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A New Characterization of Semibent and Bent Functions on Finite Fields
 Information,” ISO/IEC International Standard
, 2004
"... We present a new characterization of semibent and bent quadratic functions on finite fields. First, we determine when a GF (2)linear combination of Gold functions T r(x ) is semibent over ), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler charact ..."
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We present a new characterization of semibent and bent quadratic functions on finite fields. First, we determine when a GF (2)linear combination of Gold functions T r(x ) is semibent over ), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semibent functions. For example, we deduce that all linear combinations of Gold functions give rise to semibent functions over GF (2 ) when p belongs to a certain class of primes. Second, we generalize our results to fields GF (p ) where p is an odd prime and n is odd. In that case, we can determine if a GF (p)linear combination of Gold functions T r(x is (generalized) semibent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these pary semibent and bent functions are provided. 1
Nonlinearity and security of selfsynchronizing stream ciphers
 In Proc. of the 2005 International Symposium on Nonlinear Theory and its Applications (NOLTA 2005
, 2005
"... Abstract—Several proposed chaos based ciphers exploit the ergodic property of chaotic orbits. As chaotic systems are unstable and have sensitive dependence on initial conditions, the main difficulty for the receiver is to reproduce the chaotic signal that has been generated by the sender in order ..."
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Abstract—Several proposed chaos based ciphers exploit the ergodic property of chaotic orbits. As chaotic systems are unstable and have sensitive dependence on initial conditions, the main difficulty for the receiver is to reproduce the chaotic signal that has been generated by the sender in order to correctly decrypt the message. This is performed by a self synchronizing device. In discrete cryptography, the closest scheme is the so called self synchronizing stream cipher (SSSC). After recalling general security models for assessing cryptographic algorithms, we present SSSC scheme and two examples of cryptanalysis. In order to resist to theses attacks, the ciphering function must satisfy high nonlinearity properties which are presented. 1.
Highly Nonlinear Balanced Sboxes with Improved Bound on Unrestricted and Generalized Nonlinearity ∗
, 2008
"... We construct two classes of balanced Sboxes with high nonlinearity 2n−1 − 2(n−1)/2 for n odd. From known results, it can be deduced that for any Sbox which has nonlinearity 2n−1−2(n−1)/2, the unrestricted nonlinearity is lower bounded by 2n−1 − 2(m+n−1)/2 while the generalized nonlinearity is lowe ..."
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Cited by 1 (0 self)
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We construct two classes of balanced Sboxes with high nonlinearity 2n−1 − 2(n−1)/2 for n odd. From known results, it can be deduced that for any Sbox which has nonlinearity 2n−1−2(n−1)/2, the unrestricted nonlinearity is lower bounded by 2n−1 − 2(m+n−1)/2 while the generalized nonlinearity is lower bounded by 2n−1 − (2m − 1)2(n−1)/2. We prove that the lower bound on the unrestricted nonlinearity of both our Sbox constructions can be increased to 2n−1 − 2(m+n)/2−1. For the first class of Sbox, the lower bound on generalized nonlinearity can be increased to 2n−1−2(n−1)/2+m−1. For the second class, the generalized nonlinearity is proven to be exactly 2n−1 − 2(m+n)/2−1, which is much higher than the lower bound for our first construction. The first class of Sboxes have low maximum differential while the second class corresponds to GMW sequences, whose algebraic structure allows us to construct a larger family of Sboxes. Moreover, both classes of Sboxes can attain high algebraic degree. We also compare our constructions with some known functions with high unrestricted and/or generalized nonlinearity.
Searching Extrema on the Regular Graph of Balanced Boolean Maps
"... Abstract — The class of Boolean functions has a structure of a Boolean algebra and adimensional vector space over the prime field of characteristic two. Here we consider the graph whose nodes are the balanced Boolean functions and whose edges are pairs of maps with sum a balanced map. This is a reg ..."
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Abstract — The class of Boolean functions has a structure of a Boolean algebra and adimensional vector space over the prime field of characteristic two. Here we consider the graph whose nodes are the balanced Boolean functions and whose edges are pairs of maps with sum a balanced map. This is a regular graph with degree ` ´. Within this graph it is posed the problem to find extreme values of the non linearity operator, calculated through the WalshHadamard transform. The search of maximally nonlinear balanced maps is of great interest in several areas of information security, e.g. Sboxes design, messages integrity and participants authentication. We present some experimental results regarding the performance of local search and some other heuristic techniques based on simulated annealing, giving all of them the same average approximation. Index Terms — Boolean Function, balanced, nonlinearity, graph.
A Composition Construction of BentLike Boolean Functions from Quadratic Polynomials
, 2003
"... Abstract: In this paper, we generalize the composition construction of Khoo et al. for highly nonlinear Boolean functions ([1]). We utilize general quadratic forms instead of the trace map in the construction. The construction composes an nvariable Boolean function and an mvariable F 2 quadratic f ..."
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Abstract: In this paper, we generalize the composition construction of Khoo et al. for highly nonlinear Boolean functions ([1]). We utilize general quadratic forms instead of the trace map in the construction. The construction composes an nvariable Boolean function and an mvariable F 2 quadratic form over to get an nmvariable Boolean function with beautiful spectrum n property and a doubled algebraic degree. Especially, the method is suitable to construct functions with 3valued spectra (bentlike functions) or ones with better spectra (nearbent functions). Our proof technique is based on classification of quadratic forms over finite fields and enumeration of solutions of quadratic equations. We also prove the pary analogy of these results for odd prime p.
A New Characterization of Semibent and Bent Functions on
, 2005
"... Abstract We present a new characterization of semibent and bent quadratic functions on finite fields.First, we determine when a GF (2)linear combination of Gold functions T r(x2i+1) is semibent over ..."
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Abstract We present a new characterization of semibent and bent quadratic functions on finite fields.First, we determine when a GF (2)linear combination of Gold functions T r(x2i+1) is semibent over