On the key exchange and multivariate encryption with nonlinear polynomial maps of stable degree

We say that the sequence g^n, n 3, n ! 1 of polynomial transformation bijective mapsof free module K^n over commutative ring K is a sequence of stable degree if the order of g^n is growingwith n and the degree of each nonidentical polynomial map of kind g^n^k^ is an independent constant c.Transformation b = tgn

Synthesis of optically active polycyclic N-heterocycles derived from L-prolinamine

The N-Boc-protected (S)-prolinamine reacts readily with formaldehyde and diverse alpha-hydroxyimino ketones to give imidazole N-oxides with an enantiomerically pure N-protected (pyrrolidin-2-yl)methyl substituent. Subsequent deprotection yields the corresponding NH derivatives. Upon treatment of these products with Ac2O at room temperature, a cascade of reactions leads to optically active tricyclic products. In all these processes, the stereogenic center is preserved. In one case, the bis-heterocyclic imidazole N-oxide was transformed into the corresponding optically active imidazole-2-thione via a sulfur-transfer reaction with 2,2,4,4-tetramethylcyclobutane-1,3-dithione

On the key expansion of D(n, K)-based cryptographical algorithm

The family of algebraic graphs D(n, K) defined over finite commutative ring K have been used in different cryptographical algorithms (private and public keys, key exchange protocols). The encryption maps correspond to special walks on this graph. We expand the class of encryption maps via the use of edge transitive automorphism group G(n, K) of D(n, K). The graph D(n, K) and related directed graphs are disconnected. So private keys corresponding to walks preserve each connected component. The group G(n, K) of transformations generated by an expanded set of encryption maps acts transitively on the plainspace. Thus we have a great difference with block ciphers, any plaintexts can be transformed to an arbitrarily chosen ciphertex by an encryption map. The plainspace for the D(n, K) graph based encryption is a free module P over the ring K. The group G(n, K) is a subgroup of Cremona group of all polynomial automorphisms. The maximal degree for a polynomial from G(n, K) is 3. We discuss the Diffie-Hellman algorithm based on the discrete logarithm problem for the group τ-1Gτ, where τ is invertible affine transformation of free module P i.e. polynomial automorphism of degree 1. We consider some relations for the discrete logarithm problem for G(n, K) and public key algorithm based on the D(n, K) graphs

From compressible Naveir-Stokes with nonlocal forces to Euler

We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system. The result is based on the joint work with Jose A. Carrillo and Ewelina Zatorska.Non UBCUnreviewedAuthor affiliation: Polish Academy of SciencesFacult

Synthesis of optically active polyheterocycles containing pyrrolidine, imidazole, and 1,2,3-triazole rings

Optically active polyheterocycles containing pyrrolidine, imidazole, and 1,2,3-triazole units were obtained via a multistep synthesis with the [3+2] cycloaddition of Boc-protected (S)-(pyrrolidin-2-yl)methyl azide with 2-ethynylimidazoles in the presence of CuI (CuAAC reaction) as the key step. Typical for terminal alkynes, the reactions occurred regioselectively and 1,4-disubstituted 1,2,3-triazoles were formed exclusively. The deprotection of the pyrrolidine N-atom was performed by treatment with TFA under standard conditions

Synthesis of optically active trifluoromethyl-substituted 2,3-dihydroimidazo[2,1-b]oxazoles

The 2-unsubstituted imidazole N-oxides with a 3,3,3-trifluoro-2-hydroxypropyl group at N(1) in the presence of acetic anhydride undergo cyclization via the intramolecular nucleophilic attack of the hydroxyl group onto C(2) of the imidazole ring to give trifluoromethylated derivatives of 2,3-dihydroimidazo[2,1-b]oxazoles. This method, starting with enantiopure substrates, allows the preparation of enantiopure products in a one-pot procedure

Studies on the Reaction of Aryl Glyoxals with L-Prolinol: Unexpected Formation of Morpholin-2-one Derivatives and Stereoselective Trifluoromethylation of the Bicyclic System

Reactions of L-prolinol with aryl glyoxals lead to labile bicyclic 2-aryloyl-1,3-oxazolidines, which smoothly undergo isomerization in the presence of an acid via a cascade of reactions involving a 1,2-aryl shift. The products formed thereby were identified as bicyclic derivatives of morpholin-2-one. Both 1,3-oxazolidines and morpholin-2-ones were used for the nucleophilic trifluoromethylation by treatment with trifluoromethyltrimethylsilane (Ruppert–Prakash reagent) in the presence of a catalytic amount of CsF. Whereas in the case of the 1,3-oxazolidines only modest diastereoselectivity was achieved, the reactions with the morpholinone derivatives occurred with complete stereoselectivity

Recent progress in the chemistry of 2-unsubstituted 1h-imidazole 3-oxides

The 2-unsubstituted 1H-imidazole 3-oxides, considered as ‘nitrone-like’ azaheterocycles, are well known as versatile building blocks, useful for the preparation of more complex systems containing an imidazole moiety. Their applications in diverse reactions, such as isomerization to corresponding imidazol-2-ones, [3+2]-cycloadditions, sulfur transfer reactions leading to imidazole-2-thiones, heterocyclizations leading to fused heterocyclic systems, as well as cross coupling reactions are discussed. New methods for the preparation of enantiomerically pure 1H-imidazole 3-oxides and their applications in asymmetric reactions, synthesis of room temperature ionic liquids and biologically active compounds are also reviewed