github.com/ethw3/go-ethereuma@v0.0.0-20221013053120-c14602a4c23c/crypto/bn256/google/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 that determines the prime.
    17  var u = bigFromBase10("4965661367192848881")
    18  
    19  // P is a prime over which we form a basic field: 36u⁴+36u³+24u²+6u+1.
    20  var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583")
    21  
    22  // Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1.
    23  // Needs to be highly 2-adic for efficient SNARK key and proof generation.
    24  // Order - 1 = 2^28 * 3^2 * 13 * 29 * 983 * 11003 * 237073 * 405928799 * 1670836401704629 * 13818364434197438864469338081.
    25  // Refer to https://eprint.iacr.org/2013/879.pdf and https://eprint.iacr.org/2013/507.pdf for more information on these parameters.
    26  var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")
    27  
    28  // xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9.
    29  var xiToPMinus1Over6 = &gfP2{bigFromBase10("16469823323077808223889137241176536799009286646108169935659301613961712198316"), bigFromBase10("8376118865763821496583973867626364092589906065868298776909617916018768340080")}
    30  
    31  // xiToPMinus1Over3 is ξ^((p-1)/3) where ξ = i+9.
    32  var xiToPMinus1Over3 = &gfP2{bigFromBase10("10307601595873709700152284273816112264069230130616436755625194854815875713954"), bigFromBase10("21575463638280843010398324269430826099269044274347216827212613867836435027261")}
    33  
    34  // xiToPMinus1Over2 is ξ^((p-1)/2) where ξ = i+9.
    35  var xiToPMinus1Over2 = &gfP2{bigFromBase10("3505843767911556378687030309984248845540243509899259641013678093033130930403"), bigFromBase10("2821565182194536844548159561693502659359617185244120367078079554186484126554")}
    36  
    37  // xiToPSquaredMinus1Over3 is ξ^((p²-1)/3) where ξ = i+9.
    38  var xiToPSquaredMinus1Over3 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556616")
    39  
    40  // xiTo2PSquaredMinus2Over3 is ξ^((2p²-2)/3) where ξ = i+9 (a cubic root of unity, mod p).
    41  var xiTo2PSquaredMinus2Over3 = bigFromBase10("2203960485148121921418603742825762020974279258880205651966")
    42  
    43  // xiToPSquaredMinus1Over6 is ξ^((1p²-1)/6) where ξ = i+9 (a cubic root of -1, mod p).
    44  var xiToPSquaredMinus1Over6 = bigFromBase10("21888242871839275220042445260109153167277707414472061641714758635765020556617")
    45  
    46  // xiTo2PMinus2Over3 is ξ^((2p-2)/3) where ξ = i+9.
    47  var xiTo2PMinus2Over3 = &gfP2{bigFromBase10("19937756971775647987995932169929341994314640652964949448313374472400716661030"), bigFromBase10("2581911344467009335267311115468803099551665605076196740867805258568234346338")}