39 research outputs found
Algebraic construction of semi bent function via known power function
The study of semi bent functions (2- plateaued Boolean function) has attracted the attention of many researchers due to their cryptographic and combinatorial properties. In this paper, we have given the algebraic construction of semi bent functions defined over the finite field Fââż (n even) using the notion of trace function and Gold power exponent. Algebraically constructed semi bent functions have some special cryptographical properties such as high nonlinearity, algebraic immunity, and low correlation immunity as expected to use them effectively in cryptosystems. We have illustrated the existence of these properties with suitable examples.Publisher's Versio
On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes
International audienceThe Walsh transform \hat{Q} of a quadratic function Q : F2^n â F2 satisfies |\hat{Q(b)}| â {0, 2 n+s 2 } for all b â F_{2^n} , where 0 †s †n â 1 is an integer depending on Q. In this article, we investigate two classes of such quadratic Boolean functions which attracted a lot of research interest. For arbitrary integers n we determine the distribution of the parameter s for both of the classes, C1 = {Q(x) = Tr_n(\sum^{(nâ1)/2}_{ i=1} a_ix^{2^i +1}) : a_i â F2}, and the larger class C2, defined for even n as C2 = {Q(x) = Tr_n(^{(n/2)â1}_ { i=1} a_ix^{2^i +1}) + Tr_n/2 (a_{n/2} x^{2^n/2 +1}) : a_i â F2}. Our results have two main consequences. We obtain the distribution of the non-linearity for the rotation symmetric quadratic Boolean functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the corresponding subcodes of the second order Reed-Muller codes
Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension
The Walsh transform QËQ^ of a quadratic function Q:FpnâFpQ:FpnâFp satisfies |QË(b)|â{0,pn+s2}|Q^(b)|â{0,pn+s2} for all bâFpnbâFpn , where 0â€sâ€nâ10â€sâ€nâ1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(ââ(nâ1)/2âi=1aix2i+1):aiâF2}C1={Q(x)=Trn(âi=1â(nâ1)/2âaix2i+1):aiâF2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(â(n/2)â1i=1aix2i+1)+Trn/2(an/2x2n/2+1):aiâF2}C2={Q(x)=Trn(âi=1(n/2)â1aix2i+1)+Trn/2(an/2x2n/2+1):aiâF2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(âân/2âi=0aixpi+1):aiâFp}D={Q(x)=Trn(âi=0ân/2âaixpi+1):aiâFp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order ReedâMuller codes corresponding to C1C1 and C2C2 in terms of a generating function
Value Distributions of Perfect Nonlinear Functions
In this paper, we study the value distributions of perfect nonlinear
functions, i.e., we investigate the sizes of image and preimage sets. Using
purely combinatorial tools, we develop a framework that deals with perfect
nonlinear functions in the most general setting, generalizing several results
that were achieved under specific constraints. For the particularly interesting
elementary abelian case, we derive several new strong conditions and
classification results on the value distributions. Moreover, we show that most
of the classical constructions of perfect nonlinear functions have very
specific value distributions, in the sense that they are almost balanced.
Consequently, we completely determine the possible value distributions of
vectorial Boolean bent functions with output dimension at most 4. Finally,
using the discrete Fourier transform, we show that in some cases value
distributions can be used to determine whether a given function is perfect
nonlinear, or to decide whether given perfect nonlinear functions are
equivalent.Comment: 28 pages. minor revisions of the previous version. The paper is now
identical to the published version, outside of formattin
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented