newhope.cpp

Go to the documentation of this file.

/*
* NEWHOPE Ring-LWE scheme
* Based on the public domain reference implementation by the
* designers (https://github.com/tpoeppelmann/newhope)
*
* Further changes
* (C) 2016 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/

#include <botan/newhope.h>
#include <botan/hash.h>
#include <botan/rng.h>
#include <botan/stream_cipher.h>
#include <botan/loadstor.h>

namespace Botan {

newhope_poly::~newhope_poly()
   {
   secure_scrub_memory(coeffs, sizeof(coeffs));
   }

typedef newhope_poly poly;

namespace {

static const uint16_t PARAM_Q = 12289;
static const size_t PARAM_N = 1024;

/* Incomplete-reduction routines; for details on allowed input ranges
 * and produced output ranges, see the description in the paper:
 * https://cryptojedi.org/papers/#newhope */

inline uint16_t montgomery_reduce(uint32_t a)
   {
   const uint32_t qinv = 12287; // -inverse_mod(p,2^18)
   const uint32_t rlog = 18;
   const uint32_t rlog_mask = ((1 << rlog) - 1);

   uint32_t u = (a * qinv);
   u &= rlog_mask;
   u *= PARAM_Q;
   u += a;
   return (u >> rlog);
   }

inline uint16_t barrett_reduce(uint16_t a)
   {
   uint32_t u = (static_cast<uint32_t>(a) * 5) >> 16;
   u *= PARAM_Q;
   a = static_cast<uint16_t>(a - u);
   return a;
   }

inline void mul_coefficients(uint16_t* poly, const uint16_t* factors)
   {
   for(size_t i = 0; i < PARAM_N; i++)
      {
      poly[i] = montgomery_reduce(poly[i] * factors[i]);
      }
   }

/* GS_bo_to_no; omegas need to be in Montgomery domain */
inline void ntt(uint16_t* a, const uint16_t* omega)
   {
   for(size_t i = 0; i < 10; i+=2)
      {
      // Even level
      size_t distance = (1ULL << i);
      for(size_t start = 0; start < distance; start++)
         {
         size_t jTwiddle = 0;
         for(size_t j = start; j < PARAM_N-1; j += 2*distance)
            {
            uint16_t W = omega[jTwiddle++];
            uint16_t temp = a[j];
            a[j] = (temp + a[j + distance]); // Omit reduction (be lazy)
            a[j + distance] = montgomery_reduce((W * (static_cast<uint32_t>(temp) + 3*PARAM_Q - a[j + distance])));
            }
         }

      // Odd level
      distance <<= 1;
      for(size_t start = 0; start < distance; start++)
         {
         size_t jTwiddle = 0;
         for(size_t j = start; j < PARAM_N-1; j += 2*distance)
            {
            uint16_t W = omega[jTwiddle++];
            uint16_t temp = a[j];
            a[j] = barrett_reduce((temp + a[j + distance]));
            a[j + distance] = montgomery_reduce((W * (static_cast<uint32_t>(temp) + 3*PARAM_Q - a[j + distance])));
            }
         }
      }
   }

inline void poly_frombytes(poly* r, const uint8_t* a)
   {
   for(size_t i = 0; i < PARAM_N/4; i++)
      {
      r->coeffs[4*i+0] =  a[7*i+0]       | ((static_cast<uint16_t>(a[7*i+1]) & 0x3f) << 8);
      r->coeffs[4*i+1] = (a[7*i+1] >> 6) | (static_cast<uint16_t>(a[7*i+2]) << 2) | (static_cast<uint16_t>
                         (a[7*i+3] & 0x0f) << 10);
      r->coeffs[4*i+2] = (a[7*i+3] >> 4) | (static_cast<uint16_t>(a[7*i+4]) << 4) | (static_cast<uint16_t>
                         (a[7*i+5] & 0x03) << 12);
      r->coeffs[4*i+3] = (a[7*i+5] >> 2) | (static_cast<uint16_t>(a[7*i+6]) << 6);
      }
   }

inline void poly_tobytes(uint8_t* r, const poly* p)
   {
   for(size_t i = 0; i < PARAM_N/4; i++)
      {
      uint16_t t0 = barrett_reduce(p->coeffs[4*i+0]); //Make sure that coefficients have only 14 bits
      uint16_t t1 = barrett_reduce(p->coeffs[4*i+1]);
      uint16_t t2 = barrett_reduce(p->coeffs[4*i+2]);
      uint16_t t3 = barrett_reduce(p->coeffs[4*i+3]);

      uint16_t m;
      int16_t c;

      m = t0 - PARAM_Q;
      c = m;
      c >>= 15;
      t0 = m ^ ((t0^m)&c); // <Make sure that coefficients are in [0,q]

      m = t1 - PARAM_Q;
      c = m;
      c >>= 15;
      t1 = m ^ ((t1^m)&c); // <Make sure that coefficients are in [0,q]

      m = t2 - PARAM_Q;
      c = m;
      c >>= 15;
      t2 = m ^ ((t2^m)&c); // <Make sure that coefficients are in [0,q]

      m = t3 - PARAM_Q;
      c = m;
      c >>= 15;
      t3 = m ^ ((t3^m)&c); // <Make sure that coefficients are in [0,q]

      r[7*i+0] =  t0 & 0xff;
      r[7*i+1] = static_cast<uint8_t>((t0 >> 8) | (t1 << 6));
      r[7*i+2] = static_cast<uint8_t>((t1 >> 2));
      r[7*i+3] = static_cast<uint8_t>((t1 >> 10) | (t2 << 4));
      r[7*i+4] = static_cast<uint8_t>((t2 >> 4));
      r[7*i+5] = static_cast<uint8_t>((t2 >> 12) | (t3 << 2));
      r[7*i+6] = static_cast<uint8_t>((t3 >> 6));
      }
   }

inline void poly_getnoise(Botan::RandomNumberGenerator& rng, poly* r)
   {
   uint8_t buf[4*PARAM_N];

   rng.randomize(buf, 4*PARAM_N);

   for(size_t i = 0; i < PARAM_N; i++)
      {
      const uint32_t t = load_le<uint32_t>(buf, i);
      uint32_t d = 0;
      for(size_t j = 0; j < 8; j++)
         {
         d += (t >> j) & 0x01010101;
         }
      const uint32_t a = ((d >> 8) & 0xff) + (d & 0xff);
      const uint32_t b = (d >> 24) + ((d >> 16) & 0xff);
      r->coeffs[i] = static_cast<uint16_t>(a + PARAM_Q - b);
      }
   }

inline void poly_pointwise(poly* r, const poly* a, const poly* b)
   {
   for(size_t i = 0; i < PARAM_N; i++)
      {
      const uint16_t t = montgomery_reduce(3186*b->coeffs[i]); /* t is now in Montgomery domain */
      r->coeffs[i] = montgomery_reduce(a->coeffs[i] * t); /* r->coeffs[i] is back in normal domain */
      }
   }

inline void poly_add(poly* r, const poly* a, const poly* b)
   {
   for(size_t i = 0; i < PARAM_N; i++)
      {
      r->coeffs[i] = barrett_reduce(a->coeffs[i] + b->coeffs[i]);
      }
   }

inline void poly_ntt(poly* r)
   {
   static const uint16_t omegas_montgomery[PARAM_N/2] =
      {
      4075, 6974, 7373, 7965, 3262, 5079, 522, 2169, 6364, 1018, 1041, 8775, 2344,
      11011, 5574, 1973, 4536, 1050, 6844, 3860, 3818, 6118, 2683, 1190, 4789,
      7822, 7540, 6752, 5456, 4449, 3789, 12142, 11973, 382, 3988, 468, 6843, 5339,
      6196, 3710, 11316, 1254, 5435, 10930, 3998, 10256, 10367, 3879, 11889, 1728,
      6137, 4948, 5862, 6136, 3643, 6874, 8724, 654, 10302, 1702, 7083, 6760, 56,
      3199, 9987, 605, 11785, 8076, 5594, 9260, 6403, 4782, 6212, 4624, 9026, 8689,
      4080, 11868, 6221, 3602, 975, 8077, 8851, 9445, 5681, 3477, 1105, 142, 241,
      12231, 1003, 3532, 5009, 1956, 6008, 11404, 7377, 2049, 10968, 12097, 7591,
      5057, 3445, 4780, 2920, 7048, 3127, 8120, 11279, 6821, 11502, 8807, 12138,
      2127, 2839, 3957, 431, 1579, 6383, 9784, 5874, 677, 3336, 6234, 2766, 1323,
      9115, 12237, 2031, 6956, 6413, 2281, 3969, 3991, 12133, 9522, 4737, 10996,
      4774, 5429, 11871, 3772, 453, 5908, 2882, 1805, 2051, 1954, 11713, 3963,
      2447, 6142, 8174, 3030, 1843, 2361, 12071, 2908, 3529, 3434, 3202, 7796,
      2057, 5369, 11939, 1512, 6906, 10474, 11026, 49, 10806, 5915, 1489, 9789,
      5942, 10706, 10431, 7535, 426, 8974, 3757, 10314, 9364, 347, 5868, 9551,
      9634, 6554, 10596, 9280, 11566, 174, 2948, 2503, 6507, 10723, 11606, 2459,
      64, 3656, 8455, 5257, 5919, 7856, 1747, 9166, 5486, 9235, 6065, 835, 3570,
      4240, 11580, 4046, 10970, 9139, 1058, 8210, 11848, 922, 7967, 1958, 10211,
      1112, 3728, 4049, 11130, 5990, 1404, 325, 948, 11143, 6190, 295, 11637, 5766,
      8212, 8273, 2919, 8527, 6119, 6992, 8333, 1360, 2555, 6167, 1200, 7105, 7991,
      3329, 9597, 12121, 5106, 5961, 10695, 10327, 3051, 9923, 4896, 9326, 81,
      3091, 1000, 7969, 4611, 726, 1853, 12149, 4255, 11112, 2768, 10654, 1062,
      2294, 3553, 4805, 2747, 4846, 8577, 9154, 1170, 2319, 790, 11334, 9275, 9088,
      1326, 5086, 9094, 6429, 11077, 10643, 3504, 3542, 8668, 9744, 1479, 1, 8246,
      7143, 11567, 10984, 4134, 5736, 4978, 10938, 5777, 8961, 4591, 5728, 6461,
      5023, 9650, 7468, 949, 9664, 2975, 11726, 2744, 9283, 10092, 5067, 12171,
      2476, 3748, 11336, 6522, 827, 9452, 5374, 12159, 7935, 3296, 3949, 9893,
      4452, 10908, 2525, 3584, 8112, 8011, 10616, 4989, 6958, 11809, 9447, 12280,
      1022, 11950, 9821, 11745, 5791, 5092, 2089, 9005, 2881, 3289, 2013, 9048,
      729, 7901, 1260, 5755, 4632, 11955, 2426, 10593, 1428, 4890, 5911, 3932,
      9558, 8830, 3637, 5542, 145, 5179, 8595, 3707, 10530, 355, 3382, 4231, 9741,
      1207, 9041, 7012, 1168, 10146, 11224, 4645, 11885, 10911, 10377, 435, 7952,
      4096, 493, 9908, 6845, 6039, 2422, 2187, 9723, 8643, 9852, 9302, 6022, 7278,
      1002, 4284, 5088, 1607, 7313, 875, 8509, 9430, 1045, 2481, 5012, 7428, 354,
      6591, 9377, 11847, 2401, 1067, 7188, 11516, 390, 8511, 8456, 7270, 545, 8585,
      9611, 12047, 1537, 4143, 4714, 4885, 1017, 5084, 1632, 3066, 27, 1440, 8526,
      9273, 12046, 11618, 9289, 3400, 9890, 3136, 7098, 8758, 11813, 7384, 3985,
      11869, 6730, 10745, 10111, 2249, 4048, 2884, 11136, 2126, 1630, 9103, 5407,
      2686, 9042, 2969, 8311, 9424, 9919, 8779, 5332, 10626, 1777, 4654, 10863,
      7351, 3636, 9585, 5291, 8374, 2166, 4919, 12176, 9140, 12129, 7852, 12286,
      4895, 10805, 2780, 5195, 2305, 7247, 9644, 4053, 10600, 3364, 3271, 4057,
      4414, 9442, 7917, 2174
      };

   static const uint16_t psis_bitrev_montgomery[PARAM_N] =
      {
      4075, 6974, 7373, 7965, 3262, 5079, 522, 2169, 6364, 1018, 1041, 8775, 2344,
      11011, 5574, 1973, 4536, 1050, 6844, 3860, 3818, 6118, 2683, 1190, 4789,
      7822, 7540, 6752, 5456, 4449, 3789, 12142, 11973, 382, 3988, 468, 6843, 5339,
      6196, 3710, 11316, 1254, 5435, 10930, 3998, 10256, 10367, 3879, 11889, 1728,
      6137, 4948, 5862, 6136, 3643, 6874, 8724, 654, 10302, 1702, 7083, 6760, 56,
      3199, 9987, 605, 11785, 8076, 5594, 9260, 6403, 4782, 6212, 4624, 9026, 8689,
      4080, 11868, 6221, 3602, 975, 8077, 8851, 9445, 5681, 3477, 1105, 142, 241,
      12231, 1003, 3532, 5009, 1956, 6008, 11404, 7377, 2049, 10968, 12097, 7591,
      5057, 3445, 4780, 2920, 7048, 3127, 8120, 11279, 6821, 11502, 8807, 12138,
      2127, 2839, 3957, 431, 1579, 6383, 9784, 5874, 677, 3336, 6234, 2766, 1323,
      9115, 12237, 2031, 6956, 6413, 2281, 3969, 3991, 12133, 9522, 4737, 10996,
      4774, 5429, 11871, 3772, 453, 5908, 2882, 1805, 2051, 1954, 11713, 3963,
      2447, 6142, 8174, 3030, 1843, 2361, 12071, 2908, 3529, 3434, 3202, 7796,
      2057, 5369, 11939, 1512, 6906, 10474, 11026, 49, 10806, 5915, 1489, 9789,
      5942, 10706, 10431, 7535, 426, 8974, 3757, 10314, 9364, 347, 5868, 9551,
      9634, 6554, 10596, 9280, 11566, 174, 2948, 2503, 6507, 10723, 11606, 2459,
      64, 3656, 8455, 5257, 5919, 7856, 1747, 9166, 5486, 9235, 6065, 835, 3570,
      4240, 11580, 4046, 10970, 9139, 1058, 8210, 11848, 922, 7967, 1958, 10211,
      1112, 3728, 4049, 11130, 5990, 1404, 325, 948, 11143, 6190, 295, 11637, 5766,
      8212, 8273, 2919, 8527, 6119, 6992, 8333, 1360, 2555, 6167, 1200, 7105, 7991,
      3329, 9597, 12121, 5106, 5961, 10695, 10327, 3051, 9923, 4896, 9326, 81,
      3091, 1000, 7969, 4611, 726, 1853, 12149, 4255, 11112, 2768, 10654, 1062,
      2294, 3553, 4805, 2747, 4846, 8577, 9154, 1170, 2319, 790, 11334, 9275, 9088,
      1326, 5086, 9094, 6429, 11077, 10643, 3504, 3542, 8668, 9744, 1479, 1, 8246,
      7143, 11567, 10984, 4134, 5736, 4978, 10938, 5777, 8961, 4591, 5728, 6461,
      5023, 9650, 7468, 949, 9664, 2975, 11726, 2744, 9283, 10092, 5067, 12171,
      2476, 3748, 11336, 6522, 827, 9452, 5374, 12159, 7935, 3296, 3949, 9893,
      4452, 10908, 2525, 3584, 8112, 8011, 10616, 4989, 6958, 11809, 9447, 12280,
      1022, 11950, 9821, 11745, 5791, 5092, 2089, 9005, 2881, 3289, 2013, 9048,
      729, 7901, 1260, 5755, 4632, 11955, 2426, 10593, 1428, 4890, 5911, 3932,
      9558, 8830, 3637, 5542, 145, 5179, 8595, 3707, 10530, 355, 3382, 4231, 9741,
      1207, 9041, 7012, 1168, 10146, 11224, 4645, 11885, 10911, 10377, 435, 7952,
      4096, 493, 9908, 6845, 6039, 2422, 2187, 9723, 8643, 9852, 9302, 6022, 7278,
      1002, 4284, 5088, 1607, 7313, 875, 8509, 9430, 1045, 2481, 5012, 7428, 354,
      6591, 9377, 11847, 2401, 1067, 7188, 11516, 390, 8511, 8456, 7270, 545, 8585,
      9611, 12047, 1537, 4143, 4714, 4885, 1017, 5084, 1632, 3066, 27, 1440, 8526,
      9273, 12046, 11618, 9289, 3400, 9890, 3136, 7098, 8758, 11813, 7384, 3985,
      11869, 6730, 10745, 10111, 2249, 4048, 2884, 11136, 2126, 1630, 9103, 5407,
      2686, 9042, 2969, 8311, 9424, 9919, 8779, 5332, 10626, 1777, 4654, 10863,
      7351, 3636, 9585, 5291, 8374, 2166, 4919, 12176, 9140, 12129, 7852, 12286,
      4895, 10805, 2780, 5195, 2305, 7247, 9644, 4053, 10600, 3364, 3271, 4057,
      4414, 9442, 7917, 2174, 3947, 11951, 2455, 6599, 10545, 10975, 3654, 2894,
      7681, 7126, 7287, 12269, 4119, 3343, 2151, 1522, 7174, 7350, 11041, 2442,
      2148, 5959, 6492, 8330, 8945, 5598, 3624, 10397, 1325, 6565, 1945, 11260,
      10077, 2674, 3338, 3276, 11034, 506, 6505, 1392, 5478, 8778, 1178, 2776,
      3408, 10347, 11124, 2575, 9489, 12096, 6092, 10058, 4167, 6085, 923, 11251,
      11912, 4578, 10669, 11914, 425, 10453, 392, 10104, 8464, 4235, 8761, 7376,
      2291, 3375, 7954, 8896, 6617, 7790, 1737, 11667, 3982, 9342, 6680, 636, 6825,
      7383, 512, 4670, 2900, 12050, 7735, 994, 1687, 11883, 7021, 146, 10485, 1403,
      5189, 6094, 2483, 2054, 3042, 10945, 3981, 10821, 11826, 8882, 8151, 180,
      9600, 7684, 5219, 10880, 6780, 204, 11232, 2600, 7584, 3121, 3017, 11053,
      7814, 7043, 4251, 4739, 11063, 6771, 7073, 9261, 2360, 11925, 1928, 11825,
      8024, 3678, 3205, 3359, 11197, 5209, 8581, 3238, 8840, 1136, 9363, 1826,
      3171, 4489, 7885, 346, 2068, 1389, 8257, 3163, 4840, 6127, 8062, 8921, 612,
      4238, 10763, 8067, 125, 11749, 10125, 5416, 2110, 716, 9839, 10584, 11475,
      11873, 3448, 343, 1908, 4538, 10423, 7078, 4727, 1208, 11572, 3589, 2982,
      1373, 1721, 10753, 4103, 2429, 4209, 5412, 5993, 9011, 438, 3515, 7228, 1218,
      8347, 5232, 8682, 1327, 7508, 4924, 448, 1014, 10029, 12221, 4566, 5836,
      12229, 2717, 1535, 3200, 5588, 5845, 412, 5102, 7326, 3744, 3056, 2528, 7406,
      8314, 9202, 6454, 6613, 1417, 10032, 7784, 1518, 3765, 4176, 5063, 9828,
      2275, 6636, 4267, 6463, 2065, 7725, 3495, 8328, 8755, 8144, 10533, 5966,
      12077, 9175, 9520, 5596, 6302, 8400, 579, 6781, 11014, 5734, 11113, 11164,
      4860, 1131, 10844, 9068, 8016, 9694, 3837, 567, 9348, 7000, 6627, 7699, 5082,
      682, 11309, 5207, 4050, 7087, 844, 7434, 3769, 293, 9057, 6940, 9344, 10883,
      2633, 8190, 3944, 5530, 5604, 3480, 2171, 9282, 11024, 2213, 8136, 3805, 767,
      12239, 216, 11520, 6763, 10353, 7, 8566, 845, 7235, 3154, 4360, 3285, 10268,
      2832, 3572, 1282, 7559, 3229, 8360, 10583, 6105, 3120, 6643, 6203, 8536,
      8348, 6919, 3536, 9199, 10891, 11463, 5043, 1658, 5618, 8787, 5789, 4719,
      751, 11379, 6389, 10783, 3065, 7806, 6586, 2622, 5386, 510, 7628, 6921, 578,
      10345, 11839, 8929, 4684, 12226, 7154, 9916, 7302, 8481, 3670, 11066, 2334,
      1590, 7878, 10734, 1802, 1891, 5103, 6151, 8820, 3418, 7846, 9951, 4693, 417,
      9996, 9652, 4510, 2946, 5461, 365, 881, 1927, 1015, 11675, 11009, 1371,
      12265, 2485, 11385, 5039, 6742, 8449, 1842, 12217, 8176, 9577, 4834, 7937,
      9461, 2643, 11194, 3045, 6508, 4094, 3451, 7911, 11048, 5406, 4665, 3020,
      6616, 11345, 7519, 3669, 5287, 1790, 7014, 5410, 11038, 11249, 2035, 6125,
      10407, 4565, 7315, 5078, 10506, 2840, 2478, 9270, 4194, 9195, 4518, 7469,
      1160, 6878, 2730, 10421, 10036, 1734, 3815, 10939, 5832, 10595, 10759, 4423,
      8420, 9617, 7119, 11010, 11424, 9173, 189, 10080, 10526, 3466, 10588, 7592,
      3578, 11511, 7785, 9663, 530, 12150, 8957, 2532, 3317, 9349, 10243, 1481,
      9332, 3454, 3758, 7899, 4218, 2593, 11410, 2276, 982, 6513, 1849, 8494, 9021,
      4523, 7988, 8, 457, 648, 150, 8000, 2307, 2301, 874, 5650, 170, 9462, 2873,
      9855, 11498, 2535, 11169, 5808, 12268, 9687, 1901, 7171, 11787, 3846, 1573,
      6063, 3793, 466, 11259, 10608, 3821, 6320, 4649, 6263, 2929
      };

   mul_coefficients(r->coeffs, psis_bitrev_montgomery);
   ntt(r->coeffs, omegas_montgomery);
   }

inline void bitrev_vector(uint16_t* poly)
   {
   static const uint16_t bitrev_table[1024] =
      {
      0, 512, 256, 768, 128, 640, 384, 896, 64, 576, 320, 832, 192, 704, 448, 960, 32, 544, 288, 800, 160, 672, 416, 928, 96, 608, 352, 864, 224, 736, 480, 992,
      16, 528, 272, 784, 144, 656, 400, 912, 80, 592, 336, 848, 208, 720, 464, 976, 48, 560, 304, 816, 176, 688, 432, 944, 112, 624, 368, 880, 240, 752, 496, 1008,
      8, 520, 264, 776, 136, 648, 392, 904, 72, 584, 328, 840, 200, 712, 456, 968, 40, 552, 296, 808, 168, 680, 424, 936, 104, 616, 360, 872, 232, 744, 488, 1000,
      24, 536, 280, 792, 152, 664, 408, 920, 88, 600, 344, 856, 216, 728, 472, 984, 56, 568, 312, 824, 184, 696, 440, 952, 120, 632, 376, 888, 248, 760, 504, 1016,
      4, 516, 260, 772, 132, 644, 388, 900, 68, 580, 324, 836, 196, 708, 452, 964, 36, 548, 292, 804, 164, 676, 420, 932, 100, 612, 356, 868, 228, 740, 484, 996,
      20, 532, 276, 788, 148, 660, 404, 916, 84, 596, 340, 852, 212, 724, 468, 980, 52, 564, 308, 820, 180, 692, 436, 948, 116, 628, 372, 884, 244, 756, 500, 1012,
      12, 524, 268, 780, 140, 652, 396, 908, 76, 588, 332, 844, 204, 716, 460, 972, 44, 556, 300, 812, 172, 684, 428, 940, 108, 620, 364, 876, 236, 748, 492, 1004,
      28, 540, 284, 796, 156, 668, 412, 924, 92, 604, 348, 860, 220, 732, 476, 988, 60, 572, 316, 828, 188, 700, 444, 956, 124, 636, 380, 892, 252, 764, 508, 1020,
      2, 514, 258, 770, 130, 642, 386, 898, 66, 578, 322, 834, 194, 706, 450, 962, 34, 546, 290, 802, 162, 674, 418, 930, 98, 610, 354, 866, 226, 738, 482, 994,
      18, 530, 274, 786, 146, 658, 402, 914, 82, 594, 338, 850, 210, 722, 466, 978, 50, 562, 306, 818, 178, 690, 434, 946, 114, 626, 370, 882, 242, 754, 498, 1010,
      10, 522, 266, 778, 138, 650, 394, 906, 74, 586, 330, 842, 202, 714, 458, 970, 42, 554, 298, 810, 170, 682, 426, 938, 106, 618, 362, 874, 234, 746, 490, 1002,
      26, 538, 282, 794, 154, 666, 410, 922, 90, 602, 346, 858, 218, 730, 474, 986, 58, 570, 314, 826, 186, 698, 442, 954, 122, 634, 378, 890, 250, 762, 506, 1018,
      6, 518, 262, 774, 134, 646, 390, 902, 70, 582, 326, 838, 198, 710, 454, 966, 38, 550, 294, 806, 166, 678, 422, 934, 102, 614, 358, 870, 230, 742, 486, 998,
      22, 534, 278, 790, 150, 662, 406, 918, 86, 598, 342, 854, 214, 726, 470, 982, 54, 566, 310, 822, 182, 694, 438, 950, 118, 630, 374, 886, 246, 758, 502, 1014,
      14, 526, 270, 782, 142, 654, 398, 910, 78, 590, 334, 846, 206, 718, 462, 974, 46, 558, 302, 814, 174, 686, 430, 942, 110, 622, 366, 878, 238, 750, 494, 1006,
      30, 542, 286, 798, 158, 670, 414, 926, 94, 606, 350, 862, 222, 734, 478, 990, 62, 574, 318, 830, 190, 702, 446, 958, 126, 638, 382, 894, 254, 766, 510, 1022,
      1, 513, 257, 769, 129, 641, 385, 897, 65, 577, 321, 833, 193, 705, 449, 961, 33, 545, 289, 801, 161, 673, 417, 929, 97, 609, 353, 865, 225, 737, 481, 993,
      17, 529, 273, 785, 145, 657, 401, 913, 81, 593, 337, 849, 209, 721, 465, 977, 49, 561, 305, 817, 177, 689, 433, 945, 113, 625, 369, 881, 241, 753, 497, 1009,
      9, 521, 265, 777, 137, 649, 393, 905, 73, 585, 329, 841, 201, 713, 457, 969, 41, 553, 297, 809, 169, 681, 425, 937, 105, 617, 361, 873, 233, 745, 489, 1001,
      25, 537, 281, 793, 153, 665, 409, 921, 89, 601, 345, 857, 217, 729, 473, 985, 57, 569, 313, 825, 185, 697, 441, 953, 121, 633, 377, 889, 249, 761, 505, 1017,
      5, 517, 261, 773, 133, 645, 389, 901, 69, 581, 325, 837, 197, 709, 453, 965, 37, 549, 293, 805, 165, 677, 421, 933, 101, 613, 357, 869, 229, 741, 485, 997,
      21, 533, 277, 789, 149, 661, 405, 917, 85, 597, 341, 853, 213, 725, 469, 981, 53, 565, 309, 821, 181, 693, 437, 949, 117, 629, 373, 885, 245, 757, 501, 1013,
      13, 525, 269, 781, 141, 653, 397, 909, 77, 589, 333, 845, 205, 717, 461, 973, 45, 557, 301, 813, 173, 685, 429, 941, 109, 621, 365, 877, 237, 749, 493, 1005,
      29, 541, 285, 797, 157, 669, 413, 925, 93, 605, 349, 861, 221, 733, 477, 989, 61, 573, 317, 829, 189, 701, 445, 957, 125, 637, 381, 893, 253, 765, 509, 1021,
      3, 515, 259, 771, 131, 643, 387, 899, 67, 579, 323, 835, 195, 707, 451, 963, 35, 547, 291, 803, 163, 675, 419, 931, 99, 611, 355, 867, 227, 739, 483, 995,
      19, 531, 275, 787, 147, 659, 403, 915, 83, 595, 339, 851, 211, 723, 467, 979, 51, 563, 307, 819, 179, 691, 435, 947, 115, 627, 371, 883, 243, 755, 499, 1011,
      11, 523, 267, 779, 139, 651, 395, 907, 75, 587, 331, 843, 203, 715, 459, 971, 43, 555, 299, 811, 171, 683, 427, 939, 107, 619, 363, 875, 235, 747, 491, 1003,
      27, 539, 283, 795, 155, 667, 411, 923, 91, 603, 347, 859, 219, 731, 475, 987, 59, 571, 315, 827, 187, 699, 443, 955, 123, 635, 379, 891, 251, 763, 507, 1019,
      7, 519, 263, 775, 135, 647, 391, 903, 71, 583, 327, 839, 199, 711, 455, 967, 39, 551, 295, 807, 167, 679, 423, 935, 103, 615, 359, 871, 231, 743, 487, 999,
      23, 535, 279, 791, 151, 663, 407, 919, 87, 599, 343, 855, 215, 727, 471, 983, 55, 567, 311, 823, 183, 695, 439, 951, 119, 631, 375, 887, 247, 759, 503, 1015,
      15, 527, 271, 783, 143, 655, 399, 911, 79, 591, 335, 847, 207, 719, 463, 975, 47, 559, 303, 815, 175, 687, 431, 943, 111, 623, 367, 879, 239, 751, 495, 1007,
      31, 543, 287, 799, 159, 671, 415, 927, 95, 607, 351, 863, 223, 735, 479, 991, 63, 575, 319, 831, 191, 703, 447, 959, 127, 639, 383, 895, 255, 767, 511, 1023
      };

   for(size_t i = 0; i < PARAM_N; i++)
      {
      const uint16_t r = bitrev_table[i];
      if(i < r)
         {
         const uint16_t tmp = poly[i];
         poly[i] = poly[r];
         poly[r] = tmp;
         }
      }
   }

inline void poly_invntt(poly* r)
   {
   static const uint16_t omegas_inv_montgomery[PARAM_N/2] =
      {
      4075, 5315, 4324, 4916, 10120, 11767, 7210, 9027, 10316, 6715, 1278, 9945,
      3514, 11248, 11271, 5925, 147, 8500, 7840, 6833, 5537, 4749, 4467, 7500,
      11099, 9606, 6171, 8471, 8429, 5445, 11239, 7753, 9090, 12233, 5529, 5206,
      10587, 1987, 11635, 3565, 5415, 8646, 6153, 6427, 7341, 6152, 10561, 400,
      8410, 1922, 2033, 8291, 1359, 6854, 11035, 973, 8579, 6093, 6950, 5446,
      11821, 8301, 11907, 316, 52, 3174, 10966, 9523, 6055, 8953, 11612, 6415,
      2505, 5906, 10710, 11858, 8332, 9450, 10162, 151, 3482, 787, 5468, 1010,
      4169, 9162, 5241, 9369, 7509, 8844, 7232, 4698, 192, 1321, 10240, 4912, 885,
      6281, 10333, 7280, 8757, 11286, 58, 12048, 12147, 11184, 8812, 6608, 2844,
      3438, 4212, 11314, 8687, 6068, 421, 8209, 3600, 3263, 7665, 6077, 7507, 5886,
      3029, 6695, 4213, 504, 11684, 2302, 1962, 1594, 6328, 7183, 168, 2692, 8960,
      4298, 5184, 11089, 6122, 9734, 10929, 3956, 5297, 6170, 3762, 9370, 4016,
      4077, 6523, 652, 11994, 6099, 1146, 11341, 11964, 10885, 6299, 1159, 8240,
      8561, 11177, 2078, 10331, 4322, 11367, 441, 4079, 11231, 3150, 1319, 8243,
      709, 8049, 8719, 11454, 6224, 3054, 6803, 3123, 10542, 4433, 6370, 7032,
      3834, 8633, 12225, 9830, 683, 1566, 5782, 9786, 9341, 12115, 723, 3009, 1693,
      5735, 2655, 2738, 6421, 11942, 2925, 1975, 8532, 3315, 11863, 4754, 1858,
      1583, 6347, 2500, 10800, 6374, 1483, 12240, 1263, 1815, 5383, 10777, 350,
      6920, 10232, 4493, 9087, 8855, 8760, 9381, 218, 9928, 10446, 9259, 4115,
      6147, 9842, 8326, 576, 10335, 10238, 10484, 9407, 6381, 11836, 8517, 418,
      6860, 7515, 1293, 7552, 2767, 156, 8298, 8320, 10008, 5876, 5333, 10258,
      10115, 4372, 2847, 7875, 8232, 9018, 8925, 1689, 8236, 2645, 5042, 9984,
      7094, 9509, 1484, 7394, 3, 4437, 160, 3149, 113, 7370, 10123, 3915, 6998,
      2704, 8653, 4938, 1426, 7635, 10512, 1663, 6957, 3510, 2370, 2865, 3978,
      9320, 3247, 9603, 6882, 3186, 10659, 10163, 1153, 9405, 8241, 10040, 2178,
      1544, 5559, 420, 8304, 4905, 476, 3531, 5191, 9153, 2399, 8889, 3000, 671,
      243, 3016, 3763, 10849, 12262, 9223, 10657, 7205, 11272, 7404, 7575, 8146,
      10752, 242, 2678, 3704, 11744, 5019, 3833, 3778, 11899, 773, 5101, 11222,
      9888, 442, 2912, 5698, 11935, 4861, 7277, 9808, 11244, 2859, 3780, 11414,
      4976, 10682, 7201, 8005, 11287, 5011, 6267, 2987, 2437, 3646, 2566, 10102,
      9867, 6250, 5444, 2381, 11796, 8193, 4337, 11854, 1912, 1378, 404, 7644,
      1065, 2143, 11121, 5277, 3248, 11082, 2548, 8058, 8907, 11934, 1759, 8582,
      3694, 7110, 12144, 6747, 8652, 3459, 2731, 8357, 6378, 7399, 10861, 1696,
      9863, 334, 7657, 6534, 11029, 4388, 11560, 3241, 10276, 9000, 9408, 3284,
      10200, 7197, 6498, 544, 2468, 339, 11267, 9, 2842, 480, 5331, 7300, 1673,
      4278, 4177, 8705, 9764, 1381, 7837, 2396, 8340, 8993, 4354, 130, 6915, 2837,
      11462, 5767, 953, 8541, 9813, 118, 7222, 2197, 3006, 9545, 563, 9314, 2625,
      11340, 4821, 2639, 7266, 5828, 6561, 7698, 3328, 6512, 1351, 7311, 6553,
      8155, 1305, 722, 5146, 4043, 12288, 10810, 2545, 3621, 8747, 8785, 1646,
      1212, 5860, 3195, 7203, 10963, 3201, 3014, 955, 11499, 9970, 11119, 3135,
      3712, 7443, 9542, 7484, 8736, 9995, 11227, 1635, 9521, 1177, 8034, 140,
      10436, 11563, 7678, 4320, 11289, 9198, 12208, 2963, 7393, 2366, 9238
      };

   static const uint16_t psis_inv_montgomery[PARAM_N] =
      {
      256, 10570, 1510, 7238, 1034, 7170, 6291, 7921, 11665, 3422, 4000, 2327,
      2088, 5565, 795, 10647, 1521, 5484, 2539, 7385, 1055, 7173, 8047, 11683,
      1669, 1994, 3796, 5809, 4341, 9398, 11876, 12230, 10525, 12037, 12253, 3506,
      4012, 9351, 4847, 2448, 7372, 9831, 3160, 2207, 5582, 2553, 7387, 6322, 9681,
      1383, 10731, 1533, 219, 5298, 4268, 7632, 6357, 9686, 8406, 4712, 9451,
      10128, 4958, 5975, 11387, 8649, 11769, 6948, 11526, 12180, 1740, 10782, 6807,
      2728, 7412, 4570, 4164, 4106, 11120, 12122, 8754, 11784, 3439, 5758, 11356,
      6889, 9762, 11928, 1704, 1999, 10819, 12079, 12259, 7018, 11536, 1648, 1991,
      2040, 2047, 2048, 10826, 12080, 8748, 8272, 8204, 1172, 1923, 7297, 2798,
      7422, 6327, 4415, 7653, 6360, 11442, 12168, 7005, 8023, 9924, 8440, 8228,
      2931, 7441, 1063, 3663, 5790, 9605, 10150, 1450, 8985, 11817, 10466, 10273,
      12001, 3470, 7518, 1074, 1909, 7295, 9820, 4914, 702, 5367, 7789, 8135, 9940,
      1420, 3714, 11064, 12114, 12264, 1752, 5517, 9566, 11900, 1700, 3754, 5803,
      829, 1874, 7290, 2797, 10933, 5073, 7747, 8129, 6428, 6185, 11417, 1631, 233,
      5300, 9535, 10140, 11982, 8734, 8270, 2937, 10953, 8587, 8249, 2934, 9197,
      4825, 5956, 4362, 9401, 1343, 3703, 529, 10609, 12049, 6988, 6265, 895, 3639,
      4031, 4087, 4095, 585, 10617, 8539, 4731, 4187, 9376, 3095, 9220, 10095,
      10220, 1460, 10742, 12068, 1724, 5513, 11321, 6884, 2739, 5658, 6075, 4379,
      11159, 10372, 8504, 4726, 9453, 3106, 7466, 11600, 10435, 8513, 9994, 8450,
      9985, 3182, 10988, 8592, 2983, 9204, 4826, 2445, 5616, 6069, 867, 3635, 5786,
      11360, 5134, 2489, 10889, 12089, 1727, 7269, 2794, 9177, 1311, 5454, 9557,
      6632, 2703, 9164, 10087, 1441, 3717, 531, 3587, 2268, 324, 5313, 759, 1864,
      5533, 2546, 7386, 9833, 8427, 4715, 11207, 1601, 7251, 4547, 11183, 12131,
      1733, 10781, 10318, 1474, 10744, 5046, 4232, 11138, 10369, 6748, 964, 7160,
      4534, 7670, 8118, 8182, 4680, 11202, 6867, 981, 8918, 1274, 182, 26, 7026,
      8026, 11680, 12202, 10521, 1503, 7237, 4545, 5916, 9623, 8397, 11733, 10454,
      3249, 9242, 6587, 941, 1890, 270, 10572, 6777, 9746, 6659, 6218, 6155, 6146,
      878, 1881, 7291, 11575, 12187, 1741, 7271, 8061, 11685, 6936, 4502, 9421,
      4857, 4205, 7623, 1089, 10689, 1527, 8996, 10063, 11971, 10488, 6765, 2722,
      3900, 9335, 11867, 6962, 11528, 5158, 4248, 4118, 5855, 2592, 5637, 6072,
      2623, 7397, 8079, 9932, 4930, 5971, 853, 3633, 519, 8852, 11798, 3441, 11025,
      1575, 225, 8810, 11792, 12218, 3501, 9278, 3081, 9218, 4828, 7712, 8124,
      11694, 12204, 3499, 4011, 573, 3593, 5780, 7848, 9899, 10192, 1456, 208,
      7052, 2763, 7417, 11593, 10434, 12024, 8740, 11782, 10461, 3250, 5731, 7841,
      9898, 1414, 202, 3540, 7528, 2831, 2160, 10842, 5060, 4234, 4116, 588, 84,
      12, 7024, 2759, 9172, 6577, 11473, 1639, 9012, 3043, 7457, 6332, 11438, 1634,
      1989, 9062, 11828, 8712, 11778, 12216, 10523, 6770, 9745, 10170, 4964, 9487,
      6622, 946, 8913, 6540, 6201, 4397, 9406, 8366, 9973, 8447, 8229, 11709, 8695,
      10020, 3187, 5722, 2573, 10901, 6824, 4486, 4152, 9371, 8361, 2950, 2177,
      311, 1800, 9035, 8313, 11721, 3430, 490, 70, 10, 1757, 251, 3547, 7529,
      11609, 3414, 7510, 4584, 4166, 9373, 1339, 5458, 7802, 11648, 1664, 7260,
      9815, 10180, 6721, 9738, 10169, 8475, 8233, 9954, 1422, 8981, 1283, 5450,
      11312, 1616, 3742, 11068, 10359, 4991, 713, 3613, 9294, 8350, 4704, 672, 96,
      7036, 9783, 11931, 3460, 5761, 823, 10651, 12055, 10500, 1500, 5481, 783,
      3623, 11051, 8601, 8251, 8201, 11705, 10450, 5004, 4226, 7626, 2845, 2162,
      3820, 7568, 9859, 3164, 452, 10598, 1514, 5483, 6050, 6131, 4387, 7649, 8115,
      6426, 918, 8909, 8295, 1185, 5436, 11310, 8638, 1234, 5443, 11311, 5127,
      2488, 2111, 10835, 5059, 7745, 2862, 3920, 560, 80, 1767, 2008, 3798, 11076,
      6849, 2734, 10924, 12094, 8750, 1250, 10712, 6797, 971, 7161, 1023, 8924,
      4786, 7706, 4612, 4170, 7618, 6355, 4419, 5898, 11376, 10403, 10264, 6733,
      4473, 639, 5358, 2521, 9138, 3061, 5704, 4326, 618, 5355, 765, 5376, 768,
      7132, 4530, 9425, 3102, 9221, 6584, 11474, 10417, 10266, 12000, 6981, 6264,
      4406, 2385, 7363, 4563, 4163, 7617, 9866, 3165, 9230, 11852, 10471, 5007,
      5982, 11388, 5138, 734, 3616, 11050, 12112, 6997, 11533, 12181, 10518, 12036,
      3475, 2252, 7344, 9827, 4915, 9480, 6621, 4457, 7659, 9872, 6677, 4465, 4149,
      7615, 4599, 657, 3605, 515, 10607, 6782, 4480, 640, 1847, 3775, 5806, 2585,
      5636, 9583, 1369, 10729, 8555, 10000, 11962, 5220, 7768, 8132, 8184, 9947,
      1421, 203, 29, 8782, 11788, 1684, 10774, 10317, 4985, 9490, 8378, 4708,
      11206, 5112, 5997, 7879, 11659, 12199, 8765, 10030, 4944, 5973, 6120, 6141,
      6144, 7900, 11662, 1666, 238, 34, 3516, 5769, 9602, 8394, 9977, 6692, 956,
      10670, 6791, 9748, 11926, 8726, 11780, 5194, 742, 106, 8793, 10034, 3189,
      10989, 5081, 4237, 5872, 4350, 2377, 10873, 6820, 6241, 11425, 10410, 10265,
      3222, 5727, 9596, 4882, 2453, 2106, 3812, 11078, 12116, 5242, 4260, 11142,
      8614, 11764, 12214, 5256, 4262, 4120, 11122, 5100, 11262, 5120, 2487, 5622,
      9581, 8391, 8221, 2930, 10952, 12098, 6995, 6266, 9673, 4893, 699, 3611,
      4027, 5842, 11368, 1624, 232, 8811, 8281, 1183, 169, 8802, 3013, 2186, 5579,
      797, 3625, 4029, 11109, 1587, 7249, 11569, 8675, 6506, 2685, 10917, 12093,
      12261, 12285, 1755, 7273, 1039, 1904, 272, 3550, 9285, 3082, 5707, 6082,
      4380, 7648, 11626, 5172, 4250, 9385, 8363, 8217, 4685, 5936, 848, 8899, 6538,
      934, 1889, 3781, 9318, 10109, 10222, 6727, 961, 5404, 772, 5377, 9546, 8386,
      1198, 8949, 3034, 2189, 7335, 4559, 5918, 2601, 10905, 5069, 9502, 3113,
      7467, 8089, 11689, 5181, 9518, 8382, 2953, 3933, 4073, 4093, 7607, 8109,
      2914, 5683, 4323, 11151, 1593, 10761, 6804, 972, 3650, 2277, 5592, 4310,
      7638, 9869, 4921, 703, 1856, 9043, 4803, 9464, 1352, 8971, 11815, 5199, 7765,
      6376, 4422, 7654, 2849, 407, 8836, 6529, 7955, 2892, 9191, 1313, 10721,
      12065, 12257, 1751, 9028, 8312, 2943, 2176, 3822, 546, 78, 8789, 11789,
      10462, 12028, 6985, 4509, 9422, 1346, 5459, 4291, 613, 10621, 6784, 9747,
      3148, 7472, 2823, 5670, 810, 7138, 8042, 4660, 7688, 6365, 6176, 6149, 2634,
      5643, 9584, 10147, 11983, 5223, 9524, 11894, 10477, 8519, 1217, 3685, 2282,
      326, 10580, 3267, 7489, 4581, 2410, 5611, 11335, 6886, 8006, 8166, 11700,
      3427, 11023, 8597, 10006, 3185, 455, 65, 5276, 7776, 4622, 5927, 7869, 9902,
      11948, 5218, 2501, 5624, 2559, 10899, 1557, 1978, 10816, 10323, 8497, 4725,
      675, 1852, 10798, 12076, 10503, 3256, 9243, 3076, 2195, 10847, 12083, 10504,
      12034, 10497
      };

   bitrev_vector(r->coeffs);
   ntt(r->coeffs, omegas_inv_montgomery);
   mul_coefficients(r->coeffs, psis_inv_montgomery);
   }

inline void encode_a(uint8_t* r, const poly* pk, const uint8_t* seed)
   {
   poly_tobytes(r, pk);
   for(size_t i = 0; i < NEWHOPE_SEED_BYTES; i++)
      {
      r[NEWHOPE_POLY_BYTES+i] = seed[i];
      }
   }

inline void decode_a(poly* pk, uint8_t* seed, const uint8_t* r)
   {
   poly_frombytes(pk, r);
   for(size_t i = 0; i < NEWHOPE_SEED_BYTES; i++)
      {
      seed[i] = r[NEWHOPE_POLY_BYTES+i];
      }
   }

inline void encode_b(uint8_t* r, const poly* b, const poly* c)
   {
   poly_tobytes(r, b);
   for(size_t i = 0; i < PARAM_N/4; i++)
      {
      r[NEWHOPE_POLY_BYTES+i] = static_cast<uint8_t>(c->coeffs[4*i] |
                                                     (c->coeffs[4*i+1] << 2) |
                                                     (c->coeffs[4*i+2] << 4) |
                                                     (c->coeffs[4*i+3] << 6));
      }
   }

inline void decode_b(poly* b, poly* c, const uint8_t* r)
   {
   poly_frombytes(b, r);
   for(size_t i = 0; i < PARAM_N/4; i++)
      {
      c->coeffs[4*i+0] =  r[NEWHOPE_POLY_BYTES+i]       & 0x03;
      c->coeffs[4*i+1] = (r[NEWHOPE_POLY_BYTES+i] >> 2) & 0x03;
      c->coeffs[4*i+2] = (r[NEWHOPE_POLY_BYTES+i] >> 4) & 0x03;
      c->coeffs[4*i+3] = (r[NEWHOPE_POLY_BYTES+i] >> 6);
      }
   }

inline int32_t ct_abs(int32_t v)
   {
   int32_t mask = v >> 31;
   return (v ^ mask) - mask;
   }

inline int32_t f(int32_t* v0, int32_t* v1, int32_t x)
   {
   int32_t xit, t, r, b;

   // Next 6 lines compute t = x/PARAM_Q;
   b = x*2730;
   t = b >> 25;
   b = x - t*12289;
   b = 12288 - b;
   b >>= 31;
   t -= b;

   r = t & 1;
   xit = (t>>1);
   *v0 = xit+r; // v0 = round(x/(2*PARAM_Q))

   t -= 1;
   r = t & 1;
   *v1 = (t>>1)+r;

   return ct_abs(x-((*v0)*2*PARAM_Q));
   }

inline void helprec(poly* c, const poly* v, RandomNumberGenerator& rng)
   {
   uint8_t rand[32];

   rng.randomize(rand, 32);

   for(size_t i = 0; i < 256; i++)
      {
      int32_t v0[4], v1[4];
      uint8_t rbit = (rand[i>>3] >> (i&7)) & 1;
      int32_t k;

      k  = f(v0+0, v1+0, 8*v->coeffs[  0+i] + 4*rbit);
      k += f(v0+1, v1+1, 8*v->coeffs[256+i] + 4*rbit);
      k += f(v0+2, v1+2, 8*v->coeffs[512+i] + 4*rbit);
      k += f(v0+3, v1+3, 8*v->coeffs[768+i] + 4*rbit);

      k = (2*PARAM_Q-1-k) >> 31;

      int32_t v_tmp[4];
      v_tmp[0] = ((~k) & v0[0]) ^ (k & v1[0]);
      v_tmp[1] = ((~k) & v0[1]) ^ (k & v1[1]);
      v_tmp[2] = ((~k) & v0[2]) ^ (k & v1[2]);
      v_tmp[3] = ((~k) & v0[3]) ^ (k & v1[3]);

      c->coeffs[  0+i] = (v_tmp[0] -   v_tmp[3]) & 3;
      c->coeffs[256+i] = (v_tmp[1] -   v_tmp[3]) & 3;
      c->coeffs[512+i] = (v_tmp[2] -   v_tmp[3]) & 3;
      c->coeffs[768+i] = (-   k    + 2*v_tmp[3]) & 3;
      }
   }

inline int32_t g(int32_t x)
   {
   int32_t t, c, b;

   // Next 6 lines compute t = x/(4*PARAM_Q);
   b = x*2730;
   t = b >> 27;
   b = x - t*49156;
   b = 49155 - b;
   b >>= 31;
   t -= b;

   c = t & 1;
   t = (t >> 1) + c; // t = round(x/(8*PARAM_Q))

   t *= 8*PARAM_Q;

   return ct_abs(t - x);
   }

inline int16_t LDDecode(int32_t xi0, int32_t xi1, int32_t xi2, int32_t xi3)
   {
   int32_t t;

   t  = g(xi0);
   t += g(xi1);
   t += g(xi2);
   t += g(xi3);

   t -= 8*PARAM_Q;
   t >>= 31;
   return t&1;
   }

inline void rec(uint8_t* key, const poly* v, const poly* c)
   {
   clear_mem(key, 32);

   for(size_t i = 0; i < 256; i++)
      {
      const int32_t tmp0 = 16*PARAM_Q + 8*static_cast<int32_t>(v->coeffs[  0+i]) - PARAM_Q * (2*c->coeffs[  0+i]+c->coeffs[768+i]);
      const int32_t tmp1 = 16*PARAM_Q + 8*static_cast<int32_t>(v->coeffs[256+i]) - PARAM_Q * (2*c->coeffs[256+i]+c->coeffs[768+i]);
      const int32_t tmp2 = 16*PARAM_Q + 8*static_cast<int32_t>(v->coeffs[512+i]) - PARAM_Q * (2*c->coeffs[512+i]+c->coeffs[768+i]);
      const int32_t tmp3 = 16*PARAM_Q + 8*static_cast<int32_t>(v->coeffs[768+i]) - PARAM_Q * (c->coeffs[768+i]);

      key[i>>3] |= LDDecode(tmp0, tmp1, tmp2, tmp3) << (i & 7);
      }
   }

void gen_a(poly* a, const uint8_t* seed, Newhope_Mode mode)
   {
   std::vector<uint8_t> buf(168*16);

   std::unique_ptr<StreamCipher> xof;

   if(mode == Newhope_Mode::BoringSSL)
      {
      xof = StreamCipher::create_or_throw("CTR-BE(AES-128)");
      xof->set_key(seed, 16);
      xof->set_iv(seed + 16, 16);
      }
   else
      {
      xof = StreamCipher::create_or_throw("SHAKE-128");
      xof->set_key(seed, NEWHOPE_SEED_BYTES);
      }

   zeroise(buf);
   xof->encrypt(buf);

   size_t pos = 0, ctr = 0;

   while(ctr < PARAM_N)
      {
      // Specialized for q = 12889
      const uint16_t val = (buf[pos] | (static_cast<uint16_t>(buf[pos+1]) << 8)) & 0x3fff;

      if(val < PARAM_Q)
         {
         a->coeffs[ctr++] = val;
         }
      pos += 2;
      if(pos >= buf.size())
         {
         zeroise(buf);
         xof->encrypt(buf);
         pos = 0;
         }
      }
   }

}

// API FUNCTIONS

void newhope_keygen(uint8_t* send, poly* sk, RandomNumberGenerator& rng,
                    Newhope_Mode mode)
   {
   poly a, e, r, pk;
   uint8_t seed[NEWHOPE_SEED_BYTES];

   rng.randomize(seed, NEWHOPE_SEED_BYTES);

   gen_a(&a, seed, mode);

   poly_getnoise(rng, sk);
   poly_ntt(sk);

   poly_getnoise(rng, &e);
   poly_ntt(&e);

   poly_pointwise(&r, sk, &a);
   poly_add(&pk, &e, &r);

   encode_a(send, &pk, seed);
   }

void newhope_sharedb(uint8_t* sharedkey, uint8_t* send, const uint8_t* received,
                     RandomNumberGenerator& rng,
                     Newhope_Mode mode)
   {
   poly sp, ep, v, a, pka, c, epp, bp;
   uint8_t seed[NEWHOPE_SEED_BYTES];

   decode_a(&pka, seed, received);
   gen_a(&a, seed, mode);

   poly_getnoise(rng, &sp);
   poly_ntt(&sp);
   poly_getnoise(rng, &ep);
   poly_ntt(&ep);

   poly_pointwise(&bp, &a, &sp);
   poly_add(&bp, &bp, &ep);

   poly_pointwise(&v, &pka, &sp);
   poly_invntt(&v);

   poly_getnoise(rng, &epp);
   poly_add(&v, &v, &epp);

   helprec(&c, &v, rng);

   encode_b(send, &bp, &c);

   rec(sharedkey, &v, &c);

   const std::string kdf_hash = (mode == Newhope_Mode::SHA3) ? "SHA-3(256)" : "SHA-256";
   std::unique_ptr<HashFunction> hash = HashFunction::create_or_throw(kdf_hash);

   hash->update(sharedkey, 32);
   hash->final(sharedkey);
   }

void newhope_shareda(uint8_t sharedkey[],
                     const poly* sk,
                     const uint8_t received[],
                     Newhope_Mode mode)
   {
   poly v, bp, c;

   decode_b(&bp, &c, received);

   poly_pointwise(&v, sk, &bp);
   poly_invntt(&v);

   rec(sharedkey, &v, &c);

   const std::string kdf_hash = (mode == Newhope_Mode::SHA3) ? "SHA-3(256)" : "SHA-256";
   std::unique_ptr<HashFunction> hash = HashFunction::create_or_throw(kdf_hash);

   hash->update(sharedkey, 32);
   hash->final(sharedkey);
   }

}