T-functions revisited: new criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and a fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz p-adic functions and to study measure-preservation/ergodicity of these.

[1]  Akhil Mathew,et al.  The p-adic Numbers , 2009 .

[2]  H. Lausch,et al.  Algebra of Polynomials , 1974 .

[3]  B. Bagchi,et al.  Latin squares , 2012 .

[4]  G. Mullen,et al.  Discrete Mathematics Using Latin Squares , 1998, The Mathematical Gazette.

[5]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[6]  Vladimir Anashin,et al.  Characterization of ergodicity of p-adic dynamical systems by using the van der Put basis , 2011 .

[7]  Vladimir Anashin,et al.  Uniformly distributed sequences of p-adic integers, II , 2002, math/0209407.

[8]  Adi Shamir,et al.  A New Class of Invertible Mappings , 2002, CHES.

[9]  Nicholas Kolokotronis.  Cryptographic properties of nonlinear pseudorandom number generators , 2008, Des. Codes Cryptogr..

[10]  N. Koblitz.  p-adic Numbers, p-adic Analysis, and Zeta-Functions , 1977 .

[11]  Andrey Bogdanov,et al.  ABC: A New Fast Flexible Stream Cipher , 2005 .

[12]  M. V. Larin,et al.  Transitive polynomial transformations of residue class rings , 2002 .

[13]  Vladimir Anashin,et al.  Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers , 1998 .

[14]  Adi Shamir,et al.  Cryptographic Applications of T-Functions , 2003, Selected Areas in Cryptography.

[15]  Tor Helleseth,et al.  Alinear weakness in the Klimov-Shamir T-function , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[16]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[17]  Vladimir Anashin,et al.  Non-Archimedean Ergodic Theory and Pseudorandom Generators , 2007, Comput. J..

[18]  Robert A. Beezer.  Discrete Mathematics Using Latin Squares, by Charles F. Laywine, Gary L. Mullen , 2002 .

[19]  Tor Helleseth,et al.  Linear Properties in T-Functions , 2006, IEEE Transactions on Information Theory.

[20]  Gerhard Goos,et al.  Fast Software Encryption , 2001, Lecture Notes in Computer Science.

[21]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[22]  Andrei Khrennikov,et al.  Applied Algebraic Dynamics , 2009 .

[23]  Vladimir Anashin,et al.  Pseudorandom number generation by p-adic ergodic transformations: an addendum , 2004, ArXiv.

[24]  Chuan-Kun Wu,et al.  The Algebraic Normal Form, Linear Complexity and k-Error Linear Complexity of Single-Cycle T-Function , 2006, SETA.

[25]  K. Mahler.  p-adic numbers and their functions , 1981 .

[26]  Vladimir Anashin,et al.  Ergodic Transformations in the Space of p‐Adic Integers , 2006, math/0602083.

[27]  Marcus Nilsson,et al.  P-adic Deterministic and Random Dynamics , 2004 .

[28]  S. V. Kozyrev,et al.  On p-adic mathematical physics , 2006, 0904.4205.

[29]  Adi Shamir,et al.  New Cryptographic Primitives Based on Multiword T-Functions , 2004, FSE.

[30]  Dongdai Lin,et al.  Ergodic theory over F2[[T]] , 2012, Finite Fields Their Appl..

[31]  A. Khrennikov.  Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena , 2004 .

[32]  Dongdai Lin,et al.  Linear Relation on General Ergodic T-Function , 2011, ArXiv.

[33]  Vladimir Anashin.  Automata finiteness criterion in terms of van der Put series of automata functions , 2011, ArXiv.

[34]  Ekaterina Yurova.  Van der Put basis and p-adic dynamics , 2010 .

[35]  Dongdai Lin,et al.  Linear Weaknesses in T-functions , 2012, SETA.

[36]  V. S. Anachin.  Uniformly distributed sequences ofp-adic integers , 1994 .

[37]  Andrei Khrennikov,et al.  Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models , 2011 .

[38]  Vladimir Anashin,et al.  The Non-Archimedean Theory of Discrete Systems , 2011, Math. Comput. Sci..

[39]  Wen-Feng Qi,et al.  Linear Equation on Polynomial Single Cycle T-Functions , 2007, Inscrypt.

[40]  Kai-Thorsten Wirt.  ASC – A Stream Cipher with Built – In MAC Functionality , 2007 .

[41]  Dong Hoon Lee,et al.  A New Class of Single Cycle T-Functions , 2005, FSE.

[42]  Adi Shamir,et al.  New Applications of T-Functions in Block Ciphers and Hash Functions , 2005, FSE.