240 research outputs found
The Splitting Field of , Two-Variable NTT and Lattice-Based Cryptography
The splitting field of the polynomial is an extension over generated by and . When () is a power-of-two integer, the degree of over is . In this paper, we lay the foundation for applying the Order-LWE in to cryptographic uses. More specifically, We will compute the Galois group and the canonical embedding of into . Then we study the trace pairings of the integral basis and obtain its dual explicitly, which will be crucial when we study the error distributions on the ideal lattices associated with .
Moreover, we design a Two-Variable Number Theoretic Transform (2NTT) algorithm for the quotient , where is a prime number such that has distinct solutions. Compared to the one-variable NTT, a crucial advantage of 2NTT is that it enjoys a quadratic saving of twiddle factors. Hence, it is very interesting to see how to leverage this quadratic saving to boost the performance of 2NTT in practical implementations
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