walsh spectrum
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2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Suman Dutta ◽  
Subhamoy Maitra ◽  
Chandra Sekhar Mukherjee

<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>


Doklady BGUIR ◽  
2021 ◽  
Vol 19 (7) ◽  
pp. 31-39
Author(s):  
A. A. Budzko ◽  
T. N. Dvornikova

The work is devoted to the development of circuits for fast Walsh transform processors of the serialparallel type. The fast Walsh transform processors are designed for decoding error-correcting codes and synchronization; their use can reduce the cost of calculating the instantaneous Walsh spectrum by almost 2 times. The class of processors for computing the instantaneous spectrum according to Walsh is called serialparallel processors. Circuits of the fast Walsh transform processors of serial-parallel type have been developed. A comparative analysis of the constructed graphs of the fast Walsh transform processors is carried out. A method and a processor for calculating the Walsh transform coefficients are proposed, which allows increasing the speed of the transformations performed. When calculating the conversion coefficients using processors of parallel, serial and serial-parallel types, it was found that controllers of the serial-parallel type require 2(N–1) operations when calculating the instantaneous spectrum according to Walsh. The results obtained can be used in the design of discrete information processing devices, in telecommunication systems when coding signals for their noise-immune transmission and decoding, which ensures the optimal number of operations, and therefore the optimal hardware costs.


Author(s):  
Jinjin Chai ◽  
Zilong Wang ◽  
Erzhong Xue
Keyword(s):  

Author(s):  
Nikolay Kaleyski

AbstractWe define a family of efficiently computable invariants for (n,m)-functions under EA-equivalence, and observe that, unlike the known invariants such as the differential spectrum, algebraic degree, and extended Walsh spectrum, in the case of quadratic APN functions over $\mathbb {F}_{2^n}$ F 2 n with n even, these invariants take on many different values for functions belonging to distinct equivalence classes. We show how the values of these invariants can be used constructively to implement a test for EA-equivalence of functions from $\mathbb {F}_{2}^{n}$ F 2 n to $\mathbb {F}_{2}^{m}$ F 2 m ; to the best of our knowledge, this is the first algorithm for deciding EA-equivalence without resorting to testing the equivalence of associated linear codes.


2021 ◽  
Vol 2 (2) ◽  
pp. 1-27
Author(s):  
Debajyoti Bera ◽  
Sapv Tharrmashastha

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires Θ (2 n ) queries to an n -bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted λ, with query complexities that depend polynomially on λ. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in n , already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of n . Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.


Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 177
Author(s):  
José Andrés Armario

One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order 2m, Hm, can be carried out by enumerating m-variable Boolean functions with an arbitrary Walsh spectrum. As a consequence, the quotient per(Hm)/22m might be a measure of the “density” of m-variable Boolean functions with high nonlinearity.


Author(s):  
Tharrmashastha SAPV ◽  
Debajyoti Bera ◽  
Arpita Maitra ◽  
Subhamoy Maitra
Keyword(s):  

2020 ◽  
Vol 15 (1) ◽  
pp. 258-265
Author(s):  
Yu Zhou ◽  
Daoguang Mu ◽  
Xinfeng Dong

AbstractS-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (n + 1)-bit S-boxes, and one necessary and sufficient conditions of (n + 1)-bit S-boxes satisfying m-order resilient. Meanwhile, we also give one characterization of (n + 1)-bit S-boxes satisfying t-order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n-bit S-box S0 and the (n + 1)-bit S-box S (which is constructed by S0).


Author(s):  
Lotfallah Pourfaray ◽  
Modjtaba Ghorbani

A Boolean function is a function $f:\Bbb{Z}_n^2 \rightarrow \{0,1\}$ and we denote the set of all $n$-variable Boolean functions by $BF_n$. For $f\in BF_n$ the vector $[{\rm W}_f(a_0),\ldots,{\rm W}_f(a_{2n-1})]$ is called the Walsh spectrum of $f$, where ${\rm W}_f(a)= \sum_{x\in V} (-1)^{f(x) \oplus ax}$, where $V_n$ is the vector space of dimension $n$ over the two-element field $F_2$. In this paper, we shall consider the Cayley graph $\Gamma_f$ associated with a Boolean function $f$. We shall also find a complete characterization of the bent Boolean functions of order $16$ and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set $BF_n$. 


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