2,765 research outputs found

    Maximum-order Complexity and Correlation Measures

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    We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order kk cannot have very small maximum-order complexity

    Powers of Hamilton cycles in pseudorandom graphs

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    We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph GG is (ε,p,k,)(\varepsilon,p,k,\ell)-pseudorandom if for all disjoint XX and YV(G)Y\subset V(G) with Xεpkn|X|\ge\varepsilon p^kn and Yεpn|Y|\ge\varepsilon p^\ell n we have e(X,Y)=(1±ε)pXYe(X,Y)=(1\pm\varepsilon)p|X||Y|. We prove that for all β>0\beta>0 there is an ε>0\varepsilon>0 such that an (ε,p,1,2)(\varepsilon,p,1,2)-pseudorandom graph on nn vertices with minimum degree at least βpn\beta pn contains the square of a Hamilton cycle. In particular, this implies that (n,d,λ)(n,d,\lambda)-graphs with λd5/2n3/2\lambda\ll d^{5/2 }n^{-3/2} contain the square of a Hamilton cycle, and thus a triangle factor if nn is a multiple of 33. This improves on a result of Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.Comment: 30 pages, 1 figur

    Verifiable Random Functions (VRFs)

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    A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private key can compute the hash, but anyone with public key can verify the correctness of the hash. VRFs are useful for preventing enumeration of hash-based data structures. This document specifies several VRF constructions that are secure in the cryptographic random oracle model. One VRF uses RSA and the other VRF uses Eliptic Curves (EC).https://datatracker.ietf.org/doc/draft-irtf-cfrg-vrf/First author draf

    Families of sequences with good family complexity and cross-correlation measure

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    In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity (ff-complexity) and the cross-correlation measure of order \ell. We consider sequences not only on binary alphabet but also on kk-symbols (kk-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the ff-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high ff-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on kk-symbols alphabet.Comment: 13 pages. Comments are welcome
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