Field element to curve point mapping used by EIP 2537

For a BLS12-381 implemented by EIP-2537 a short Weierstrass curve equation y^2 = x^3 + A * x + B has a property that a product AB = 0, so to implement a mapping function two step algorithms is performed:

  • Field element is mapped to a some other curve with AB != 0
  • Isogeny is applied to one to one map a point of this other curve to a point on BLS12-381
  • Cofactor is cleared for a point now on BLS12-381

Below we describe generic algorithms for mapping and isogeny application, and later on give concrete parameters for the algorithms

Helper function to clear a cofactor

Later on we use a helper function to clear a cofactor of the curve point. It’s implemented as

    clear_cofactor(P) := h_eff * P

where values of h_eff are given below in parameters sections

Simplified SWU for AB != 0

The function map_to_curve_simple_swu(u) implements a simplification of the Shallue-van de Woestijne-Ulas mapping described by Brier et al., which they call the “simplified SWU” map. Wahby and Boneh generalize and optimize this mapping.

Preconditions: A Weierstrass curve y^2 = g(x) x^3 + A * x + B where A != 0 and B != 0.

Constants:

  • A and B, the parameters of the Weierstrass curve.

  • Z, an element of F meeting the below criteria. The criteria are:

    1. Z is non-square in F,
    2. Z != -1 in F,
    3. the polynomial g(x) - Z is irreducible over F, and
    4. g(B / (Z * A)) is square in F.

Sign of y: Inputs u and -u give the same x-coordinate. Thus, we set sgn0(y) == sgn0(u).

Exceptions: The exceptional cases are values of u such that Z^2 * u^4 + Z * u^2 == 0. This includes u == 0, and may include other values depending on Z. Implementations must detect this case and set x1 = B / (Z * A), which guarantees that g(x1) is square by the condition on Z given above.

Operations:

1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
2.  x1 = (-B / A) * (1 + tv1)
3.  If tv1 == 0, set x1 = B / (Z * A)
4. gx1 = x1^3 + A * x1 + B
5.  x2 = Z * u^2 * x1
6. gx2 = x2^3 + A * x2 + B
7.  If is_square(gx1), set x = x1 and y = sqrt(gx1)
8.  Else set x = x2 and y = sqrt(gx2)
9.  If sgn0(u) != sgn0(y), set y = -y
10. return (x, y)

Simplified SWU for AB == 0

Wahby and Boneh show how to adapt the simplified SWU mapping to Weierstrass curves having A == 0 or B == 0, which the mapping of simple SWU does not support.

This method requires finding another elliptic curve E’ given by the equation

    y'^2 = g'(x') = x'^3 + A' * x' + B'

that is isogenous to E and has A’ != 0 and B’ != 0. This isogeny defines a map iso_map(x’, y’) given by a pair of rational functions. iso_map takes as input a point on E’ and produces as output a point on E.

Once E’ and iso_map are identified, this mapping works as follows: on input u, first apply the simplified SWU mapping to get a point on E’, then apply the isogeny map to that point to get a point on E.

Note that iso_map is a group homomorphism, meaning that point addition commutes with iso_map. Thus, when using this mapping in the hash_to_curve construction of , one can effect a small optimization by first mapping u0 and u1 to E’, adding the resulting points on E’, and then applying iso_map to the sum. This gives the same result while requiring only one evaluation of iso_map.

Preconditions: An elliptic curve E’ with A’ != 0 and B’ != 0 that is isogenous to the target curve E with isogeny map iso_map from E’ to E.

So the full mapping algorithm looks as:

  • map_to_curve_simple_swu is the simple SWU mapping to E’
  • iso_map is the isogeny map from E’ to E

Sign of y: for this map, the sign is determined by map_to_curve_simple_swu. No further sign adjustments are necessary.

Exceptions: map_to_curve_simple_swu handles its exceptional cases. Exceptional cases of iso_map are inputs that cause the denominator of either rational function to evaluate to zero; such cases MUST return the identity point on E.

Full algorithm restated

1. (x', y') = map_to_curve_simple_swu(u)    # (x', y') is on E'
2.   (x, y) = iso_map(x', y')               # (x, y) is on E
3. (x, y) = clear_cofactor((x, y))          # clears cofactor for point (x, y) on E
4. return (x, y)

Parameters for EIP-2537

Fp-to-G1 mapping

  • Z: 11
  • E’: y’^2 = x’^3 + A’ * x’ + B’, where
    • A’ = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aefd881ac98936f8da0e0f97f5cf428082d584c1d
    • B’ = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14fcef35ef55a23215a316ceaa5d1cc48e98e172be0
  • h_eff: 0xd201000000010001

The 11-isogeny map from (x’, y’) on E’ to (x, y) on E is given by the following rational functions:

  • x = x_num / x_den, where
    • x_num = k_(1,11) * x’^11 + k_(1,10) * x’^10 + k_(1,9) * x’^9 + … + k_(1,0)
    • x_den = x’^10 + k_(2,9) * x’^9 + k_(2,8) * x’^8 + … + k_(2,0)
  • y = y’ * y_num / y_den, where
    • y_num = k_(3,15) * x’^15 + k_(3,14) * x’^14 + k_(3,13) * x’^13 + … + k_(3,0)
    • y_den = x’^15 + k_(4,14) * x’^14 + k_(4,13) * x’^13 + … + k_(4,0)

The constants used to compute x_num are as follows:

  • k_(1,0) = 0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7
  • k_(1,1) = 0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e33c70d1e86b4838f2a6f318c356e834eef1b3cb83bb
  • k_(1,2) = 0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e501ec68e25c958c3e3d2a09729fe0179f9dac9edcb0
  • k_(1,3) = 0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b33289f1b330835336e25ce3107193c5b388641d9b6861
  • k_(1,4) = 0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb65982fac18985a286f301e77c451154ce9ac8895d9
  • k_(1,5) = 0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab9097e68f90a0870d2dcae73d19cd13c1c66f652983
  • k_(1,6) = 0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e218f9c86b2a8da25128c1052ecaddd7f225a139ed84
  • k_(1,7) = 0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c62ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e
  • k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c928c5d1de4fa295f296b74e956d71986a8497e317
  • k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314baf4bb1b7fa3190b2edc0327797f241067be390c9e
  • k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af36003b14866f69b771f8c285decca67df3f1605fb7b
  • k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229

The constants used to compute x_den are as follows:

  • k_(2,0) = 0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c9588617fc8ac62b558d681be343df8993cf9fa40d21b1c
  • k_(2,1) = 0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2588c48bf5713daa8846cb026e9e5c8276ec82b3bff
  • k_(2,2) = 0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e00b11aceacd6a3d0967c94fedcfcc239ba5cb83e19
  • k_(2,3) = 0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8abc28d6fd04976d5243eecf5c4130de8938dc62cd8
  • k_(2,4) = 0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44e833b306da9bd29ba81f35781d539d395b3532a21e
  • k_(2,5) = 0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bcef24b8982f7400d24bc4228f11c02df9a29f6304a5
  • k_(2,6) = 0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073062aede9cea73b3538f0de06cec2574496ee84a3a
  • k_(2,7) = 0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e
  • k_(2,8) = 0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da67f39883503826692abba43704776ec3a79a1d641
  • k_(2,9) = 0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533978f31c1593174e4b4b7865002d6384d168ecdd0a

The constants used to compute y_num are as follows:

  • k_(3,0) = 0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6c956543d3cd0c7aee9b3ba3c2be9845719707bb33
  • k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f906343eb67ad34d6c56711962fa8bfe097e75a2e41c696
  • k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b91400da7d26d521628b00523b8dfe240c72de1f6
  • k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3ec03251cf9de405aba9ec61deca6355c77b0e5f4cb
  • k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd221351adc2ee7f8dc099040a841b6daecf2e8fedb
  • k_(3,5) = 0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c3d3ad5544e203f6326c95a807299b23ab13633a5f0
  • k_(3,6) = 0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413c4d634f3747a87ac2460f415ec961f8855fe9d6f2
  • k_(3,7) = 0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da6c26c842642f64550fedfe935a15e4ca31870fb29
  • k_(3,8) = 0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6a1f20cabe69d65201c78607a360370e577bdba587
  • k_(3,9) = 0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30
  • k_(3,10) = 0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183e416389e61031bf3a5cce3fbafce813711ad011c132
  • k_(3,11) = 0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bfe7f911f643249d9cdf41b44d606ce07c8a4d0074d8e
  • k_(3,12) = 0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710e807b4633f06c851c1919211f20d4c04f00b971ef8
  • k_(3,13) = 0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e6632d3c40659cc6cf90ad1c232a6442d9d3f5db980133
  • k_(3,14) = 0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f5d396a7ce46ba1049b6579afb7866b1e715475224b
  • k_(3,15) = 0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add4fa95af01b2b665027efec01c7704b456be69c8b604

The constants used to compute y_den are as follows:

  • k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec3232b5be72e7a07f3688ef60c206d01479253b03663c1
  • k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a10356f453e01f78a4260763529e3532f6102c2e49a03d
  • k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5dd279cd2eca6757cd636f96f891e2538b53dbf67f2
  • k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da489e726af41727364f2c28297ada8d26d98445f5416
  • k_(4,4) = 0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0fc9dec916a20b15dc0fd2ededda39142311a5001d
  • k_(4,5) = 0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac
  • k_(4,6) = 0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365bc400a0051d5fa9c01a58b1fb93d1a1399126a775c
  • k_(4,7) = 0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34fd206357132b920f5b00801dee460ee415a15812ed9
  • k_(4,8) = 0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc5750c4bf39b4852cfe2f7bb9248836b233d9d55535d4a
  • k_(4,9) = 0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb5308592e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55
  • k_(4,10) = 0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b36e636a5c871a5c29f4f83060400f8b49cba8f6aa8
  • k_(4,11) = 0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a88cea7913516f968986f7ebbea9684b529e2561092
  • k_(4,12) = 0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5ceef9a00d9b8693000763e3b90ac11e99b138573345cc
  • k_(4,13) = 0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748df27942480e420517bd8714cc80d1fadc1326ed06f7
  • k_(4,14) = 0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efcd6356caa205ca2f570f13497804415473a1d634b8f

Fp2-to-G2 mapping

Symbol I means a non-residue used to make an extension field Fp2

  • Z: -(2 + I)
  • E’: y’^2 = x’^3 + A’ * x’ + B’, where
    • A’ = 240 * I
    • B’ = 1012 * (1 + I)
  • h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff031508ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689f6a359894c0adebbf6b4e8020005aaa95551

The 3-isogeny map from (x’, y’) on E’ to (x, y) on E is given by the following rational functions:

  • x = x_num / x_den, where
    • x_num = k_(1,3) * x’^3 + k_(1,2) * x’^2 + k_(1,1) * x’ + k_(1,0)
    • x_den = x’^2 + k_(2,1) * x’ + k_(2,0)
  • y = y’ * y_num / y_den, where
    • y_num = k_(3,3) * x’^3 + k_(3,2) * x’^2 + k_(3,1) * x’ + k_(3,0)
    • y_den = x’^3 + k_(4,2) * x’^2 + k_(4,1) * x’ + k_(4,0)

The constants used to compute x_num are as follows:

  • k_(1,0) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 + 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 * I
  • k_(1,1) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71a * I
  • k_(1,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71e + 0x8ab05f8bdd54cde190937e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ffffffffe38d * I
  • k_(1,3) = 0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d6108f142b85757098e38d0f671c7188e2aaaaaaaa5ed1

The constants used to compute x_den are as follows:

  • k_(2,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa63 * I
  • k_(2,1) = 0xc + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f * I

The constants used to compute y_num are as follows:

  • k_(3,0) = 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 + 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 * I
  • k_(3,1) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaaaaaaa97be * I
  • k_(3,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71c + 0x8ab05f8bdd54cde190937e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ffffffffe38f * I
  • k_(3,3) = 0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977c69aa274524e79097a56dc4bd9e1b371c71c718b10

The constants used to compute y_den are as follows:

  • k_(4,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb * I
  • k_(4,1) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3 * I
  • k_(4,2) = 0x12 + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99 * I