• The Pure and Applied Mathematics
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• v.6 no.2
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• pp.79-86
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• 1999

A Boolean function generates a binary sequence which is frequently used in a stream cipher. There are number of critical concepts which a Boolean function, as a key stream generator in a stream cipher, satisfies. These are nonlinearity, correlation immunity, balancedness, SAC (strictly avalanche criterion), PC (propagation criterion) and so on. In this paper we construct an algorithm for finding the correlation immune order of a Boolean function, and check how long to find the correlation immune order of a given Boolean function in our algorithm.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.14 no.4
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• pp.49-59
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• 2004

Boolean function synthesize problem is to find a boolean expression with in/outputs of original function. This problem can be modeled into a 0-1 integer programming. In this paper, we find a boolean expressions of S-boxes of DES for an example, whose algebraic structure has been unknown for many years. The results of this paper can be used for efficient hardware implementation of a function and cryptanalysis using algebraic structure of a block cipher.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.561-574
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• 2023

In this paper, we focus on establishing the MacWilliams-type identities on vectorial Boolean functions with bent component functions. As their applications, we provide a bound for the non-existence of vectorial dual-bent functions with prescribed minimum degree, and several Gleason-type theorems are presented as well.

• The Transactions of the Korean Institute of Electrical Engineers P
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• v.53 no.4
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• pp.210-215
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• 2004

This paper proposes an implicit method for computing the minimum cost feedback vertex set for a graph. For an arbitrary graph, a Boolean function is derived, whose satisfying assignments directly correspond to feedback vertex sets of the graph. Importantly, cycles in the graph are never explicitly enumerated, but rather, are captured implicitly in this Boolean function. This function is then used to determine the minimum cost feedback vertex set. Even though computing the minimum cost satisfying assignment for a Boolean function remains an NP-hard problem, it is possible to exploit the advances made in the area of Boolean function representation in logic synthesis to tackle this problem efficiently in practice for even reasonably large sized graphs. The algorithm has obvious application in flip-flop selection for partial scan. The algorithm proposed in this paper is the first to obtain the MFVS solutions for many benchmark circuits.

• Journal of the Korean Institute of Telematics and Electronics
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• v.21 no.4
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• pp.43-51
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• 1984

In the case of Quine Mccluskey's methode for Boolean function minimization, we have to examine each bits of binary represented minterms. In this paper, cube relations between misterms that are represented by means of decimal number, and all sorts of rules for Boolean function minimization are described as theorems, and they are verified. And based on these theorems, the new fast algorithm for Boolean function minimization is proposed. An example of manual operation is sholvn, and the process is writed out by a FORTRAN program. In this program, the essential pl.imp implicants of the Boolean function that has 100 each of minterms including redundant minterms, are finked and printed out, (the more minterms can be treated if we take the more larger size of arrays) and the outputs are coincided with the results of manual operation.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.515-526
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• 2021

Denote by ��+ the set of probability measures supported on ℝ+. Suppose V�� is the variance function of the Cauchy-Stieltjes Kernel (CSK) family ��-(��) generated by a non degenerate probability measure �� ∈ ��+. We determine the formula for variance function under boolean multiplicative convolution power. This formula is used to identify the relation between variance functions under the map ${\nu}{\mapsto}{\mathbb{M}}_t({\nu})=({\nu}^{{\boxtimes}(t+1)})^{{\uplus}{\frac{1}{t+1}}}$ from ��+ onto itself.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.11 no.6
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• pp.127-134
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• 2001

At PKC\`98, SangUk Shin et al. proposed a new hash function based on advantages of SHA-1, RIPEMD-160, and HAVAL. They claimed that the Boolean functions of the hash function have good properties including the SAC(Strict Avalanche Criterion). In this paper, we first show that some of Boolean functions which are used in Shin\`s hash function does not satisfy the SAC, and then argue that satisfying the SAC may not be a good property of Boolean functions, when it is used for constructing compress functions of a hash function.

• Kyungpook Mathematical Journal
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• v.48 no.2
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• pp.195-206
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• 2008

The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties.

• Proceedings of the KIEE Conference
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• 1987.07a
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• pp.226-229
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• 1987

This paper was aimed to, using a new data structure, develop a set of algorithms to execute the output function of Digital System. These functions were represented as directed, acyclic graphs. by applying many restrictions on vertices on graph, the efficient manipulation of boolean function was accomplished. The results were as follows; 1. A canonical representation of a boolean function was created by the reduction algorithm. 2. The operation of two functions was accomplished using t he apply algorithm, according to the binary operator. 3. The arguments having 1 as the value nf function were enumerated using the satisfy algorithm. 4. Composing TTL 74181 4-bit ALU and 74182 look-ahead carry generator, the ALU having 4-bit and 16-bit as word size was implemented.

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.47-56
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• 2004

The nonlinearity of a Boolean function f on $GF(2)^n$ is the minimum hamming distance between f and all affine functions on $GF(2)^n$ and it measures the ability of a cryptographic system using the functions to resist against being expressed as a set of linear equations. Finding out the exact value of the nonlinearity of given Boolean functions is not an easy problem therefore one wants to estimate the nonlinearity using extra information on given functions, or wants to find a lower bound or an upper bound on the nonlinearity. In this paper we extend the notion of auto-correlations of Boolean functions to vector Boolean functions and obtain upper bounds and a lower bound on the nonlinearity of vector Boolean functions in the context of their auto-correlations. Also we can describe avalanche characteristics of vector Boolean functions by examining the extended notion of auto-correlations.