THE FROBENIUS PROBLEM FOR NUMERICAL SEMIGROUPS GENERATED BY THE THABIT NUMBERS OF THE FIRST, SECOND KIND BASE b AND THE CUNNINGHAM NUMBERS

The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers

New orthogonality criterion for shortest vector of lattices and its applications

The security of most lattice based cryptography relies on the hardness of computing a shortest nonzero vector of lattices. We say that a lattice basis is SV-reduced if it contains a shortest nonzero vector of the lattice. In this paper, we prove that, π∕6 orthogonality between the shortest vector of the basis and the vector space spanned by other vectors of the basis is enough to be SV-reduced under the assumption that a plausible condition Cn holds. By using the π∕6 orthogonality under C2, we prove a new complexity bound log3[Formula presented]+1 for Gauss–Lagrange algorithm which clarifies why the currently known complexity is so far fall short to expose the efficiency of the algorithm we experience in practice. Our experiments suggest that our complexity bound of Gauss–Lagrange algorithm is somewhat close to actual efficiency of the algorithm. We also show that LLL(δ) algorithm outputs a SV-reduced basis if δ≥1∕3 for two dimensional lattice. We present an efficient three dimensional SV-reduction algorithm by using the condition C3 and π∕6 orthogonality and how to generalize the algorithm for higher dimension. © 2020 Elsevier B.V