We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings f:X→P(V) of Fano manifolds X. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding f. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space B or, equivalently, as the derived category of the sheaf of A-modules on B, where A is an A_∞ algebra. This latter interpretation corresponds to a noncommutative resolution of B. We compute explicitly the algebra A by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an A_∞ algebra with a cyclic group. Then we apply our results to the HPD of f corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in P^n.

We use the learning with errors (LWE) problem to build a new simple and provably secure key exchange scheme. The basic idea of the construction can be viewed as certain extension of Diffie-Hellman problem with errors. The mathematical structure behind comes from the commutativity of computing a bilinear form in two different ways due to the associativity of the matrix multiplications: (x^t \times A)\times y = x^t \times (A\times y), where x,y are column vectors and A is a square matrix. We show that our new schemes are more efficient in terms of communication and computation complexity compared with key exchange schemes or key transport schemes via encryption schemes based on the LWE problem. Furthermore, we extend our scheme to the ring learning with errors (RLWE) problem, resulting in small key size and better efficiency.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!