Extreme k-families

AbstractLet P be a poset. A subset A of P is a k-family iff A contains no (k + 1)-element chain. For i ⩾ 1, let Ai be the set of elements of A at depth i − 1 in A. The k-families of P can be ordered by defining A ⩽ B iff, for all i, Ai is included in the order ideal generated by Bi. This paper examines minimal r-element k-families, defined as k-families A such that |A| = r and for every B < A, |B| < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure, which have been used to obtain extremal results for families of sets with width restrictions. Let Mk,r be the set of minimal r-element k-families and let Mk = ∪r ≥ 0 Mk,r. It is shown that Mk is a join-subsemilattice by the lattice Ak of k-families. Mk is a lower semimodular lattice, where the rth rank is given by Mk,r. If wk is the maximum size of a k-family, then |Mk,r| ⩽ (wrk)and |∪Mk| ⩽ Σi = 1wk ⌈i/k⌉. Let D(A) = max{|B| − |A| | B is a k-family and B ⩽ A}. For k-families A and B, D(A v B) ⩽ D(A) + D(B). This result shows that {A | D(A) = 0} is also a join-subsemilattice of Ak

Generic attacks on iterated hash functions

Includes bibliographical references (leaves 126-132).We survery the existing generic attacks on hash functions based on the Merkle­Damgard construction: that is, attacks in which the compression function is treated as a black box

Two applications of the Theory of Currents

In the first part of the thesis we find an adapted version of the Rademacher theorem of differentiability of Lipschitz functions, when the Lebesgue measure on the euclidean space is replaced by a generical Radon measure. In the second part of the thesis we explain how to understand the Steiner tree problem as a mass minimization problem in a family of rectifiable currents with coefficients in a normed group and we exhibit some calibrations in order to prove the absolute minimaity of some concrete configurations. The common point of this problems is a substantial use of the Theory of Currents as a tool for proof