Two main computational problems serve as security foundations of current fully homomorphic encryption schemes: Regev’s Learning With Errors problem (LWE) and HowgraveGraham’s Approximate Greatest Common Divisor problem (AGCD). Our first contribution is a reduction from LWE to AGCD. As a second contribution, we describe a new AGCD-based fully homomorphic encryption scheme, which outperforms all ...

Two main computational problems serve as security foundations of current fully homomorphic encryption schemes: Regev’s Learning With Errors problem (LWE) and HowgraveGraham’s Approximate Greatest Common Divisor problem (AGCD). Our first contribution is a reduction from LWE to AGCD. As a second contribution, we describe a new AGCD-based fully homomorphic encryption scheme, which outperforms all ...

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 6= d ∈ D, there exists an n = n(d) with dn = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that xnD ∩ ynD is a principal ideal. We prove that the polynomial ring D[X...

Fully homomorphic encryption over the integers (FHE-OI) is currently the only alternative to lattice-based FHE. FHE-OI includes a family of schemes whose security is based on the hardness of different variants of the approximate greatest common divisor (AGCD) problem. The majority of these works is based on the noise-free variant of AGCD which is potentially weaker than the general one. In part...

We have engineered the use of 2D, 1D and 0D carbon allotropes as spacers to achieve in excess of 1000-fold fluorescence enhancements in a Surface plasmon-coupled emission (SPCE) platform. We also have demonstrated the femtomolar sensitivity of silver decorated carbon dots (AgCD) in the detection of a radiating dipole.

Let R be an integral domain in which every two nonzero elements have a greatest common divisor. Let (an)n>1 be a sequence of nonzero elements in R. We prove that gcd(an, am) = agcd(n,m) for all n,m > 1 if and only if an = ∏