scholarly journals On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Brajesh Kumar Singh
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Coding Theory ◽  
Block Ciphers ◽  
Covering Radius ◽  
Third Order ◽  
Complicated Problem

The rth-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complicated problem, especially when r>1. This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form Tr1nλxd for all x∈𝔽2n, λ∈𝔽2n*, where a d=2i+2j+2k+1, where i, j, and   k are integers such that i>j>k≥1 and n>2i, and b d=23ℓ+22ℓ+2ℓ+1, where ℓ is a positive integer such that gcdℓ,𝓃=1 and n>6.

ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

2014 ◽  
Vol 25 (03) ◽  
pp. 343-353 ◽  
Author(s):  
YU ZHOU ◽  
LIN WANG ◽  
WEIQIONG WANG ◽  
XINFENG DONG ◽  
XIAONI DU
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Sum Of Squares ◽  
Necessary Condition ◽  
Resilient Function ◽  
The Absolute ◽  
Global Avalanche Characteristics ◽  
Sufficient And Necessary Condition

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

LOWER BOUNDS ON THE SECOND ORDER NONLINEARITY OF BOOLEAN FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1331-1349 ◽  
Author(s):  
XUELIAN LI ◽  
YUPU HU ◽  
JUNTAO GAO
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order ◽  
Algebraic Degree ◽  
Positive Integers ◽  
Second Order Nonlinearity ◽  
Better Than ◽  
Monomial Functions

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

scholarly journals Crypto-Archaeology: unearthing design methodology of DES s-boxes

2017 ◽  
Author(s):  
Sankhanil Dey ◽  
Ranjan Ghosh
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Design Methodology ◽  
Block Cipher ◽  
Block Ciphers ◽  
Public Domain ◽  
Differential Cryptanalysis ◽  
Linear Cryptanalysis ◽  
Strict Avalanche Criterion ◽  
Independence Criterion

US defence sponsored the DES program in 1974 and released it in 1977. It remained as a well-known and well accepted block cipher until 1998. Thirty-two 4-bit DES S-Boxes are grouped in eight each with four and are put in public domain without any mention of their design methodology. S-Boxes, 4-bit, 8-bit or 32-bit, find a permanent seat in all future block ciphers. In this paper, while looking into the design methodology of DES S-Boxes, we find that S-Boxes have 128 balanced and non-linear Boolean Functions, of which 102 used once, while 13 used twice and 92 of 102 satisfy the Boolean Function-level Strict Avalanche Criterion. All the S-Boxes satisfy the Bit Independence Criterion. Their Differential Cryptanalysis exhibits better results than the Linear Cryptanalysis. However, no S-Boxes satisfy the S-Box-level SAC analyses. It seems that the designer emphasized satisfaction of Boolean-Function-level SAC and S-Box-level BIC and DC, not the S-Box-level LC and SAC.

scholarly journals On self-correcting logic circuits of unreliable gates

2021 ◽  
pp. 1-18
Author(s):  
Kirill Andreevich Popkov
Keyword(s):  
Positive Integer ◽  
Boolean Function ◽  
Boolean Functions ◽  
Arbitrary Number ◽  
Logic Circuits ◽  
Logic Circuit ◽  
Complete Basis

The following statements are proved: 1) for any integer m ≥ 3 there is a basis consisting of Boolean functions of no more than m variables, in which any Boolean function can be implemented by a logic circuit of unreliable gates that self-corrects relative to certain faults in an arbitrary number of gates; 2) for any positive integer k there are bases consisting of Boolean functions of no more than two variables, in each of which any Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to certain faults in no more than k gates; 3) there is a functionally complete basis consisting of Boolean functions of no more than two variables, in which almost no Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to at least some faults in no more than one gate.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

scholarly journals Lower Bounds for $q$-ary Codes with Large Covering Radius

10.37236/222 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Immanuel Halupczok ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Efficient Method ◽  
Winning Strategy ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Original Proof ◽  
Sphere Covering

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

On the number of threshold functions

1993 ◽  
Vol 3 (4) ◽  
Author(s):  
A.A. Irmatov
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Lower Bounds ◽  
Threshold Function ◽  
Threshold Functions ◽  
Large N

AbstractA Boolean function is called a threshold function if its truth domain is a part of the n-cube cut off by some hyperplane. The number of threshold functions of n variables P(2, n) was estimated in [1, 2, 3]. Obtaining the lower bounds is a problem of special difficulty. Using a result of the paper [4], Zuev in [3] showed that for sufficiently large nP(2, n) > 2In the present paper a new proof which gives a more precise lower bound of P(2, n) is proposed, namely, it is proved that for sufficiently large nP(2, n) > 2

Quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence baed on MUBs

2021 ◽  
Author(s):  
ZHANG Fu Gang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Mutually Unbiased Bases ◽  
Uncertainty Relations ◽  
Quantum Uncertainty ◽  
Single Qubit ◽  
Qubit System ◽  
Special Case

Abstract In this paper, we discuss quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence for a single qubit system based on three mutually unbiased bases. For $\alpha\in[\frac{1}{2},1)\cup(1,2]$, the upper and lower bounds of sums of coherence are obtained. However, the above results cannot be verified directly for any $\alpha\in(0,\frac{1}{2})$. Hence, we only consider the special case of $\alpha=\frac{1}{n+1}$, where $n$ is a positive integer, and we obtain the upper and lower bounds. By comparing the upper and lower bounds, we find that the upper bound is equal to the lower bound for the special $\alpha=\frac{1}{2}$, and the differences between the upper and the lower bounds will increase as $\alpha$ increases. Furthermore, we discuss the tendency of the sum of coherence, and find that it has the same tendency with respect to the different $\theta$ or $\varphi$, which is opposite to the uncertainty relations based on the R\'{e}nyi entropy and Tsallis entropy.

scholarly journals Integer Points in Knapsack Polytopes and $s$-Covering Radius

10.37236/2887 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Iskander Aliev ◽  
Martin Henk ◽  
Eva Linke
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bounds ◽  
Covering Radius ◽  
Frobenius Number ◽  
Lower And Upper Bounds ◽  
Integer Points ◽  
Knapsack Polytope ◽  
Optimal Lower Bound

Given a matrix $A\in \mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider for a positive integer $s$ the set ${\mathcal F}_s(A)\subset \mathbb{Z}^m$ of all vectors $b\in \mathbb{Z}^m$ such that the associated knapsack polytope\begin{equation*}P(A, b)=\{ x \in \mathbb{R}^n_{\ge 0}: A x= b\}\end{equation*}contains at least $s$ integer points. We present lower and upper bounds on the so called diagonal $s$-Frobenius number associated to the set ${\mathcal F}_s(A)$. In the case $m=1$ we prove an optimal lower bound for the $s$-Frobenius number, which is the largest integer $b$ such that $P(A,b)$ contains less than $s$ integer points.  

Some classes of concatenated quantum codes: constructions and lower bounds

2015 ◽  
pp. 385-405
Author(s):  
Hachiro Fujita
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Coding Theory ◽  
Minimum Distance ◽  
Concatenated Codes ◽  
Quantum Codes ◽  
Decoding Algorithm ◽  
Quantum Analogue

In classical coding theory code concatenation is successfully used to construct good errorcorrecting codes and most of the asymptotically good codes known so far are based on concatenation. In this paper we present some classes of asymptotically good concatenated quantum codes, which are a quantum analogue of classical concatenated codes, and derive lower bounds on the minimum distance and the rate of the codes. Our bounds improve on the best lower bound of Ashikhmin–Litsyn–Tsfasman and Matsumoto for rates smaller than about one half. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.

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