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crypto/bn256: refine comments according to #19577, #21595, and #21836 #21847

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Nov 17, 2020
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crypto/bn256: refine comments according to #19577, #21595, and #21836 #21847

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9 changes: 7 additions & 2 deletions crypto/bn256/cloudflare/bn256.go
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Expand Up @@ -9,8 +9,13 @@
//
// This package specifically implements the Optimal Ate pairing over a 256-bit
// Barreto-Naehrig curve as described in
// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible
// with the implementation described in that paper.
// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is not
// compatible with the implementation described in that paper, as different
// parameters are chosen.
//
// (This package previously claimed to operate at a 128-bit security level.
// However, recent improvements in attacks mean that is no longer true. See
// https://moderncrypto.org/mail-archive/curves/2016/000740.html.)
package bn256

import (
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5 changes: 3 additions & 2 deletions crypto/bn256/google/bn256.go
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Expand Up @@ -12,8 +12,9 @@
//
// This package specifically implements the Optimal Ate pairing over a 256-bit
// Barreto-Naehrig curve as described in
// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible
// with the implementation described in that paper.
// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is not
// compatible with the implementation described in that paper, as different
// parameters are chosen.
//
// (This package previously claimed to operate at a 128-bit security level.
// However, recent improvements in attacks mean that is no longer true. See
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2 changes: 2 additions & 0 deletions crypto/bn256/google/constants.go
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Expand Up @@ -20,7 +20,9 @@ var u = bigFromBase10("4965661367192848881")
var P = bigFromBase10("21888242871839275222246405745257275088696311157297823662689037894645226208583")

// Order is the number of elements in both G₁ and G₂: 36u⁴+36u³+18u²+6u+1.
// Needs to be highly 2-adic for efficient SNARK key and proof generation.
// Order - 1 = 2^28 * 3^2 * 13 * 29 * 983 * 11003 * 237073 * 405928799 * 1670836401704629 * 13818364434197438864469338081.
// Refer to https://eprint.iacr.org/2013/879.pdf and https://eprint.iacr.org/2013/507.pdf for more information on these parameters.
var Order = bigFromBase10("21888242871839275222246405745257275088548364400416034343698204186575808495617")

// xiToPMinus1Over6 is ξ^((p-1)/6) where ξ = i+9.
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