289 research outputs found
On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators
In this paper we study a class of dynamical systems generated by iterations
of multivariate polynomials and estimate the degreegrowth of these iterations.
We use these estimates to bound exponential sums along the orbits of these
dynamical systems and show that they admit much stronger estimates than in the
general case and thus can be of use for pseudorandom number generation.Comment: Mathematics of Computation (to appear
Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation
Let be a positive integer. In this paper we provide a general theory to
produce full orbit sequences in the affine -dimensional space over a finite
field. For our construction covers the case of the Inversive Congruential
Generators (ICG). In addition, for we show that the sequences produced
using our construction are easier to compute than ICG sequences. Furthermore,
we prove that they have the same discrepancy bounds as the ones constructed
using the ICG.Comment: To appear in Mathematics of Computatio
Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators
ABSTRACT: We present several general results that show how algebraic dynamical systems with a slow degree growth and also rational automorphisms can be used to construct stronger pseu-dorandom number generators. We then give several concrete constructions that illustrate the applicability of these general results
Fractional jumps: complete characterisation and an explicit infinite family
In this paper we provide a complete characterisation of transitive fractional
jumps by showing that they can only arise from transitive projective
automorphisms. Furthermore, we prove that such construction is feasible for
arbitrarily large dimension by exhibiting an infinite class of projectively
primitive polynomials whose companion matrix can be used to define a full orbit
sequence over an affine space
An Algebraic System for Constructing Cryptographic Permutations over Finite Fields
In this paper we identify polynomial dynamical systems over finite fields as
the central component of almost all iterative block cipher design strategies
over finite fields. We propose a generalized triangular polynomial dynamical
system (GTDS), and give a generic algebraic definition of iterative (keyed)
permutation using GTDS. Our GTDS-based generic definition is able to describe
widely used and well-known design strategies such as substitution permutation
network (SPN), Feistel network and their variants among others. We show that
the Lai-Massey design strategy for (keyed) permutations is also described by
the GTDS. Our generic algebraic definition of iterative permutation is
particularly useful for instantiating and systematically studying block ciphers
and hash functions over aimed for multiparty computation and
zero-knowledge based cryptographic protocols. Finally, we provide the
discrepancy analysis a technique used to measure the (pseudo-)randomness of a
sequence, for analyzing the randomness of the sequence generated by the generic
permutation or block cipher described by GTDS
Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences
Automatic sequences are not suitable sequences for cryptographic applications
since both their subword complexity and their expansion complexity are small,
and their correlation measure of order 2 is large. These sequences are highly
predictable despite having a large maximum order complexity. However, recent
results show that polynomial subsequences of automatic sequences, such as the
Thue--Morse sequence, are better candidates for pseudorandom sequences. A
natural generalization of automatic sequences are morphic sequences, given by a
fixed point of a prolongeable morphism that is not necessarily uniform. In this
paper we prove a lower bound for the maximum order complexity of the sum of
digits function in Zeckendorf base which is an example of a morphic sequence.
We also prove that the polynomial subsequences of this sequence keep large
maximum order complexity, such as the Thue--Morse sequence.Comment: 23 pages, 5 figures, 4 table
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