github.com/jimmyx0x/go-ethereum@v1.10.28/crypto/bn256/cloudflare/constants.go (about)

     1  // Copyright 2012 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package bn256
     6  
     7  import (
     8  	"math/big"
     9  )
    10  
    11  func bigFromBase10(s string) *big.Int {
    12  	n, _ := new(big.Int).SetString(s, 10)
    13  	return n
    14  }
    15  
    16  // u is the BN parameter.
    17  var u = bigFromBase10("4965661367192848881")
    18  
    19  // Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1.
    20  // Needs to be highly 2-adic for efficient SNARK key and proof generation.
    21  // Order - 1 = 2^28 * 3^2 * 13 * 29 * 983 * 11003 * 237073 * 405928799 * 1670836401704629 * 13818364434197438864469338081.
    22  // Refer to https://eprint.iacr.org/2013/879.pdf and https://eprint.iacr.org/2013/507.pdf for more information on these parameters.
    23  var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")
    24  
    25  // P is a prime over which we form a basic field: 36u⁴+36u³+24u²+6u+1.
    26  var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583")
    27  
    28  // p2 is p, represented as little-endian 64-bit words.
    29  var p2 = [4]uint64{0x3c208c16d87cfd47, 0x97816a916871ca8d, 0xb85045b68181585d, 0x30644e72e131a029}
    30  
    31  // np is the negative inverse of p, mod 2^256.
    32  var np = [4]uint64{0x87d20782e4866389, 0x9ede7d651eca6ac9, 0xd8afcbd01833da80, 0xf57a22b791888c6b}
    33  
    34  // rN1 is R^-1 where R = 2^256 mod p.
    35  var rN1 = &gfP{0xed84884a014afa37, 0xeb2022850278edf8, 0xcf63e9cfb74492d9, 0x2e67157159e5c639}
    36  
    37  // r2 is R^2 where R = 2^256 mod p.
    38  var r2 = &gfP{0xf32cfc5b538afa89, 0xb5e71911d44501fb, 0x47ab1eff0a417ff6, 0x06d89f71cab8351f}
    39  
    40  // r3 is R^3 where R = 2^256 mod p.
    41  var r3 = &gfP{0xb1cd6dafda1530df, 0x62f210e6a7283db6, 0xef7f0b0c0ada0afb, 0x20fd6e902d592544}
    42  
    43  // xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9.
    44  var xiToPMinus1Over6 = &gfP2{gfP{0xa222ae234c492d72, 0xd00f02a4565de15b, 0xdc2ff3a253dfc926, 0x10a75716b3899551}, gfP{0xaf9ba69633144907, 0xca6b1d7387afb78a, 0x11bded5ef08a2087, 0x02f34d751a1f3a7c}}
    45  
    46  // xiToPMinus1Over3 is ξ^((p-1)/3) where ξ = i+9.
    47  var xiToPMinus1Over3 = &gfP2{gfP{0x6e849f1ea0aa4757, 0xaa1c7b6d89f89141, 0xb6e713cdfae0ca3a, 0x26694fbb4e82ebc3}, gfP{0xb5773b104563ab30, 0x347f91c8a9aa6454, 0x7a007127242e0991, 0x1956bcd8118214ec}}
    48  
    49  // xiToPMinus1Over2 is ξ^((p-1)/2) where ξ = i+9.
    50  var xiToPMinus1Over2 = &gfP2{gfP{0xa1d77ce45ffe77c7, 0x07affd117826d1db, 0x6d16bd27bb7edc6b, 0x2c87200285defecc}, gfP{0xe4bbdd0c2936b629, 0xbb30f162e133bacb, 0x31a9d1b6f9645366, 0x253570bea500f8dd}}
    51  
    52  // xiToPSquaredMinus1Over3 is ξ^((p²-1)/3) where ξ = i+9.
    53  var xiToPSquaredMinus1Over3 = &gfP{0x3350c88e13e80b9c, 0x7dce557cdb5e56b9, 0x6001b4b8b615564a, 0x2682e617020217e0}
    54  
    55  // xiTo2PSquaredMinus2Over3 is ξ^((2p²-2)/3) where ξ = i+9 (a cubic root of unity, mod p).
    56  var xiTo2PSquaredMinus2Over3 = &gfP{0x71930c11d782e155, 0xa6bb947cffbe3323, 0xaa303344d4741444, 0x2c3b3f0d26594943}
    57  
    58  // xiToPSquaredMinus1Over6 is ξ^((1p²-1)/6) where ξ = i+9 (a cubic root of -1, mod p).
    59  var xiToPSquaredMinus1Over6 = &gfP{0xca8d800500fa1bf2, 0xf0c5d61468b39769, 0x0e201271ad0d4418, 0x04290f65bad856e6}
    60  
    61  // xiTo2PMinus2Over3 is ξ^((2p-2)/3) where ξ = i+9.
    62  var xiTo2PMinus2Over3 = &gfP2{gfP{0x5dddfd154bd8c949, 0x62cb29a5a4445b60, 0x37bc870a0c7dd2b9, 0x24830a9d3171f0fd}, gfP{0x7361d77f843abe92, 0xa5bb2bd3273411fb, 0x9c941f314b3e2399, 0x15df9cddbb9fd3ec}}