Parallel Fast Möbius (Reed-Muller) Transform and its Implementation with CUDA on GPUs

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

2020

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.

Section: Preliminary Resultsmentioning

confidence: 63%

A Method for Fast Computing the Algebraic Degree of Boolean Functions

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

2020

“…But the experimental results show that the bitwise version of the algorithm is about 25 times faster in comparison to the byte-wise version 3 . Analogous research concerning the parallel bitwise implementation of the ANFT is represented in [4] and similar results about its efficiency are obtained. After these results it is natural to think about a bitwise implementation of the last algorithm.…”

Section: Bitwise Approachmentioning

confidence: 80%

Fast Computing the Algebraic Degree of Boolean Functions

2019

Algebraic Informatics

Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the vectors of the n-dimensional Boolean cube. Byte-wise and bitwise versions of a search based on the WLO and their implementations are discussed. They are compared with the usual exhaustive search applied in computing the algebraic degree. For Boolean functions of n variables, the bitwise implementation of the search by WLO has total time complexity O(n.2 n ). When such a function is given by its truth table vector and its algebraic degree is computed by the bitwise versions of the algorithms discussed, the total time complexity is Θ((9n − 2).2 n−7 ) = Θ(n.2 n ). All algorithms discussed have time complexities of the same type, but with big differences in the constants hidden in the Θ-notation. The experimental results after numerous tests confirm the theoretical results-the running times of the bitwise implementation are dozens of times better than the running times of the byte-wise algorithms.

“…Comparing the results obtained here (for example, milliseconds for computing the ANF of one function) with these in [2], the reader can choose easily when to use a parallel version instead of a serial (non-parallel) version and vice versa. Finally, as we mentioned at the beginning of this work, we comment on the numerical results from Section 4 of [6].…”

Section: Methodology Of Testingmentioning

confidence: 85%

“…There the authors write: "For n = 13; 14 we have not enough memory in our computer to obtain the ANF" (here n is the number of variables). This section (Table 2) and the results from [2] show why this result is very questionable for us. Obviously the implementation of the ANFT in [6] is not efficient enough and hence the comparison results are not realistic.…”

Section: Methodology Of Testingmentioning

confidence: 86%

See 1 more Smart Citation

Fast Bitwise Implementation of the Algebraic Normal Form Transform

2017

SJC

The representation of Boolean functions by their algebraic normalforms (ANFs) is very important for cryptography, coding theory andother scientific areas. The ANFs are used in computing the algebraic degreeof S-boxes, some other cryptographic criteria and parameters of errorcorrectingcodes. Their applications require these criteria and parameters tobe computed by fast algorithms. Hence the corresponding ANFs should alsobe obtained by fast algorithms. Here we continue our previous work on fastcomputing of the ANFs of Boolean functions. We present and investigatethe full version of bitwise implementation of the ANF transform. The experimental results show that this implementation ismore than 25 times faster in comparison to the well-known byte-wise ANFtransform.ACM Computing Classification System (1998): F.2.1, F.2.2.