# Introducing Arithmetic Failures to Accelerate QC-MDPC Code-Based Cryptography

Antonio Guimarães,
Edson Borin,
Diego F. Aranha

January 2019

### Abstract

In this work, we optimize the performance of QC-MDPC code-based cryptosystems through the insertion of configurable failure rates in their arithmetic procedures. We present constant time algorithms with a configurable failure rate for multiplication and inversion over binary polynomials, the two most expensive subroutines used in QC-MDPC implementations. Using a failure rate negligible compared to the security level ($$2^-128$$), our multiplication is 2 times faster than NTL on sparse polynomials and 1.6 times faster than a naive constant-time sparse polynomial multiplication. Our inversion algorithm, based on Wu et al., is 2 times faster than the original algorithm and 12 times faster than Itoh-Tsujii using the same modulus polynomial ($$x^32749 - 1$$). By inserting these algorithms in a version of QcBits at the 128-bit quantum security level, we were able to achieve a speedup of 1.9 on the key generation and up to 1.4 on the decryption time. Comparing with variant 2 of the BIKE suite, which also implements the Niederreiter Cryptosystem using QC-MDPC codes, our final version of QcBits performs the uniform decryption 2.7 times faster.

Publication

*Code-Based Cryptography*