Paper 2017/612 Original: eprint.iacr.org/2017/612 ↗

Large Modulus Ring-LWE $\geq$ Module-LWE

Abstract

We present a reduction from the module learning with errors problem (MLWE) in dimension \(d\) and with modulus \(q\) to the ring learning with errors problem (RLWE) with modulus \(q^{d}\). Our reduction increases the LWE error rate \(\alpha\) by a factor of \( n^{c+1/2} \cdot \sqrt{d} \) for ring dimension \(n\), module rank \(d\) and any constant \(c>0\) in the case of power-of-two cyclotomics. Since, on the other hand, MLWE is at least as hard as RLWE, we conclude that the two problems are polynomial-time equivalent. As a corollary, we obtain that the RLWE instance described above is equivalent to solving lattice problems on module lattices. We also present a self reduction for power-of-two cyclotomic RLWE that reduces the ring dimension \(n\) by a power-of-two factor \(2^i\), while increasing the modulus by a power of \(2^i\) and the error rate by a factor of \( 2^{i\cdot (1-c)} \cdot n^{c+1/2} \) for any constant \(c>0\). Our results suggest that when discussing hardness to drop the RLWE/MLWE distinction in favour of distinguishing problems by the module rank required to solve them.