Attacks on Search RLWE

We describe a new attack on the Search Ring Learning-With-Errors (RLWE) problem based on the chi-square statistical test, and give examples of RLWE instances in Galois number fields which are vulnerable to our attack. We prove a search-to-decision reduction for Galois fields which applies for any unramified prime modulus q, regardless of the residue degree f of q, and we use this in our attacks. The time complexity of our attack is O(q ), where f is the residue degree of q in K. We also show an attack on the RLWE problem in general cyclotomic rings (non 2-power cyclotomic rings) which works when the modulus is a ramified prime. We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks.

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