Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA new class of hyper-bent Boolean functions in binomial forms
87
0
0.0
(
0
)
Added by
Chunming Tang
Publication date
2011
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}pm 2^{frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $mathbb{F}_{2^n}$ by $f_{a,b}:=mathrm{Tr}_{1}^{n}(ax^{(2^m-1)})+mathrm{Tr}_{1}^{4}(bx^{frac{2^n-1}{5}})$, where $n=2m$, $mequiv 2pmod 4$, $ain mathbb{F}_{2^m}$ and $binmathbb{F}_{16}$. When $ain mathbb{F}_{2^m}$ and $(b+1)(b^4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x^5+x+a^{-1}$, we present a characterization of hyper-bentness of $f_{a,b}$. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of $f_{a,b}$ in the case $ainmathbb{F}_{2^{frac{m}{2}}}$.
rate research
Read More
Let $mathbb{F}_{p^{n}}$ be the finite field with $p^n$ elements and $operatorname{Tr}(cdot)$ be the trace function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, where $p$ is a prime and $n$ is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form $f(x)=g(x)+F(operatorname{Tr}(u_1x),operatorname{Tr}(u_2x),cdots,operatorname{Tr}(u_{tau}x))$, where $g(x)$ is a function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, $taugeq2$ is an integer, $F(x_1,cdots,x_n)$ is a reduced polynomial in $mathbb{F}_{p}[x_1,cdots,x_n]$ and $u_iin mathbb{F}^{*}_{p^n}$ for $1leq i leq tau$. As a consequence, we obtain a generic result on the Walsh transform of $f(x)$ and characterize the bentness of $f(x)$ when $g(x)$ is bent for $p=2$ and $p>2$ respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions $f(x)$ when $g(x)$ is not bent for the first time and present a class of bent functions from non-bent Gold functions.
Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $alpha$. A recent conjecture postulated by Courtade and Kumar states that for any Boolean function $f:{0,1}^nto{0,1}$, $I(f(X^n);Y^n)le 1-H(alpha)$. Although the conjecture has been proved to be true in the dimension-free high noise regime by Samorodnitsky, here we present a calculus-based approach to show a dimension-dependent result by examining the second derivative of $H(alpha)-H(f(X^n)|Y^n)$ at $alpha=1/2$. Along the way, we show that the dictator function is the most informative function in the high noise regime.
In this paper, we investigate the power functions $F(x)=x^d$ over the finite field $mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$. It is proved that $F(x)=x^d$ is APcN at certain $c$s in $mathbb{F}_{2^{4n}}$, and it is the second class of APcN power functions over finite fields of even characteristic. Further, the $c$-differential spectrum of these power functions is also determined.
Differential uniformity is a significant concept in cryptography as it quantifies the degree of security of S-boxes respect to differential attacks. Power functions of the form $F(x)=x^d$ with low differential uniformity have been extensively studied in the past decades due to their strong resistance to differential attacks and low implementation cost in hardware. In this paper, we give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the differential uniformity of $F(x)=x^d$ over $mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$, and we completely determine its differential spectrum.
In this paper we present a new class of convolutional codes that admits an efficient al- gebraic decoding algorithm. We study some of its properties and show that it can decode interesting sequences of errors patterns. The second part of the paper is devoted to in- vestigate its use in a variant of the McEliece cryptosystem. In contrast to the classical McEliece cryptosystems, where block codes are used, we propose the use of a convolu- tional encoder to be part of the public key. In this setting the message is a sequence of messages instead of a single block message and the errors are added randomly throughout the sequence. We conclude the paper providing some comments on the security. Although there is no obvious security threats to this new scheme, we point out several possible adaptations of existing attacks and discuss the difficulties of such attacks to succeed in breaking this cryptosystem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
