Preimage Entropies of Semi-Flows

In this paper, several preimage entropies for semi-flows on compact metric spaces are introduced and studied. We prove that most of these entropies are invariant in a certain sense under conjugacy when the semi-flows under consideration are free of fixed points. The relation between these entropies is studied and an inequality relating them is given. It is also shown that most of these entropies for semi-flow are consistent with that for the time-1 mapping.

A generalized preimage theorem in global analysis

The concept of locally fine point and generalized regular valueof a C1 map between Banach spaces were carried over C1 map between Banach manifolds. Hence the preimage theorem, a principle constructing Banach manifolds in global analysis, is generalized.

Collision Attack on the Full Extended MD4 and Pseudo-Preimage Attack on RIPEMD

The cryptographic hash functions Extended MD4 and RIPEMD are double-branch hash functions, which consist of two parallel branches. Extended MD4 was proposed by Rivest in 1990, and RIPEMD was devised in the framework of the RIPE project （RACE Integrity Primitives Evaluation, 1988-1992）. On the basis of differential analysis and meet-in-the- middle attack principle, this paper proposes a collision attack on the full Extended MD4 and a pseudo-preimage attack on the full RIPEMD respectively. The collision attack on Extended MD4 holds with a complexity of 237, and a collision instance is presented. The pseudo-preimage attack on RIPEMD holds with a complexity of 21254, which optimizes the complexity order for brute-force attack. The results in this study will also be beneficial to the analysis of other double-branch hash functions such as RIPEMD-160.

Improved Preimage Attack on 3-Pass HAVAL

HAVAL is a hash function proposed by Zheng et al.in 1992,including 3-,4-and 5-pass versions.We improve pseudo-preimage and preimage attacks on 3-pass HAVAL at the complexity of 2 172 and 2 209.6,respectively,as compared to the previous best known results:2 192 and 2 225 by Sasaki et al.in 2008.We extend the skip interval for partial-patching and apply the initial structure technique to find the better message chunks,and combine the indirect-partial-matching,partial-fixing and multi-neutral-word partial-fixing techniques to improve the attacks based on the meet-in-the-middle method.These are the best pseudo-preimage and preimage attacks on 3-pass HAVAL.

Forward Expansiveness and Entropies for Subsystems of Z^(k)_(+)-actions

In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.i.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.

On an Entropy of Zk＋-Actions

In this paper,a definition of entropy for Z＋k（k≥2）-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite（even zero） entropy as single transformations.Some basic properties are investigated and its value for the Z＋k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z＋k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the ＂inverse orbits＂ structure of the system.