The anti-Ramsey number, AR(n, G), for a graph G and an integer n ≥ |V (G)|, is defined to be the minimal integer r such that in any edge-colouring of K n by at least r colours there is a multicoloured copy of G, namely, a copy of G that each of its edges has a distinct colour. In this paper we determine, for large enough n, AR(n, L ∪ t P 2 ) and AR(n, L ∪ k P 3 ) for any large enough t and k, and a graph L satisfying some conditions. Consequently, we determine AR(n, G), for large enough n, where G is P 3 ∪ t P 2 for any t ≥ 3, P 4 ∪ t P 2 and C 3 ∪ t P 2 for any t ≥ 2, k P 3 for any k ≥ 3, t P 2 ∪ k P 3 for any t ≥ 1, k ≥ 2, and P t+1 ∪ k P 3 for any t ≥ 3, k ≥ 1. Furthermore, we obtain upper and lower bounds for AR(n, G), for large enough n, where G is P k+1 ∪ t P 2 and C k ∪ t P 2 for any k ≥ 4, t ≥ 1.
Abstract:The r-rounds Even-Mansour block cipher is a generalization of the well known Even-Mansour block cipher to r iterations. Attacks on this construction were described by Nikolić et al. and Dinur et al. for r = 2, 3. These attacks are only marginally better than brute force but are based on an interesting observation (due to Nikolić et al.): for a "typical" permutation P, the distribution of P(x) ⊕ x is not uniform. This naturally raises the following question. Let us call permutations for which the distribution of P(x) ⊕ x is uniformly "balanced" -is there a sufficiently large family of balanced permutations, and what is the security of the resulting Even-Mansour block cipher? We show how to generate families of balanced permutations from the Luby-Rackoff construction and use them to define a 2n-bit block cipher from the 2-round Even-Mansour scheme. We prove that this cipher is indistinguishable from a random permutation of {0, 1} 2n , for any adversary who has oracle access to the public permutations and to an encryption/decryption oracle, as long as the number of queries is o(2 n/2 ). As a practical example, we discuss the properties and the performance of a 256-bit block cipher that is based on our construction, and uses the Advanced Encryption Standard (AES), with a fixed key, as the public permutation.
Constructing a Pseudo Random Function (PRF) from a pseudorandom permutation is a fundamental problem in cryptology. Such a construction, implemented by truncating the last m bits of permutations of {0, 1} n was suggested by Hall et al. (1998). They conjectured that the distinguishing advantage of an adversary with q quesires, Advn,m(q), is small if q = o(2 (m+n)/2 ), established an upper bound on Advn,m(q) that confirms the conjecture for m < n/7, and also declared a general lower bound Advn,m(q) = Ω(q 2 /2 n+m ). The conjecture was essentially confirmed by Bellare and Impagliazzo in 1999. Nevertheless, the problem of estimating Advn,m(q) remained open. Combining the trivial bound 1, the birthday bound, and a result by Stam (1978) leads to the following upper bound:In this paper we show that this upper bound is tight for every m < n and q > 1. This, in turn, verifies that the converse to the conjecture of Hall et al. is also correct, i.e., that Advn,m(q) is negligible only for q = o(2 (m+n)/2 ).
A Chebyshev-type quadrature for a given weight function is a quadrature formula with equal weights. In this work we show that a method presented by Kane may be used to determine the order of magnitude of the minimal number of nodes required in Chebyshev-type quadratures for doubling weight functions. This extends a long line of research on Chebyshev-type quadratures starting with the 1937 work of Bernstein.
Abstract. An oracle chooses a function f from the set of n bits strings to itself, which is either a randomly chosen permutation or a randomly chosen function. When queried by an n-bit string w, the oracle computes f (w), truncates the m last bits, and returns only the first n − m bits of f (w). How many queries does a querying adversary need to submit in order to distinguish the truncated permutation from the (truncated) function? In 1998, Hall et al. [4] showed an algorithm for determining (with high probability) whether or not f is a permutation, using O(2 m+n 2 ) queries. They also showed that if m < n/7, a smaller number of queries will not suffice. For m > n/7, their method gives a weaker bound. In this note, we first show how a modification of the approximation method used by Hall et al. can solve the problem completely. It extends the result to practically any m, showing that Ω(2 m+n 2 ) queries are needed to get a non-negligible distinguishing advantage. However, more surprisingly, a better bound for the distinguishing advantage can be obtained from a result of Stam [8] published, in a different context, already in 1978. We also show that, at least in some cases, the bound in [8] is tight.
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