# On the RLWE/PLWE equivalence for cyclotomic number fields

## Abstract

We study the equivalence between the ring learning with errors and polynomial learning with errors problems for cyclotomic number fields, namely: we prove that both problems are equivalent via a polynomial noise increase as long as the number of distinct primes dividing the conductor is kept constant. We refine our bound in the case where the conductor is divisible by at most three primes and we give an asymptotic subexponential formula for the condition number of the attached Vandermonde matrix valid for arbitrary degree.

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## Notes

1. At https://www.safecrypto.eu/pqclounge/ a summary of candidates and the history of all submissions, attacks and withdrawals is available to filter and check.

2. This is the definition of RLWE/PLWE in search version. As all this material is nowadays well known to the specialist we are sparing as many details as possible. We are taking this version as starting point, as it is more suitable for our argument. We refer the reader to [10] for the decisional version of the problem.

3. For $$p(x)=\displaystyle \sum _{i=0}^np_ix^i\in {\mathbb {R}}[x]$$, the 1-norm is defined as $$||p||_1=\displaystyle \sum _{i=0}^n|p_i|$$

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## Acknowledgements

Partially supported by MTM2016-79400-P.

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Correspondence to Iván Blanco-Chacón.

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Blanco-Chacón, I. On the RLWE/PLWE equivalence for cyclotomic number fields. AAECC 33, 53–71 (2022). https://doi.org/10.1007/s00200-020-00433-z

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• DOI: https://doi.org/10.1007/s00200-020-00433-z

### Keywords

• Ring learning with errors
• Polynomial learning with errors
• Cyclotomic polynomials
• Condition number