Cryptanalysis of PLWE based on zero-trace quadratic roots

Beatriz Barbero-Lucas; Iván Blanco-Chacón; Raúl Durán-Díaz; Rodrigo Martín Sánchez-Ledesma; Rahinatou Yuh Njah Nchiwo

Paper

Cryptanalysis of PLWE based on zero-trace quadratic roots

Abstract

We extend two of the attacks on the PLWE problem presented in (Y. Elias, K. E. Lauter, E. Ozman, and K. E. Stange, Ring-LWE Cryptography for the Number Theorist, in Directions in Number Theory, E. E. Eischen, L. Long, R. Pries, and K. E. Stange, eds., vol. 3 of Association for Women in Mathematics Series, Cham, 2016, Springer International Publishing, pp. 271-290) to a ring

where the irreducible monic polynomial

has an irreducible quadratic factor over

of the form

with

of suitable multiplicative order in

. Our attack exploits the fact that the trace of the root is zero, and has overwhelming success probability as a function of the number of samples taken as input. An implementation in Maple and some examples of our attack are also provided.