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Abstract: In this paper, we introduce the concept of k-integral graphs. A graph Γ is called k-integral if the extension degree of the splitting field of the characteristic polynomial of Γ over rational field ℚ is equal to k. We prove that for any positive integers k and Δ, the set of all finite connected graphs with algebraic degree at most k and maximum degree at most Δ is finite. We study 2-integral Cayley graphs over finite groups G with respect to Cayley sets which are a union of conjugacy classes of G. Among other general results, we completely characterize all finite abelian groups having a connected 2-integral Cayley graph with valency 2, 3, 4 and 5. Furthermore, we classify the finite groups G that al Cayley graphs over G with bounded valency are 2-integral.
Keywords: Cayley graph, algebraic degree, characters of groups, integral eigenvalue
Published in RUP: 18.08.2026; Views: 342; Downloads: 5
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Abstract: In this paper the possibilities of an iterative concatenation method towards construction of Boolean functions resistant to algebraic cryptanalysis are investigated. The notion of ▫$\mathcal{AAR}$▫ (Algebraic Attack Resistant) function is introduced as a unified measure of protection against classical algebraic attacks as well as fast algebraic attacks. Then, it is shown that functions that posses the highest resistance to fast algebraic attacks are necessarily of maximum ▫$\mathcal{AI}$▫ (Algebraic Immunity), the notion defined as a minimum degree of functions that annihilate either ▫$f$▫ or ▫$1+f$▫. More precisely, if for any non-annihilating function ▫$g$▫ of degree ▫$e$▫ an optimum degreerelation ▫$e+d \ge n$▫ is satisfied in the product ▫$fg=h$▫ (denoting ▫$deg(h)=d$▫), then the function ▫$f$▫ in ▫$n$▫ variables must have maximum ▫$\mathcal{AI}$▫, i.e. for nonzero function ▫$g$▫ the relation ▫$fg=0$▫ or ▫$(1+f)g=0$▫ implies. The presented theoretical framework allows us to iteratively construct functions with maximum ▫$\mathcal{AI}$▫ satisfying ▫$e+d=n-1$▫, thus almost optimized resistance to fast algebraic cryptanalysis. This infinite class for the first time, apart from almost optimal resistance to algebraic cryptanalysis, in addition generates the functions that possess high nonlinearity (superior to previous constructions) and maximum algebraic degree, thus unifying most of the relevant cryptographic criteria.
Keywords: algebraic cryptoanalysis, fast algebraic attacks, algebraic immunity, annihilators, algebraic attack resistant, high degree product, stream ciphers, Boolean function
Published in RUP: 15.10.2013; Views: 5826; Downloads: 147
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