Parity and Group Theory

D. Cervantes, Sandra L. Quiroga, Luis José Perissinotti +2 more

We first give a geometrical description of the action of the parity operator ( ˆ P) on non relativistic spin 1 2 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2) ⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). ˆ P is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys ˆP 2 = −1. Then, from the direct product of O(3) by Z2, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator ( ˆ T) acting on spinors, and its product with ˆ P. Both, ˆ P and ˆ T, generate the group Z4 × Z2. As in the case of parity, ˆT is the non relativistic limit of the corresponding Dirac matrix operator T = γ 3 γ 1, and obeys ˆ T 2 = −1. Key words: parity transformation; time reversal; non relativistic spinors; group theory. 1.

Gilbert Baumslag, Nelly Fazio, Antonio R. Nicolosi +2 more

Abstract. We propose a generalization of the learning parity with noise (LPN) and learning with errors (LWE) problems to an abstract class of group-theoretic learning problems that we term learning homomorphisms with noise (LHN). This class of problems contains LPN and LWE as special cases, but is much more general. It allows, for example, instantiations based on non-abelian groups, resulting in a new avenue for the application of combinatorial group theory to the development of cryptographic primitives. We then study a particular instantiation using relatively free groups and construct a symmetric cryptosystem based upon it. Keywords. Learning with errors. Post-quantum cryptography. Non-commutative cryptography. Burnside groups. 1

Healy David

Demetrios N. Christodoulides, Jianke Yang