scholarly journals Laced Boolean functions and subset sum problems in finite fields

2011 ◽  
Vol 159 (11) ◽  
pp. 1059-1069 ◽  
Author(s):  
David Canright ◽  
Sugata Gangopadhyay ◽  
Subhamoy Maitra ◽  
Pantelimon Stănică
Keyword(s):  
Finite Fields ◽  
Boolean Functions ◽  
Subset Sum ◽  
Subset Sum Problems

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

scholarly journals Two Boolean Functions with Five-Valued Walsh Spectra and High Nonlinearity

2015 ◽  
Vol 26 (05) ◽  
pp. 537-556 ◽  
Author(s):  
Xiwang Cao ◽  
Lei Hu
Keyword(s):  
Finite Fields ◽  
Boolean Functions ◽  
Exponential Sums ◽  
New Method ◽  
High Nonlinearity ◽  
Walsh Spectra ◽  
And Diffusion ◽  
Confusion And Diffusion

For cryptographic systems the method of confusion and diffusion is used as a fundamental technique to achieve security. Confusion is reflected in nonlinearity of certain Boolean functions describing the cryptographic transformation. In this paper, we present two Boolean functions which have low Walsh spectra and high nonlinearity. In the proof of the nonlinearity, a new method for evaluating some exponential sums over finite fields is provided.

scholarly journals Pairs of Quadratic Forms over Finite Fields

10.37236/4072 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Alexander Pott ◽  
Kai-Uwe Schmidt ◽  
Yue Zhou
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Quadratic Forms ◽  
Symmetric Matrices ◽  
Explicit Expressions

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.

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