modular_equations 1.0.5

//! Elliptic curve computation.
//!
//! These are needed in Lenstra elliptic-curve factorization method.
//!
use rand::Rng;

use itertools::Itertools;

use crate::{
    arith::{Arith, CoreArith},
    UInt,
};

const BYTES_10K_LEN: usize = 1806;

/// Bytes for lcm(1,...,B), where the upper bound B equals 10_000.
///
/// To reproduce the following array:
/// 1) Compute lcm(1,...,B) for B = 10_000, and convert the result to base 2 (binary)
/// 2) Decompose the base 2 representation to an array of bytes (each element has 8 bits)
///
/// E.g., in the following array hexadecimal 0x9b contains the first 8 bits of the base 2
/// representation, 0x2c the next 8 bits etc.
static BYTES_10K: [u8; BYTES_10K_LEN] = [
    0x9b, 0x2c, 0xc9, 0x32, 0x95, 0x1f, 0x3e, 0x64, 0x97, 0xc2, 0x46, 0x3b, 0xa9, 0xf2, 0xdb, 0x91,
    0x2e, 0xda, 0x8e, 0x89, 0x25, 0x06, 0xa4, 0xab, 0xbc, 0x33, 0x3e, 0x7d, 0x6f, 0x3a, 0x25, 0xed,
    0x61, 0xb0, 0xe3, 0xff, 0xca, 0x04, 0x57, 0x5d, 0x7f, 0xf2, 0x2d, 0xc3, 0xc6, 0x2c, 0xc5, 0x47,
    0x2c, 0x1d, 0x82, 0xd3, 0x55, 0x55, 0x6e, 0x25, 0xd8, 0x6d, 0xd4, 0x8d, 0x4b, 0x61, 0x79, 0xaa,
    0xf1, 0x05, 0x2b, 0x70, 0x4f, 0x83, 0x13, 0x3e, 0xe9, 0x42, 0xe6, 0x80, 0x26, 0xc1, 0xce, 0x4e,
    0x93, 0xa2, 0xf5, 0xfd, 0x75, 0xa0, 0x61, 0xe9, 0x2c, 0x5e, 0xa5, 0x6c, 0x27, 0x8a, 0xc5, 0x00,
    0xcb, 0x13, 0x78, 0xf6, 0x79, 0x5c, 0x86, 0xef, 0x75, 0xdf, 0x10, 0x05, 0xc5, 0xae, 0xa2, 0xbb,
    0x61, 0x9a, 0x51, 0x92, 0x51, 0x03, 0x63, 0x01, 0x23, 0x1e, 0x38, 0x30, 0x1a, 0x3f, 0x7e, 0xcf,
    0xc0, 0x6c, 0x2a, 0x71, 0x79, 0x8d, 0xa1, 0xed, 0xbc, 0xda, 0xdf, 0x40, 0x31, 0x1f, 0xeb, 0x0f,
    0x45, 0x58, 0xd2, 0x7c, 0xe3, 0xeb, 0xf9, 0xa8, 0x80, 0x1e, 0xbc, 0xc7, 0xe4, 0xb8, 0x8f, 0x77,
    0x50, 0x7b, 0xb8, 0x17, 0x60, 0xb0, 0x5e, 0x9c, 0x85, 0xb8, 0xaa, 0x31, 0xca, 0xc7, 0xd3, 0x25,
    0xd7, 0x9a, 0x89, 0xae, 0x6a, 0xd1, 0xa7, 0xa6, 0x0f, 0x82, 0x61, 0x87, 0xfb, 0xf7, 0xc3, 0xfe,
    0xed, 0x86, 0x83, 0x07, 0x7d, 0xaf, 0xd8, 0xcb, 0x4f, 0xe8, 0xae, 0xb2, 0xd4, 0x4b, 0xbc, 0x20,
    0x46, 0x6b, 0x8a, 0x18, 0x3d, 0xf5, 0x88, 0x69, 0xfb, 0xcb, 0x97, 0x3c, 0x62, 0xed, 0x35, 0xb4,
    0xc1, 0x64, 0x44, 0xd0, 0x5c, 0x18, 0x5f, 0x7f, 0xc3, 0x85, 0x9d, 0x5f, 0x70, 0x24, 0xef, 0x18,
    0xcb, 0xb6, 0x01, 0xd8, 0x08, 0x99, 0x76, 0x12, 0x39, 0xa6, 0x26, 0xa5, 0x0d, 0x1e, 0x23, 0x15,
    0x67, 0xc1, 0x31, 0x2e, 0x92, 0x37, 0x7d, 0xe3, 0xd0, 0x7d, 0x52, 0xb4, 0xe7, 0x77, 0x77, 0x65,
    0xdb, 0x25, 0xe6, 0xeb, 0x98, 0x3e, 0x81, 0xf2, 0x3b, 0x48, 0x95, 0x2c, 0x63, 0x43, 0xf7, 0xdc,
    0x41, 0x40, 0x04, 0x80, 0xbf, 0xdf, 0xf1, 0x83, 0xc6, 0x53, 0x21, 0x5d, 0x81, 0x42, 0xe5, 0x3e,
    0xb4, 0x03, 0x9c, 0x00, 0x00, 0xb7, 0x9e, 0x40, 0x1b, 0xa1, 0x3d, 0x58, 0x06, 0xef, 0xc7, 0x8d,
    0x54, 0x06, 0x5e, 0x64, 0x69, 0x35, 0xa8, 0xbf, 0x9f, 0x39, 0x18, 0x56, 0x9e, 0xb3, 0x32, 0x45,
    0x38, 0xe0, 0xc4, 0x3a, 0xfe, 0x7b, 0xca, 0x28, 0xce, 0xf6, 0x81, 0xaf, 0xfd, 0x09, 0xbd, 0x3b,
    0x29, 0x9a, 0xdb, 0x83, 0x07, 0x2b, 0xf5, 0x9f, 0x7a, 0xef, 0xb8, 0xcb, 0xf7, 0xf2, 0x04, 0x86,
    0xb9, 0xf9, 0x17, 0x31, 0x3b, 0xce, 0x64, 0x72, 0xd6, 0xf7, 0x2c, 0xef, 0xf7, 0xed, 0x0d, 0x4a,
    0xdb, 0xd3, 0x9c, 0xfb, 0xb1, 0xe5, 0xaf, 0x7d, 0x01, 0x0a, 0x98, 0x7c, 0x44, 0x62, 0xba, 0x46,
    0x05, 0x6c, 0xde, 0x95, 0x95, 0x34, 0x22, 0x49, 0x25, 0x61, 0xa6, 0xb1, 0x04, 0x68, 0x23, 0x7f,
    0x0a, 0xc1, 0x06, 0xf6, 0x2f, 0xca, 0xa3, 0x16, 0x1b, 0xea, 0x57, 0x2e, 0x3e, 0x83, 0x0a, 0xe2,
    0x25, 0x3b, 0x9e, 0x4b, 0x02, 0x1e, 0x7a, 0xf1, 0x0a, 0x01, 0x9a, 0xd5, 0xfc, 0x78, 0x83, 0x3f,
    0x58, 0x4e, 0xdd, 0xfb, 0x30, 0x42, 0xb4, 0x3a, 0x49, 0xf9, 0x67, 0x04, 0xa9, 0xe2, 0xa5, 0xab,
    0xf4, 0x58, 0x6a, 0x50, 0x25, 0xb0, 0x13, 0xe9, 0xf9, 0xc4, 0x41, 0xed, 0x61, 0xaf, 0x07, 0x76,
    0x00, 0xdf, 0x8b, 0x2c, 0xf4, 0xfa, 0x32, 0x31, 0xcd, 0x6a, 0x6f, 0xae, 0x2a, 0x68, 0x92, 0xe8,
    0x88, 0x1b, 0xfd, 0x54, 0x98, 0x6d, 0xc4, 0x41, 0x3f, 0x4a, 0xbe, 0x89, 0xa5, 0x80, 0xd7, 0x4e,
    0xd4, 0xd5, 0x97, 0x49, 0xee, 0x88, 0x3e, 0xac, 0xc7, 0xa7, 0x28, 0x4c, 0x1e, 0xdf, 0x30, 0x31,
    0x2a, 0x94, 0xf9, 0x7b, 0xfe, 0xe6, 0x14, 0x92, 0x86, 0x17, 0xae, 0xec, 0x11, 0x97, 0x6d, 0xd2,
    0xbf, 0xd9, 0xe8, 0xc7, 0x17, 0xdd, 0xc6, 0x1d, 0xe2, 0xf5, 0xb7, 0xc0, 0xfb, 0xad, 0x85, 0xa5,
    0x28, 0x9d, 0x22, 0x89, 0x5e, 0xfe, 0x22, 0xd8, 0x28, 0x7a, 0x29, 0xad, 0x4f, 0x80, 0xcc, 0x0a,
    0x49, 0x48, 0x4f, 0xd6, 0xd3, 0xac, 0x33, 0x3b, 0xa3, 0x5a, 0xda, 0x8c, 0xec, 0x1a, 0x1d, 0x1c,
    0xa7, 0x98, 0xba, 0xe4, 0x3e, 0x93, 0x81, 0x64, 0xfa, 0x66, 0xcd, 0x4a, 0x51, 0xf9, 0xea, 0x1a,
    0x1e, 0x47, 0x2e, 0x32, 0x8c, 0xd9, 0x42, 0x66, 0x81, 0x2b, 0x15, 0xa1, 0x32, 0x87, 0x27, 0x2e,
    0x7a, 0x13, 0x89, 0x88, 0x38, 0x97, 0x0b, 0x77, 0x22, 0xe6, 0x84, 0x73, 0x2f, 0x5d, 0x04, 0x59,
    0xdf, 0x0b, 0xdf, 0x72, 0xc3, 0x89, 0xca, 0x96, 0x9f, 0x11, 0xd6, 0xcc, 0x16, 0x1d, 0xae, 0x96,
    0xc3, 0x1e, 0x30, 0xb0, 0xdc, 0x0d, 0x18, 0xd9, 0x22, 0x9b, 0xe6, 0xb5, 0x36, 0xc5, 0x21, 0x7f,
    0xa0, 0xaa, 0x36, 0xd2, 0xc8, 0x43, 0x40, 0x3c, 0xa3, 0x9e, 0x3b, 0xc2, 0x66, 0x45, 0x54, 0x4e,
    0x7f, 0x71, 0x42, 0x93, 0x3e, 0x81, 0x94, 0x7b, 0x85, 0xff, 0xd0, 0x3d, 0x3d, 0xc9, 0x38, 0x27,
    0xd1, 0xeb, 0xe8, 0x25, 0x5a, 0x8c, 0x1b, 0x55, 0x52, 0x32, 0x8d, 0xc2, 0x43, 0xd2, 0x3e, 0x96,
    0x42, 0xa5, 0x78, 0x25, 0x82, 0x44, 0x2b, 0x5b, 0x98, 0x3f, 0xbd, 0xe9, 0x58, 0x7f, 0x9c, 0x66,
    0x54, 0xa2, 0xe7, 0xda, 0xd3, 0xd3, 0xdd, 0x4e, 0x6f, 0xc6, 0x3d, 0xea, 0x59, 0x83, 0x85, 0x02,
    0x98, 0x75, 0x7a, 0xba, 0xb8, 0xc5, 0x41, 0xd5, 0xe4, 0xb9, 0x3b, 0x02, 0x8d, 0x96, 0x44, 0xa3,
    0xed, 0xc8, 0x3e, 0xee, 0x6a, 0x1b, 0x9a, 0xa7, 0x33, 0xbe, 0x75, 0x0f, 0xa2, 0xfe, 0xaa, 0xc3,
    0x76, 0xbb, 0x07, 0xff, 0x6d, 0xab, 0x0e, 0x6d, 0xeb, 0xb8, 0xcd, 0x14, 0xdd, 0x20, 0xbc, 0xa4,
    0x17, 0x6d, 0x3a, 0x9f, 0x37, 0x7c, 0x9b, 0xca, 0x30, 0xf2, 0x66, 0xdf, 0xa9, 0xdd, 0x0f, 0xbc,
    0xd2, 0x4f, 0x49, 0x77, 0x3d, 0x49, 0x1b, 0x8a, 0xca, 0xde, 0xb6, 0xe7, 0xb7, 0x35, 0x71, 0x8d,
    0xa2, 0xe6, 0xfa, 0xf1, 0x46, 0xb9, 0x28, 0x1f, 0xba, 0x03, 0x3c, 0x39, 0xe1, 0x37, 0x40, 0x1d,
    0x97, 0x08, 0x55, 0xf8, 0xd9, 0x8d, 0x0b, 0x84, 0x53, 0xab, 0x74, 0x82, 0xa0, 0x06, 0x3d, 0x74,
    0x11, 0x41, 0x0a, 0x7d, 0xd1, 0xb9, 0x18, 0x6c, 0xdc, 0xd8, 0x55, 0x98, 0x37, 0xc3, 0x1f, 0x79,
    0x82, 0x41, 0x0a, 0xea, 0x0b, 0x0c, 0xe1, 0x1c, 0xbd, 0xfd, 0x9e, 0xa8, 0xf4, 0xa9, 0x3b, 0x4e,
    0x0a, 0xac, 0xc7, 0x17, 0x45, 0x80, 0x4a, 0x32, 0xe2, 0x01, 0x9b, 0x6a, 0x04, 0x52, 0x85, 0x2a,
    0x25, 0xd0, 0x42, 0x4e, 0xed, 0xd0, 0xdb, 0xed, 0xdb, 0x89, 0xf0, 0x4c, 0x88, 0x0d, 0xd7, 0xeb,
    0x4d, 0x90, 0xd7, 0x89, 0xab, 0x1b, 0xa0, 0x3d, 0xa7, 0x3e, 0x3f, 0xc0, 0x9f, 0x02, 0x93, 0x08,
    0x2d, 0x9d, 0xf3, 0x96, 0xd7, 0xff, 0x27, 0xe3, 0xf8, 0x23, 0x8c, 0xf6, 0xce, 0xaf, 0x1a, 0x49,
    0xf4, 0xe4, 0x0b, 0xd6, 0x34, 0xb3, 0x42, 0x07, 0x90, 0x40, 0xbd, 0x6b, 0x3e, 0xed, 0x5d, 0xb3,
    0xf8, 0x95, 0x2c, 0x66, 0x88, 0x91, 0xcd, 0x25, 0x72, 0xd5, 0xff, 0xef, 0xeb, 0x84, 0xae, 0xbc,
    0xa2, 0xed, 0x89, 0x6d, 0x05, 0x6b, 0x64, 0x2d, 0x2d, 0xa1, 0x76, 0xb4, 0xa1, 0xeb, 0xa6, 0x4f,
    0x56, 0x53, 0xec, 0xd5, 0x5a, 0x00, 0x6d, 0x55, 0x1a, 0xba, 0x9b, 0x65, 0x97, 0x7b, 0xdd, 0x0d,
    0x60, 0xfc, 0x94, 0x79, 0x1a, 0xf2, 0x2e, 0x19, 0xd7, 0x52, 0xf2, 0xfa, 0x5b, 0x6f, 0xad, 0xd0,
    0x4d, 0x55, 0x0d, 0x0b, 0x47, 0x23, 0x60, 0x4a, 0xd9, 0x0a, 0xdd, 0xa2, 0x60, 0xdf, 0xda, 0xd4,
    0x47, 0x9f, 0x66, 0x60, 0xd8, 0x0e, 0xbf, 0x14, 0x54, 0x36, 0xd1, 0x9d, 0xf1, 0xa1, 0x03, 0x4d,
    0x75, 0x98, 0x20, 0x6c, 0xf2, 0x45, 0x5c, 0xef, 0x4b, 0x37, 0x4e, 0xae, 0xf2, 0x6b, 0x46, 0xea,
    0x7c, 0xc0, 0x4a, 0x0a, 0x9e, 0xb3, 0x1a, 0xc0, 0xbe, 0x25, 0xc8, 0x93, 0x65, 0x76, 0xb6, 0xb9,
    0xaa, 0x78, 0x7c, 0x9b, 0xb4, 0x5e, 0x92, 0x59, 0x78, 0x12, 0xc8, 0x22, 0xa3, 0x7c, 0x8f, 0x79,
    0x80, 0x08, 0xca, 0xab, 0x9e, 0x7a, 0x91, 0x67, 0x64, 0x03, 0xa7, 0x29, 0xde, 0xe2, 0x9e, 0x2a,
    0x80, 0xd2, 0xd3, 0xa0, 0xca, 0x78, 0x14, 0x3b, 0xad, 0x9e, 0x48, 0xd8, 0x25, 0x0f, 0x53, 0xd2,
    0xe8, 0xc3, 0x2e, 0xb7, 0xef, 0xa0, 0x0a, 0x42, 0xe8, 0x49, 0x1e, 0x57, 0xa4, 0x87, 0xde, 0x5c,
    0xb4, 0xea, 0xfa, 0x8c, 0x6b, 0x2e, 0xaf, 0x1e, 0x14, 0x5b, 0xa0, 0x74, 0x1c, 0xd1, 0x53, 0xd3,
    0x68, 0x7b, 0x4e, 0xb6, 0x8b, 0xab, 0x1e, 0x0e, 0xb4, 0x5d, 0x54, 0x6f, 0x03, 0xc3, 0xf6, 0xda,
    0xb8, 0x11, 0x2d, 0xf0, 0x82, 0xeb, 0x22, 0xb1, 0xfd, 0x62, 0xb9, 0x0b, 0xcb, 0xcc, 0xae, 0xcf,
    0x82, 0x81, 0x3e, 0x06, 0x17, 0x5f, 0xf2, 0x6a, 0xcb, 0xa6, 0x3d, 0x47, 0x91, 0x2d, 0xe8, 0x14,
    0x02, 0xc6, 0xc0, 0x6b, 0x0a, 0x62, 0x92, 0x0e, 0x0d, 0x11, 0xd8, 0x3c, 0xf0, 0x65, 0x9a, 0x6f,
    0xb1, 0x0d, 0x1f, 0x93, 0x1a, 0xc0, 0x31, 0x77, 0xa2, 0x7c, 0xc1, 0x22, 0x28, 0x1a, 0x28, 0x5c,
    0x46, 0x03, 0x26, 0xa3, 0xab, 0xf6, 0x7a, 0x29, 0x9e, 0xc6, 0x41, 0x29, 0x1e, 0xe0, 0x07, 0x58,
    0x9d, 0x25, 0x53, 0xaa, 0xcd, 0x1b, 0x85, 0xfd, 0xed, 0x41, 0xa6, 0xb0, 0xbc, 0x06, 0xe3, 0xc1,
    0xf6, 0x33, 0xad, 0xcf, 0x19, 0xc6, 0x43, 0xbd, 0x7c, 0x20, 0x63, 0x4e, 0x03, 0x40, 0x3e, 0xea,
    0x9f, 0x17, 0x2c, 0xbc, 0x40, 0x8a, 0xc9, 0x06, 0xb8, 0x2b, 0xbb, 0x4d, 0xf6, 0x1c, 0x4d, 0xca,
    0x19, 0xd0, 0xf6, 0x87, 0x3c, 0xb2, 0xb0, 0x1f, 0xc0, 0xa2, 0x9c, 0x1f, 0x22, 0xc8, 0x02, 0x35,
    0xc8, 0x79, 0x03, 0x56, 0xa7, 0x61, 0xca, 0x59, 0x3a, 0x0d, 0xb2, 0xfb, 0x78, 0x84, 0x18, 0xdd,
    0xfa, 0x78, 0x1e, 0x1b, 0xe6, 0x94, 0x05, 0x39, 0xe7, 0xde, 0xec, 0x98, 0xed, 0x4a, 0x80, 0x8b,
    0x2a, 0xe1, 0x02, 0xcf, 0x89, 0x93, 0xec, 0xc8, 0xe8, 0xda, 0xb2, 0x64, 0x7d, 0x64, 0x0f, 0xac,
    0xad, 0xc7, 0x1d, 0x36, 0xf2, 0xfb, 0x87, 0xa5, 0x5a, 0x9e, 0x50, 0x38, 0x54, 0x52, 0xfe, 0xaa,
    0x9c, 0x78, 0x32, 0x1f, 0x6a, 0x9d, 0x4a, 0x2d, 0x4b, 0x77, 0xc0, 0x35, 0x94, 0x17, 0xd7, 0x5f,
    0x26, 0xc5, 0xe1, 0x55, 0xb3, 0x5a, 0xae, 0x49, 0xd6, 0x4e, 0x0a, 0x85, 0x2b, 0x67, 0x78, 0xd2,
    0x05, 0x14, 0x9d, 0x89, 0xc8, 0x9f, 0x41, 0x0e, 0xca, 0x12, 0xed, 0x58, 0xe8, 0xa9, 0xca, 0x0c,
    0x2e, 0xe0, 0xa4, 0xe0, 0xe8, 0x97, 0x45, 0x30, 0x87, 0xc4, 0xe7, 0x80, 0x64, 0xaf, 0xff, 0x9b,
    0x00, 0x6b, 0x2a, 0xb6, 0xfc, 0x5d, 0x73, 0x54, 0x6a, 0xa8, 0x9c, 0x04, 0xd9, 0xe0, 0xb6, 0x58,
    0x31, 0xda, 0x20, 0x4f, 0x84, 0xd0, 0xab, 0x1c, 0xb9, 0xc7, 0xd3, 0x35, 0x7a, 0x09, 0x9d, 0x06,
    0x50, 0x01, 0xc8, 0x02, 0x53, 0x51, 0x71, 0xe7, 0x10, 0xf5, 0x6f, 0xc2, 0xa3, 0x77, 0x9d, 0xb8,
    0xce, 0x87, 0x6a, 0xdb, 0x20, 0xc2, 0x71, 0xfb, 0x78, 0x8d, 0x92, 0x4c, 0xb2, 0x44, 0x5e, 0x0d,
    0x71, 0xc0, 0xff, 0xa6, 0x5d, 0x40, 0xbc, 0xaf, 0xa9, 0x5e, 0x8c, 0x5d, 0x69, 0x05, 0x78, 0xce,
    0x45, 0x7b, 0xca, 0xc6, 0x1e, 0x3c, 0x21, 0xf1, 0x2f, 0x4b, 0xab, 0x91, 0x01, 0x97, 0x77, 0xbb,
    0xdb, 0xe1, 0x61, 0x5d, 0x2b, 0xc8, 0x25, 0xbb, 0xdb, 0x1f, 0xc7, 0x4b, 0x75, 0xde, 0x32, 0x41,
    0x46, 0x6a, 0x85, 0x31, 0x41, 0xf0, 0x01, 0xaf, 0x5c, 0x4c, 0x26, 0xc6, 0x8a, 0x9f, 0xf7, 0xc0,
    0xf8, 0x4e, 0xce, 0x5b, 0xf8, 0xf7, 0xdd, 0x9a, 0xf4, 0x0b, 0xce, 0x2d, 0x6e, 0x14, 0x7d, 0x72,
    0xe6, 0x4e, 0x0c, 0x90, 0x83, 0x6f, 0x69, 0x0b, 0x77, 0x8a, 0xfe, 0xa9, 0x50, 0x77, 0x9e, 0x36,
    0x70, 0x6e, 0x2f, 0xb0, 0x86, 0x41, 0x28, 0x37, 0xff, 0x36, 0xf5, 0x34, 0xe4, 0x8f, 0xc4, 0x20,
    0x76, 0xf3, 0x9e, 0xc2, 0xfb, 0x00, 0x66, 0xec, 0xcf, 0xe3, 0xf1, 0xe8, 0xc6, 0xd1, 0x5c, 0x91,
    0x1a, 0xed, 0x99, 0xf4, 0x1b, 0x8e, 0xdb, 0x9c, 0x73, 0x98, 0xa3, 0xdc, 0x9e, 0x5d, 0xbc, 0xbd,
    0x67, 0xcb, 0x43, 0x6a, 0x85, 0xb8, 0xbd, 0x49, 0x53, 0x19, 0xad, 0x93, 0x6f, 0x7d, 0x8b, 0x14,
    0xe5, 0x0a, 0x56, 0x08, 0x1f, 0xa4, 0x93, 0x1e, 0xd2, 0xe2, 0xfa, 0x8d, 0x3a, 0x18, 0x0d, 0xff,
    0x75, 0x26, 0x1c, 0xdf, 0x7d, 0x2d, 0x63, 0x03, 0x37, 0xc5, 0xb3, 0x0b, 0xa8, 0xa7, 0x78, 0xeb,
    0x3a, 0xa1, 0x2b, 0xee, 0xa8, 0xae, 0x5a, 0x47, 0x55, 0xdc, 0x82, 0x10, 0xf8, 0x14, 0x3d, 0x18,
    0x34, 0xfe, 0xad, 0x80, 0x2d, 0x00, 0x93, 0xda, 0xe6, 0xb4, 0x84, 0xc1, 0x8e, 0x10, 0x26, 0xbf,
    0xfe, 0xd6, 0xb1, 0x87, 0x9f, 0xd6, 0xe4, 0x9f, 0xf2, 0xfa, 0x07, 0xfc, 0x20, 0x90, 0x0c, 0x13,
    0x66, 0x28, 0x0e, 0x56, 0x73, 0x9d, 0x12, 0x18, 0xb4, 0xc5, 0xbf, 0x4a, 0x08, 0x0d, 0x3f, 0x58,
    0x76, 0xe1, 0xc3, 0xff, 0x40, 0x2d, 0x32, 0x8f, 0x0e, 0x2f, 0x70, 0xf2, 0x40, 0x00,
];

/// Type to represent elliptic curves.
///
/// Elliptic curves are considered in projective coordinates
/// in Montgomery form b*y^2*z = x^3 + a*x^2*z + x*z^2.
///
/// The previous form is useful because it allows to compute
/// elliptic point additions and doubling without the y-coordinate.
pub struct EllipticCurve<T: UInt> {
    x: T,
    z: T,
}

impl<T: UInt> EllipticCurve<T> {
    /// Compute a prime factor candidate from the elliptic curve.
    pub fn compute_maybe_factor_from_curve(modu: T) -> T {
        let mut curve = EllipticCurve {
            x: T::one(),
            z: T::one(),
        };

        match curve.init_rnd_point(modu) {
            (true, a) => {
                // Return factor candidate gcd(k*P.z, modu)
                T::gcd_mod(curve.montgomery_ladder(a, modu), modu)
            }
            (false, a) => a,
        }
    }

    /// Get random point on the elliptic curve using Suyama's parametrization.
    fn init_rnd_point(&mut self, modu: T) -> (bool, T) {
        let sigma = rand::thread_rng().gen_range(6..u8::MAX).into();

        let u = T::sub_mod(T::mult_mod(sigma, sigma, modu), 5.into(), modu);
        let u3 = T::exp_mod_unsafe(u, 3.into(), modu);
        let v = T::mult_mod(sigma, 4.into(), modu);

        self.x = u3;
        self.z = T::exp_mod_unsafe(v, 3.into(), modu);

        let vu_diff = T::exp_mod_unsafe(T::sub_mod(v, u, modu), 3.into(), modu);
        let uv_add = T::add_mod_unsafe(T::mult_mod_unsafe(u, 3.into(), modu), v, modu);

        let a_numer = T::mult_mod_unsafe(vu_diff, uv_add, modu);
        let a_denumer = T::mult_mod_unsafe(T::mult_mod(u3, 4.into(), modu), v, modu);
        let a_denumer_inv = T::multip_inv(a_denumer, modu);

        if a_denumer_inv == T::zero() {
            // No multiplicative inverse for `a_denumer`
            return (false, T::gcd_mod(a_denumer, modu));
        }

        let mut a = T::sub_mod_unsafe(
            T::mult_mod_unsafe(a_numer, a_denumer_inv, modu),
            2.into(),
            modu,
        );

        a = T::mult_mod_unsafe(
            T::add_mod_unsafe(a, 2.into(), modu),
            T::multip_inv(4.into(), modu),
            modu,
        );

        (true, a)
    }

    /// Double a point P (`self`) on the elliptic curve in-place.
    fn elliptic_double(&mut self, a: T, modu: T) {
        let psum = T::add_mod(self.x, self.z, modu);
        let psub = T::sub_mod(self.x, self.z, modu);

        let psum_square = T::mult_mod_unsafe(psum, psum, modu);
        let psub_square = T::mult_mod_unsafe(psub, psub, modu);

        let pmix = T::sub_mod_unsafe(psum_square, psub_square, modu);

        self.x = T::mult_mod_unsafe(psum_square, psub_square, modu);

        self.z = T::mult_mod_unsafe(
            pmix,
            T::add_mod_unsafe(psub_square, T::mult_mod_unsafe(a, pmix, modu), modu),
            modu,
        );
    }

    /// Add two points P (`self`) and Q (`point`) on the elliptic curve,
    /// updating the point P in-place.
    ///
    /// Difference between the points equals the initial point `point0`.
    fn elliptic_add(&mut self, point: &Self, point0: &Self, modu: T) {
        let lp_sum = T::add_mod(self.x, self.z, modu);
        let lp_sub = T::sub_mod(self.x, self.z, modu);

        let rp_sum = T::add_mod(point.x, point.z, modu);
        let rp_sub = T::sub_mod(point.x, point.z, modu);

        let lterm = T::mult_mod_unsafe(lp_sub, rp_sum, modu);
        let rterm = T::mult_mod_unsafe(lp_sum, rp_sub, modu);

        let term_add = T::add_mod_unsafe(lterm, rterm, modu);
        let term_sub = T::sub_mod_unsafe(lterm, rterm, modu);

        self.x = T::mult_mod_unsafe(point0.z, T::mult_mod_unsafe(term_add, term_add, modu), modu);

        self.z = T::mult_mod_unsafe(point0.x, T::mult_mod_unsafe(term_sub, term_sub, modu), modu);
    }

    /// Multiply a point P on elliptic curve by a scalar k.
    ///
    /// This multiplication k*P is computed with Montgomery ladder algorithm
    /// where parameter k equals lcm(1,...,10_000) of which byte representation
    /// has been saved into static array `BYTES_10K`.
    fn montgomery_ladder(&self, a: T, modu: T) -> T {
        let mut q = EllipticCurve {
            x: self.x,
            z: self.z,
        };
        let mut p = EllipticCurve {
            x: self.x,
            z: self.z,
        };

        p.elliptic_double(a, modu);

        let it_bits_rev = (0..u8::BITS).rev();
        let it = BYTES_10K.iter().cartesian_product(it_bits_rev);

        // First and last bits of `BYTES_10K_LEN` must be left out
        let take_count = BYTES_10K_LEN * u8::BITS as usize - 1;

        for (byte_val, cbit) in it.take(take_count).skip(1) {
            if (*byte_val >> cbit) & 1 == 1 {
                q.elliptic_add(&p, self, modu);
                p.elliptic_double(a, modu);
            } else {
                p.elliptic_add(&q, self, modu);
                q.elliptic_double(a, modu);
            }
        }

        q.z
    }
}

#[cfg(test)]
mod tests;