Hamid Usefi

Non-matrix polynomial identity enveloping algebras

Abstract Let L be a restricted Lie superalgebra with its restricted enveloping algebra u ( L ) over a field F of characteristic p > 2 . A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2 × 2 matrices over F . We characterize L when u ( L ) satisfies a non-matrix polynomial identity.

The classification of p-nilpotent restricted Lie algebras of dimension at most 4

Abstract In this paper we obtain the classification of p-nilpotent restricted Lie algebras of dimension at most 4 over a perfect field of positive characteristic p.

Database query privacy using homomorphic encryptions

Homomorphic encryption is a novel encryption method for it enables computing over encrypted data. This has a wide range of real world ramifications such as being able to blindly compute a search result sent to a remote server without revealing its content. In this paper we summarize how SQL queries can be made secure using a homomorphic encryption scheme based on the ideas of Gahi, et al [1]. Gahis model is based on the DGHV scheme which is a homomorphic encryption scheme which acts on plaintext bits and produce corresponding ciphertext. We use Gahis blueprint to propose an improved model which is based on the more recent Ring based homomorphic encryption scheme by Braserski, et al [2]. Our method is more general in the sense that it can be extended to use with any fully homomorphic encryption scheme rather than being restricted to the DGHV scheme as in Gahis method.

Lie solvable enveloping algebras of characteristic two

Abstract Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable.

Lie identities on symmetric elements of restricted enveloping algebras

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

The restricted isomorphism problem for metacyclic restricted Lie algebras

Let L be a restricted Lie algebra with the restricted enveloping algebra u(L) over a perfect field of positive characteristic p. The restricted isomorphism problem asks what invariants of L are determined by u(L). This problem is the analogue of the modular isomorphism problem for finite p-groups. Baginski and Sandling have given a positive answer to the modular isomorphism problem for metacyclic p-groups. In this paper, we provide a positive answer to the restricted isomorphism problem in case L is metacyclic and p-nilpotent.

Homomorphic Evaluation of Database Queries

Homomorphic encryption is an encryption method that enables computing over encrypted data. This has a wide range of real world ramifications such as being able to blindly compute a search result sent to a remote server without revealing its content. This paper discusses how database search queries can be made secure using a homomorphic encryption scheme. We propose a new database search technique that can be used with the ring-based fully homomorphic encryption scheme proposed by Braserski.

Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian

Abstract Let L be a restricted Lie algebra over a field of characteristic p > 2

Higher dimensional varieties in coding theory

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