Mesnager,et al.  Bent Functions: Fundamentals and Results , 2016 .

[72]  Claude Carlet,et al.  Boolean Functions for Cryptography and Error-Correcting Codes , 2010, Boolean Models and Methods.

[73]  Shuhong Gao,et al.  Optimal normal bases , 1992, Des. Codes Cryptogr..

[74]  Richard J. Turyn,et al.  Hadamard Matrices, Baumert-Hall Units, Four-Symbol Sequences, Pulse Compression, and Surface Wave Encodings , 1974, J. Comb. Theory, Ser. A.

[75]  Yoshinori Hamahata.  A Note on Modular Forms on Finite Upper Half Planes , 2007, WAIFI.

[76]  Joseph H. Silverman,et al.  The arithmetic of elliptic curves , 1986, Graduate texts in mathematics.

[77]  Albert C. Claus,et al.  Number Theory in Science and Communications , 1998 .

[78]  Igor E. Shparlinski,et al.  On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators , 2009, Math. Comput..

[79]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[80]  D. V. Sarwate.  Mean-square correlation of shift-register sequences , 1984 .

[81]  Motoko Qiu Kawakita.  Certain sextics with many rational points , 2017, Adv. Math. Commun..

[82]  Yuliang Zheng,et al.  On plateaued functions , 1999, IEEE Trans. Inf. Theory.

[83]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[84]  Torleiv Kløve,et al.  Some Codes Correcting Asymmetric Errors of Limited Magnitude , 2011, IEEE Transactions on Information Theory.

[85]  Cunsheng Ding,et al.  Several classes of binary sequences with three-level autocorrelation , 1999, IEEE Trans. Inf. Theory.

[86]  M. Kawakita.  ON QUOTIENT CURVES OF THE FERMAT CURVE OF DEGREE TWELVE ATTAINING THE SERRE BOUND , 2009 .

[87]  Gérard D. Cohen,et al.  On q-ary plateaued functions over Fq and their explicit characterizations , 2019, Eur. J. Comb..

[88]  Dmitri Maslov,et al.  Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning , 2013, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[89]  Igor E. Shparlinski,et al.  Dynamical Systems Generated by Rational Functions , 2003, AAECC.

[90]  Shuhong Gao,et al.  Density of Normal Elements , 1997 .

[91]  Jérôme Lacan,et al.  Systematic MDS erasure codes based on Vandermonde matrices , 2004, IEEE Communications Letters.

[92]  Lara Dolecek,et al.  Channel Coding for Nonvolatile Memory Technologies: Theoretical Advances and Practical Considerations , 2017, Proceedings of the IEEE.

[93]  Christian Mauduit.  On Finite Pseudorandom Binary Sequences , 1998 .

[94]  Miura Shinji,et al.  Algebraic geometric codes on certain plane curves , 1993 .

[95]  Peter Selinger,et al.  Quantum circuits of T-depth one , 2012, ArXiv.

[96]  Hans Dobbertin,et al.  Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity , 1994, FSE.

[97]  Sihem Mesnager,et al.  On the p-ary (cubic) bent and plateaued (vectorial) functions , 2018, Des. Codes Cryptogr..

[98]  Deng Tang,et al.  Construction of n-variable (n ≡ 2 mod 4) balanced Boolean functions with maximum absolute value in autocorrelation spectra < 2n/2 , 2016, IACR Cryptol. ePrint Arch..

[99]  Jan-Hendrik Evertse,et al.  Linear Structures in Blockciphers , 1987, EUROCRYPT.

[100]  Guang Gong,et al.  Design space exploration of the lightweight stream cipher WG-8 for FPGAs and ASICs , 2013, WESS '13.

[101]  Enes Pasalic,et al.  Efficient implementation of generalized Maiorana–McFarland class of cryptographic functions , 2016, Journal of Cryptographic Engineering.

[102]  Pantelimon Stanica,et al.  Partial spread and vectorial generalized bent functions , 2017, Des. Codes Cryptogr..

[103]  Torleiv Kløve,et al.  Codes Correcting Single Errors of Limited Magnitude , 2012, IEEE Transactions on Information Theory.

[104]  P. Bantay.  A trace formula for vector-valued modular forms , 2012, 1211.5911.

[105]  Pantelimon Stanica,et al.  Generalized Bent Functions and Their Gray Images , 2016, WAIFI.

[106]  B. Schneier.  Twofish : A 128-bit block cipher , 1998 .

[107]  Christopher Wolf,et al.  Solving Underdetermined Systems of Multivariate Quadratic Equations Revisited , 2012, Public Key Cryptography.

[108]  Robert S. Coulter,et al.  Bent polynomials over finite fields , 1997, Bulletin of the Australian Mathematical Society.

[109]  Dean G. Hoffman,et al.  A Note on a Conjecture Concerning Symmetric Resilient Functions , 1993, Inf. Process. Lett..

[110]  Daniel Gajski,et al.  Introduction to high-level synthesis , 1994, IEEE Design & Test of Computers.

[111]  Martin Hell,et al.  Grain: a stream cipher for constrained environments , 2007, Int. J. Wirel. Mob. Comput..

[112]  Claude Carlet,et al.  Two New Classes of Bent Functions , 1994, EUROCRYPT.

[113]  Taizo Shirai,et al.  Public-Key Identification Schemes Based on Multivariate Quadratic Polynomials , 2011, CRYPTO.

[114]  K. Conrad.  Finite Fields , 2004, Series and Products in the Development of Mathematics.

[115]  Guang Gong,et al.  Generating Good Span n Sequences Using Orthogonal Functions in Nonlinear Feedback Shift Registers , 2014, Open Problems in Mathematics and Computational Science.

[116]  T. Itoh,et al.  A Fast Algorithm for Computing Multiplicative Inverses in GF(2^m) Using Normal Bases , 1988, Inf. Comput..

[117]  Bo-Yin Yang,et al.  HMFEv - An Efficient Multivariate Signature Scheme , 2017, PQCrypto.

[118]  Cunsheng Ding,et al.  Duadic sequences of prime lengths , 2000, Discret. Math..

[119]  Thomas Peyrin,et al.  FOAM: Searching for Hardware-Optimal SPN Structures and Components with a Fair Comparison , 2014, CHES.

[120]  Torleiv Kløve.  Codes of Length 2 Correcting Single Errors of Limited Size , 2015, IMACC.

[121]  Sihem Mesnager,et al.  Results on Characterizations of Plateaued Functions in Arbitrary Characteristic , 2015, BalkanCryptSec.

[122]  Lilya Budaghyan.  Construction and Analysis of Cryptographic Functions , 2014, Springer International Publishing.

[123]  Wiman’s and Edge’s sextic attaining Serre’s bound II , 2014 .

[124]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[125]  Igor E. Shparlinski,et al.  Recent Advances in the Theory of Nonlinear Pseudorandom Number Generators , 2002 .

[126]  G. Mason,et al.  Vector-valued modular forms and Poincaré series , 2004 .

[127]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[128]  G. Mason,et al.  Generalized modular forms , 2003 .

[129]  Ko Stoffelen,et al.  Shorter Linear Straight-Line Programs for MDS Matrices , 2017, IACR Cryptol. ePrint Arch..

[130]  Claude Carlet,et al.  Partially-bent functions , 1992, Des. Codes Cryptogr..

[131]  Igor E. Shparlinski,et al.  On pseudorandom numbers from multivariate polynomial systems , 2010, Finite Fields Their Appl..

[132]  W. Eholzer,et al.  Modular invariance and uniqueness of conformal characters , 1994, hep-th/9407074.

[133]  J. Silvester.  Determinants of block matrices , 2000, The Mathematical Gazette.

[134]  M. Golay.  Static multislit spectrometry and its application to the panoramic display of infrared spectra. , 1951, Journal of the Optical Society of America.

[135]  Torleiv Kløve,et al.  On Non-Linear Codes Correcting Errors of Limited Size , 2017, GLOBECOM 2017 - 2017 IEEE Global Communications Conference.

[136]  Martin Hell,et al.  The Grain Family of Stream Ciphers , 2008, The eSTREAM Finalists.

[137]  Torleiv Kløve,et al.  Systematic, Single Limited Magnitude Error Correcting Codes for Flash Memories , 2011, IEEE Transactions on Information Theory.

[138]  Bella Bose,et al.  Optimal, Systematic, $q$-Ary Codes Correcting All Asymmetric and Symmetric Errors of Limited Magnitude , 2010, IEEE Transactions on Information Theory.

[139]  Pham Do Tuan,et al.  On the estimation of Fourier coefficients. , 1969 .

[140]  Aline Gouget,et al.  On the Propagation Criterion of Boolean Functions , 2004 .

[141]  Gregory V. Bard,et al.  Algebraic Cryptanalysis , 2009 .

[142]  Moshe Schwartz,et al.  Quasi-Cross Lattice Tilings With Applications to Flash Memory , 2011, IEEE Transactions on Information Theory.

[143]  Jonathan Jedwab,et al.  The L_4 norm of Littlewood polynomials derived from the Jacobi symbol , 2011, ArXiv.

[144]  Jonathan I. Hall,et al.  Construction of Even Length Binary Sequences With Asymptotic Merit Factor $6$ , 2008, IEEE Transactions on Information Theory.

[145]  Sihem Mesnager,et al.  Decomposing Generalized Bent and Hyperbent Functions , 2017, IEEE Transactions on Information Theory.

[146]  Marcel J. E. Golay.  A class of finite binary sequences with alternate auto-correlation values equal to zero (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[147]  Matthew G. Parker,et al.  Two binary sequence families with large merit factor , 2009, Adv. Math. Commun..

[148]  Petr Savický.  On the Bent Boolean Functions That are Symmetric , 1994, Eur. J. Comb..

[149]  Elinor Velasquez,et al.  Special functions on finite upper half planes , 1992 .

[150]  Kai-Uwe Schmidt,et al.  Quaternary Constant-Amplitude Codes for Multicode CDMA , 2006, IEEE Transactions on Information Theory.

[151]  Frank R. Kschischang,et al.  Fast encoding and decoding of Gabidulin codes , 2009, 2009 IEEE International Symposium on Information Theory.

[152]  Ian F. Blake,et al.  Specific irreducible polynomials with linearly independent roots over finite fields , 1997 .

[153]  Guang Gong,et al.  Communication System Security , 2012 .

[154]  H. Shapiro,et al.  Extremal problems for polynomials and power series , 1951 .

[155]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[156]  Kris Gaj,et al.  Comparison of FPGA-Targeted Hardware Implementations of eSTREAM Stream Cipher Candidates , 2008 .

[157]  Yongqiang Li,et al.  On the Construction of Lightweight Circulant Involutory MDS Matrices , 2016, FSE.

[158]  Claude Carlet.  Partially-bent functions , 1993, Des. Codes Cryptogr..

[159]  Guang Gong,et al.  Resilience to distinguishing attacks on WG-7 cipher and their generalizations , 2013, Cryptography and Communications.

[160]  Jean-Charles Faugère,et al.  On the complexity of solving quadratic Boolean systems , 2011, J. Complex..

[161]  Antoine Joux,et al.  A Crossbred Algorithm for Solving Boolean Polynomial Systems , 2017, NuTMiC.

[162]  Pantelimon Stanica,et al.  Rotation Symmetric Boolean Functions -; Count and Cryptographic Properties , 2003, Electron. Notes Discret. Math..

[163]  M. Mosca,et al.  A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits , 2012, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[164]  Pantelimon Stanica,et al.  Cryptographic Boolean Functions and Applications , 2009 .

[165]  Igor E. Shparlinski,et al.  On the linear and nonlinear complexity profile of nonlinear pseudorandom number generators , 2003, IEEE Trans. Inf. Theory.

[166]  A. Sebbar,et al.  ON THE EXISTENCE OF VECTOR-VALUED AUTOMORPHIC FORMS , 2017 .

[167]  Frédérique E. Oggier,et al.  Lightweight MDS Involution Matrices , 2015, FSE.

[168]  Siang Meng Sim,et al.  Lightweight MDS Generalized Circulant Matrices , 2016, FSE.

[169]  R. Spillman.  The effect of DON'T CARES on the complexity of combinational circuits , 1980, Proceedings of the IEEE.

[170]  Pantelimon Stanica,et al.  Bisecting binomial coefficients , 2016, Discret. Appl. Math..

[171]  Martin Rötteler,et al.  Quantum Resource Estimates for Computing Elliptic Curve Discrete Logarithms , 2017, ASIACRYPT.

[172]  Alina Ostafe,et al.  Pseudorandom Vector Sequences Derived from Triangular Polynomial Systems with Constant Multipliers , 2010, WAIFI.

[173]  Philippe Langevin.  On Generalized Bent Functions , 1993 .

[174]  Jonathan I. Hall,et al.  Modifications of Modified Jacobi Sequences , 2011, IEEE Transactions on Information Theory.

[175]  A. Atkin,et al.  Modular Forms , 2017 .

[176]  Enes Pasalic,et al.  Full Characterization of Generalized Bent Functions as (Semi)-Bent Spaces, Their Dual, and the Gray Image , 2018, IEEE Transactions on Information Theory.

[177]  Ann M. Boehmer,et al.  Binary pulse compression codes , 1967, IEEE Trans. Inf. Theory.

[178]  Nigel Jefferies.  Sporadic partitions of binomial coefficients , 1991 .

[179]  Yongqiang Li,et al.  On Constructions of a Sort of MDS Block Diffusion Matrices for Block Ciphers and Hash Functions , 2015, IACR Cryptol. ePrint Arch..

[180]  Gilles Brassard,et al.  Tight bounds on quantum searching , 1996, quant-ph/9605034.

[181]  Sugata Gangopadhyay,et al.  Bent and generalized bent Boolean functions , 2013, Des. Codes Cryptogr..

[182]  A. Youssef.  On the Design of Linear Transformations for Substitution Permutation Encryption Networks , 2007 .

[183]  Keqin Feng,et al.  Complete Characterization of Generalized Bent and 2k-Bent Boolean Functions , 2017, IEEE Transactions on Information Theory.

[184]  Torleiv Kløve,et al.  Some Codes Correcting Unbalanced Errors of Limited Magnitude for Flash Memories , 2013, IEEE Transactions on Information Theory.