Fast Computing the Algebraic Degree of Boolean Functions

Bakoev, Valentin

doi:10.1007/978-3-030-21363-3_5

Cited by 4 publications

(12 citation statements)

References 6 publications

(13 reference statements)

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“…1. If we compare it with the corresponding Figure 1 in [4] we will notice the evolution and improvements we discuss here. When Boolean functions (BFs) of 6 and more variables are tested, 2 n−6 consecutive integers are taken from the dynamic array and so they form the serial Boolean function.…”

Section: Resultsmentioning

confidence: 97%

“…We have conducted a lot of tests to verify and understand what these theoretical time complexities mean in practice. To obtain more precise results in comparison with these in [4], we have done some essential changes in the tests' parameters, methodology of testing, etc. So, we used the largest test file (of size ≈ 14 GB) which contains 10 9 randomly generated unsigned integers, each in a single 64-bit computer word.…”

Section: Resultsmentioning

confidence: 99%

“…But using the same algorithms for the remaining half f ∈ B n is not efficient enough-this conclusion follows from the analysis of the numerical results (in Sect. 4 of [10]) and after comparing them with the results from [4]. That is why for this half of Boolean functions we will use the first way of computing the algebraic degree and thus we combine both ways.…”

Section: A Methods For Fast Computing Thementioning

confidence: 99%

“…In [4] we outlined a new idea for another bitwise algorithm-to check the bits of A f according to the WLO sequence. So it will be analogous to the byte-wise WLO algorithm and it will have the same time complexity O(2 n ).…”

Section: A Methods For Fast Computing Thementioning

confidence: 99%

“…In [4] we consider two approaches in computing the algebraic degree: a byte-wise and a bitwise. The first approach includes ES algorithm and the so called Byte-wise WLO algorithm.…”

Section: Fast Computing the Algebraic Degree Of Boolean Functionsmentioning

confidence: 99%

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A Method for Fast Computing the Algebraic Degree of Boolean Functions

Bakoev

2020

Proceedings of the 21st International Conference on Computer Systems and Technologies

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The algebraic degree of Boolean functions (or vectorial Boolean functions) is an important cryptographic parameter that should be computed by fast algorithms. They work in two main ways: (1) by computing the algebraic normal form and then searching the monomial of the highest degree in it, or (2) by examination the algebraic properties of the true table vector of a given function. We have already done four basic steps in the study of the first way, and the second one has been studied by other authors. Here we represent a method for fast computing (the fastest way we know) the algebraic degree of Boolean functions. It is a combination of the most efficient components of these two ways and the corresponding algorithms. The theoretical time complexities of the method are derived in each of the cases when the Boolean function is represented in a byte-wise or in a bitwise manner. They are of the same type Θ(n.2 n) for a Boolean function of n variables, but they have big differences between the constants in Θ-notation. The theoretical and experimental results shown here demonstrate the advantages of the bitwise approach in computing the algebraic degree-they are dozens of times faster than the byte-wise approaches. CCS CONCEPTS • Security and privacy → Software and application security; • Theory of computation → Cryptographic primitives; Design and analysis of algorithms; • Mathematics of computing → Combinatorial algorithms.

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Section: Resultsmentioning

confidence: 97%

Section: Resultsmentioning

confidence: 99%

Section: A Methods For Fast Computing Thementioning

confidence: 99%

Section: A Methods For Fast Computing Thementioning

confidence: 99%

“…In [4] we consider two approaches in computing the algebraic degree: a byte-wise and a bitwise. The first approach includes ES algorithm and the so called Byte-wise WLO algorithm.…”

Section: Fast Computing the Algebraic Degree Of Boolean Functionsmentioning

confidence: 99%

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A Method for Fast Computing the Algebraic Degree of Boolean Functions

Bakoev

2020

Proceedings of the 21st International Conference on Computer Systems and Technologies

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Complexity Evaluation of Constructing Method for Saturated Best Resilient Functions in Stream Cipher Design

Sadkhan

Jawad

2019

2019 1st AL-Noor International Conference for Science and Technology (NICST)

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Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

Bakoev

2020

Discrete Math. Algorithm. Appl.

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The problem “Given a Boolean function [Formula: see text] of [Formula: see text] variables by its truth table vector. Find (if exists) a vector [Formula: see text] of maximal (or minimal) weight, such that [Formula: see text].” is considered here. It is closely related to the problem of computing the algebraic degree of Boolean functions which is an important cryptographic parameter. To solve this problem efficiently, we explore the orders of the vectors of the [Formula: see text]-dimensional Boolean cube [Formula: see text] according to their weights. The notion of “[Formula: see text]th layer” of [Formula: see text] is involved in the definition and examination of the “weight order” relation. It is compared with the known relation “precedes”. Several enumeration problems concerning these relations are solved and the relevant notes were added to three sequences in the on-line encyclopedia of integer sequences (OEIS). One special weight order is defined and examined in detail. In it, the lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called weight-lexicographic order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. The results obtained by them were used in creating two new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed — the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results from numerous tests confirm the efficiency of these algorithms. Other applications of the obtained algorithms are also discussed — when representing, generating and ranking other combinatorial objects.

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Fast Computing the Algebraic Degree of Boolean Functions

Cited by 4 publications

References 6 publications

A Method for Fast Computing the Algebraic Degree of Boolean Functions

A Method for Fast Computing the Algebraic Degree of Boolean Functions

Complexity Evaluation of Constructing Method for Saturated Best Resilient Functions in Stream Cipher Design

Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube

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