Fast multiplication and the PLWE–RLWE equivalence for an infinite family of maximal real subfields of cyclotomic fields

Abstract We prove the equivalence between the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems for the maximal totally real subfield of the $$2^r 3^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>r</mml:mi> </mml:msup> <mml:msup> <mml:mn>3</mml:mn> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:math> th cyclotomic field for $$r \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and $$s \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Moreover, we describe a fast algorithm for computing the product of two elements in the ring of integers of these subfields. This multiplication algorithm has quasilinear complexity in the dimension of the field, as it makes use of the fast Discrete Cosine Transform (DCT). Our approach assumes that the two input polynomials are given in a basis of Chebyshev-like polynomials, in contrast to the customary power basis. To validate this assumption, we prove that the change of basis from the power basis to the Chebyshev-like basis can be computed with $${\mathcal {O}}(n \log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> arithmetic operations, where n is the problem dimension. Finally, we provide a heuristic and theoretical comparison of the vulnerability to some attacks for the p th cyclotomic field versus the maximal totally real subextension of the 4 p th cyclotomic field for a reasonable set of parameters of cryptographic size.