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/*
* CC0 1.0 Universal or the following MIT License
*
* MIT License
*
* Copyright (c) 2023: Hanno Becker, Vincent Hwang, Matthias J. Kannwischer, Bo-Yin Yang, and Shang-Yi Yang
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*/
// root of unity: 1753
// Q1
// omegaQ1 = 1753 mod Q1
// invomegaQ1 = omegaQ^{-1} mod Q1
// R = 2^32 below
// RmodQ1 = 2^32 mod^{+-} Q1
// Q1prime = Q1^{-1} mod^{+-} 2^32
// invNQ1 = NTT_N^{-1} mod Q1
// invNQ1R2modQ1 = -NTT_N^{-1} 2^32 2^32 mod^{+-} Q1 below
// invNQ1R2modQ1_prime = invNQ1R2modQ1 (Q1^{-1} mod^{+-} 2^32) mod^{+-} 2^32
// invNQ1R2modQ1_prime_half = (invNQ1R2modQ1 / 2) (Q1^{-1} mod^{+-} 2^32) mod^{+-} 2^32
// invNQ1R2modQ1_doubleprime = (invNQ1R2modQ1_prime Q1 - invNQ1R2modQ1) / 2^32
// invNQ1_final_R2modQ1 = -invNQ1R2modQ1 invomegaQ1^{128} mod q
// invNQ1_final_R2modQ1_prime = invNQ1_final_R2modQ1 (Q1^{-1} mod^{+-} 2^32) mod^{+-} 2^32
// invNQ1_final_R2modQ1_prime_half = (invNQ1_final_R2modQ1 / 2) (Q1^{-1} mod^{+-} 2^32) mod^{+-} 2^32
// invNQ1_final_R2modQ1_doubleprime = (invNQ1_final_R2modQ1_prime Q1 - invNQ1_final_R2modQ1) / 2^32
// RmodQ1_prime = -(RmodQ1 + Q1) Q1prime mod^{+-} 2^32
// RmodQ1_prime_half = ( -(RmodQ1 + Q1) / 2) Q1prime mod^{+-} 2^32
// RmodQ1_doubleprime = (RmodQ1_prime Q1 - RmodQ1_prime ) / 2^32