Abstract
Number theoretic transform (NTT) is a basic mathematic operation, and is particularly fundamental to the practical implementations of cryptographic algorithms based on lattices with algebraic structures. In this work, we make a systematic and comprehensive study of NTT and its variants. We first review the NTT technique and the recent advances raised in the implementations of practical lattice-based cryptography. We clarify the relationship of some existing NTT variants, and prove their computational equivalence. We then make the generalizations of NTT, analyze their exact computational complexity, and derive the optimal bounds. Finally, we show the applications of our results to some prominent practical lattice-based algorithms.
This work is supported in part by National Key Research and Development Program of China under Grant No. 2017YFB0802000, National Natural Science Foundation of China under Grant Nos. 61472084 and U1536205, Shanghai Innovation Action Project under Grant No. 16DZ1100200, Shanghai Science and Technology Development Funds under Grant No. 16JC1400801, and Shandong Provincial Key Research and Development Program of China under Grant Nos. 2017CXGC0701 and 2018CXGC0701.
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Liang, Z. et al. (2021). Number Theoretic Transform: Generalization, Optimization, Concrete Analysis and Applications. In: Wu, Y., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2020. Lecture Notes in Computer Science(), vol 12612. Springer, Cham. https://doi.org/10.1007/978-3-030-71852-7_28
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