On the Constructions of New Symmetric Ciphers Based on Nonbijective Multivariate Maps of Prescribed Degree

The main purpose of this paper is to introduce stream ciphers with the nonbijective encryption function of multivariate nature constructed in terms of algebraic graph theory. More precisely, we describe the two main symmetric algorithms for creation of multivariate encryption transformations based on three families of bipartite graphs with partition sets isomorphic to Kn, where K is selected as the finite commutative ring. The plainspace of the algorithm is Ω={x∣∑xi∈K⁎, x∈Kn}⊂Kn,Ω≅K⁎×Kn-1. The second algorithm is a generalization of the first one with using the jump operator, where generalized encryption map has an essentially higher degree in comparison with the previous version. Moreover, the degree of this generalized map is not bounded by some constant. This property guarantees resistance of the cipher to linearization attacks.

Variational characterization of Hp

In this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.

JUMP OPERATIONS FOR BOREL GRAPHS

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

A fixed point for the jump operator on structures

Assuming that 0# exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure such thatwhere is the set of Turing degrees which compute a copy of More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full “nth-order arithmetic for all n, cannot prove the existence of such a structure.

Jump inversions inside effectively closed sets and applications to randomness

We study inversions of the jump operator on classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails.Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable.