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This paper focuses on high-speed NEON-based constant-time implementations of multiplication of large polynomials in the NIST PQC KEM Finalists: NTRU, Saber, and CRYSTALS-Kyber. We use the Number Theoretic Transform (NTT)-based multiplication in Kyber, the Toom-Cook algorithm in NTRU, and both types of multiplication in Saber. Following these algorithms and using Apple M1, we improve the decapsulation performance of the NTRU, Kyber, and Saber-based KEMs at the security level 3 by the factors of 8.4, 3.0, and 1.6, respectively, compared to the reference implementations. On Cortex-A72, we achieve the speed-ups by factors varying between 5.7 and 7.5x for the Forward/Inverse NTT in Kyber, and between 6.0 and 7.8x for Toom-Cook in NTRU, over the best existing implementations in pure C. For Saber, when using NEON instructions on Cortex-A72, the implementation based on NTT outperforms the implementation based on the Toom-Cook algorithm by 14% in the case of the MatrixVectorMul function but is slower by 21% in the case of the InnerProduct function. Taking into account that in Saber, keys are not available in the NTT domain, the overall performance of the NTT-based version is very close to the performance of the Toom-Cook version. The differences for the entire decapsulation at the three major security levels (1, 3, and 5) are −4, −2, and +2%, respectively. Our benchmarking results demonstrate that our NEON-based implementations run on an Apple M1 ARM processor are comparable to those obtained using the best AVX2-based implementations run on an AMD EPYC 7742 processor. Our work is the first NEON-based ARMv8 implementation of each of the three NIST PQC KEM finalists. 

Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in GL(n,Z). We compare the strengths of various methods to sample random elements of GL(n,Z), finding some are stronger than others with respect to the problem of recognizing rotations of the Zn lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma’s RandomSLnZ command) generates instances of this last problem which can be efficiently broken, even in dimensions nearing 1,500. Likewise, we find that the random basis generation method in one of the NIST Post-Quantum Cryptography competition submissions (DRS) generates instances which can be efficiently broken, even at its 256-bit security settings. Other random basis generation algorithms (some older, some newer) are described which appear to be much stronger.