The emergence of quantum computers is pushing an unprecedented transition in the public key cryptography field. Conventional algorithms, mostly represented by elliptic curves and RSA, are vulnerable to attacks using quantum computers and need, therefore, to be replaced. Cryptosystems based on error-correcting codes are considered some of the most promising candidates to replace them for encryption schemes. Among the code families, QC-MDPC codes achieve the smallest key sizes while maintaining the desired security properties. Their performance, however, still needs to be greatly improved to reach a competitive level. In this work, we focus on optimizing the performance of QC-MDPC code-based cryptosystems through improvements concerning both their implementations and algorithms. We first present a new enhanced version of QcBits' key encapsulation mechanism, which is a constant time implementation of the Niederreiter cryptosystem using QC-MDPC codes. In this version, we updated the implementation parameters to meet the 128-bit quantum security level, replaced some of the core algorithms avoiding slower instructions, vectorized the entire code using the AVX 512 instruction set extension and introduced some other minor improvements. Comparing with the current state-of-the-art implementation for QC-MDPC codes, the BIKE implementation, our code performs 1.9 times faster when decrypting messages. We then optimize the performance of QC-MDPC code-based cryptosystems through the insertion of a configurable failure rate 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 the one used in the NTL library on sparse polynomials and 1.6 times faster than a naive constant-time sparse polynomial multiplication. Our inversion algorithm, based on the inversion algorithm of Wu et al., is 2 times faster than the original and 12 times faster than the inversion algorithm of Itoh and Tsujii using the same modulus polynomial (x^{32749} - 1). By inserting these algorithms in our enhanced version of QcBits, 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 BIKE, our final version of QcBits performs the uniform decryption 2.7 times faster. Moreover, the techniques presented in this work can also be applied to BIKE, opening new possibilities for further Improvements.
Secure and efficient software implementation of QC-MDPC code-based cryptography
