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ECGFp.pas
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ECGFp.pas
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{License, info, etc
------------------
This implementation is made by me, Walied Othman, to contact me
mail to Walied.Othman@belgacom.net or Triade@ulyssis.org,
always mention wether it 's about the FGInt for Delphi or for
FreePascal, or wether it 's about the 6xs, preferably in the subject line.
If you 're going to use these implementations, at least mention my
name or something and notify me so I may even put a link on my page.
This implementation is freeware and according to the coderpunks'
manifesto it should remain so, so don 't use these implementations
in commercial software. Encryption, as a tool to ensure privacy
should be free and accessible for anyone. If you plan to use these
implementations in a commercial application, contact me before
doing so, that way you can license the software to use it in commercial
Software. If any algorithm is patented in your country, you should
acquire a license before using this software. Modified versions of this
software must contain an acknowledgement of the original author (=me).
This implementation is available at
http://triade.studentenweb.org
copyright 2000, Walied Othman
This header may not be removed.
}
Unit ECGFp;
{$H+}
Interface
Uses FGInt, math;
Type
TECPoint = Record
XCoordinate, YCoordinate : TFGInt;
Infinity : Boolean;
End;
TOrderList = Record
order1 : TFGInt;
order2 : TFGInt;
order3 : TFGInt;
order4 : TFGInt;
order5 : TFGInt;
order6 : TFGInt;
End;
Const
primes : Array[1..1228] Of integer =
(3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127,
131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251,
257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389,
397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541,
547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677,
683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839,
853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009,
1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123,
1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279,
1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429,
1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553,
1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693,
1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847,
1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997,
1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287,
2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417,
2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593,
2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719,
2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861,
2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037,
3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209,
3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359,
3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527,
3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659,
3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821,
3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967,
3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129,
4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273,
4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457,
4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637,
4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789,
4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957,
4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101,
5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281,
5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443,
5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623,
5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779,
5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903,
5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101,
6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269,
6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397,
6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599,
6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779,
6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947,
6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103,
7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283,
7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487,
7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607,
7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789,
7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951,
7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161,
8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311,
8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521,
8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681,
8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831,
8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007,
9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181,
9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343,
9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491,
9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679,
9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839,
9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973);
Procedure ECPointToECPointString(ECPoint : TECPoint; Prime : TFGInt; Compression : boolean; Out ECPointString : RawByteString);
Procedure ECPointStringToECPoint(ECPointString : RawByteString; Prime, a, b : TFGInt; Out ECPoint : TECPoint);
Procedure ECPointCopy(P : TECPoint; Out Copied : TECPoint);
Procedure ECPointDestroy(Var P : TECPoint);
Procedure ECDoublePoint(P : TECPoint; Prime, a : TFGInt; Var Doubled : TECPoint);
Procedure ECAddPoints(P, Q : TECPoint; Prime, a : TFGInt; Out Sum : TECPoint);
Procedure ECPointKMultiple(P : TECPoint; Prime, a, k : TFGInt; Out Multiple : TECPoint);
Procedure ECPointInverse(P : TECPoint; Prime : TFGInt; Var Inverse : TECPoint);
Procedure ECInbedStringOnEC(InString : RawByteString; Prime, a, b : TFGInt; Out P : TECPoint; Out DidItWork : Boolean);
Procedure ECExtractInbeddedString(P : TECPoint; Var InBeddedString : RawByteString);
Procedure ECPrimeSearch(Var GInt : TFGInt; nrRMtests : byte);
Procedure ECFindNextPointOnEC(Var Prime, a, b : TFGInt; Var P : TECPoint);
Procedure ConstructCurveWithCMD(D : byte; Var a0, b0 : TFGInt);
Procedure IsCMD(Const D, p : TFGInt; Out W, V : TFGInt; Out IsCMD : boolean);
Procedure FindNextCMCandidate(Const p : TFGInt; Var D : byte; Out Found : boolean);
Procedure FindOrders(Const p : TFGInt; Out D : byte; Out Orders : TOrderList; Out Found : boolean);
Procedure IsNearlyPrime(Var n : TFGInt; leastsize : longint; Var k, r : TFGInt; Var INP : boolean);
Procedure ConstructCurveAndPointWithGoodORder(Var p, k, r : TFGInt; D : byte; Var a, b : TFGInt; Var G : TECPoint);
Procedure ConstructCurveAndPoint(Var p : TFGINt; leastsize : longint; Var a, b, k, r : TFGInt; Var G : TECPoint; Var DidItWork : boolean);
Procedure FindPrimeGoodCurveAndPoint(Var p, a, b, k, r : TFGInt; leastsize : longint; Var G : TECPoint);
Procedure IsECSuperSingular(p, a, b : TFGInt; Var SS : boolean);
Implementation
// Convert a point (ECPoint) on an EC y^2 = x^3 + a*x + b over GF(Prime)
// to a string (ECPointString)
Procedure ECPointToECPointString(ECPoint : TECPoint; Prime : TFGInt; Compression : boolean; Out ECPointString : RawByteString);
Var
l : longint;
temp : RawByteString;
Begin
If ECPoint.infinity Then
Begin
ECPointString := chr(0);
exit;
End;
FGIntToBase256String(Prime, temp);
l := length(temp);
FGIntToBase256String(ECPoint.XCoordinate, ECPointString);
While length(ECPointString) < l Do ECPointString := chr(0) + ECPointString;
If Compression Then
Begin
ECPointString := chr((ECPoint.YCoordinate.Number[1] Mod 2) + 2) + ECPointString;
End
Else
Begin
FGIntToBase256String(ECPoint.YCoordinate, temp);
While length(temp) < l Do temp := chr(0) + temp;
ECPointString := chr(4) + ECPointString + temp;
End
End;
// Does the opposite from the procedure above, ECPointString MUST be
// created by the above procedure
Procedure ECPointStringToECPoint(ECPointString : RawByteString; Prime, a, b : TFGInt; Out ECPoint : TECPoint);
Var
temp : RawByteString;
temp1, temp2, temp3 : TFGInt;
l : longint;
Begin
If ECPointString = chr(0) Then
Begin
ECPoint.Infinity := true;
Base2StringToFGInt('0', ECPoint.XCoordinate);
Base2StringToFGInt('0', ECPoint.YCoordinate);
End
Else
Begin
ECPoint.Infinity := false;
FGIntToBase256String(Prime, temp);
l := length(temp);
If ECPointString[1] = chr(4) Then
Begin
Delete(ECPointString, 1, 1);
temp := copy(ECPointString, 1, l);
Base256StringToFGInt(temp, ECPoint.XCoordinate);
temp := copy(ECPointString, l + 1, l);
Base256StringToFGInt(temp, ECPoint.YCoordinate);
End
Else
Begin
temp := copy(ECPointString, 2, l);
Base256StringToFGInt(temp, ECPoint.XCoordinate);
FGIntSquareMod(ECPoint.XCoordinate, Prime, temp1);
FGIntMulMod(temp1, ECPoint.XCoordinate, Prime, temp2);
FGIntDestroy(temp1);
FGIntMulMod(ECPoint.XCoordinate, a, Prime, temp1);
FGIntAddMod(temp1, temp2, Prime, temp3);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
FGIntAddMod(temp3, b, Prime, temp1);
FGIntDestroy(temp3);
FGIntSquareRootModP(temp1, Prime, temp2);
FGIntDestroy(temp1);
If ((temp2.Number[1] Mod 2) + 2) = ord(ECPointString[1]) Then
Begin
FGIntCopy(temp2, ECPoint.YCoordinate);
FGIntDestroy(temp2);
End
Else
Begin
FGIntCopy(Prime, ECPoint.YCoordinate);
FGIntSubBis(ECPoint.YCoordinate, temp2);
FGIntDestroy(temp2);
End
End;
End
End;
// Make a copy of an ECPoint
// Copied := P
Procedure ECPointCopy(P : TECPoint; Out Copied : TECPoint);
Begin
Copied.Infinity := P.Infinity;
FGIntCopy(P.XCoordinate, Copied.XCoordinate);
FGIntCopy(P.YCoordinate, Copied.YCoordinate);
End;
// Free the memory used by an ECPoint
Procedure ECPointDestroy(Var P : TECPoint);
Begin
FGIntDestroy(P.XCoordinate);
FGIntDestroy(P.YCoordinate);
End;
// Double a Point P on an Elliptic Curve y^2 = x^3 + a*x + b, over the
// finite prime field with "Prime" elements
// 2 * P = Doubled
Procedure ECDoublePoint(P : TECPoint; Prime, a : TFGInt; Var Doubled : TECPoint);
Var
temp1, temp2, temp3, zero : TFGInt;
Begin
If P.Infinity Then
Begin
ECPointCopy(P, Doubled);
exit;
End;
Base2StringToFGInt('0', zero);
If FGIntCompareAbs(P.YCoordinate, zero) = Eq Then
Begin
Doubled.Infinity := true;
FGIntCopy(zero, Doubled.XCoordinate);
FGIntCopy(zero, Doubled.YCoordinate);
FGIntDestroy(zero);
exit;
End;
FGIntDestroy(zero);
Doubled.Infinity := false;
FGIntSquareMod(P.XCoordinate, Prime, temp1);
FGIntAddMod(temp1, temp1, Prime, temp2);
FGIntAddMod(temp2, temp1, Prime, temp3);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
FGIntAddMod(temp3, a, Prime, temp2);
FGIntDestroy(temp3);
FGIntAddMod(P.YCoordinate, P.YCoordinate, Prime, temp1);
FGIntModInv(temp1, Prime, temp3);
FGIntDestroy(temp1);
FGIntMulMod(temp3, temp2, Prime, temp1);
FGIntDestroy(temp2);
FGIntDestroy(temp3);
FGIntAddmod(P.XCoordinate, P.XCoordinate, Prime, temp2);
FGIntSquareMod(temp1, Prime, temp3);
FGIntChangeSign(temp2);
FGIntAddMod(temp3, temp2, Prime, Doubled.XCoordinate);
FGIntDestroy(temp2);
FGIntDestroy(temp3);
FGIntCopy(Doubled.XCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(P.XCoordinate, temp2, Prime, temp3);
FGIntDestroy(temp2);
FGIntMulMod(temp3, temp1, Prime, temp2);
FGIntDestroy(temp1);
FGIntDestroy(temp3);
FGIntCopy(P.YCoordinate, temp1);
FGIntChangeSign(temp1);
FGIntAddMod(temp2, temp1, Prime, Doubled.YCoordinate);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
End;
// Add 2 Points, P and Q, on an Elliptic Curve y^2 = x^3 + a*x + b, over the
// finite prime field with "Prime" elements
// Q + P = Sum
Procedure ECAddPoints(P, Q : TECPoint; Prime, a : TFGInt; Out Sum : TECPoint);
Var
temp1, temp2, temp3, temp4 : TFGInt;
Begin
If P.Infinity Then
Begin
ECPointCopy(Q, Sum);
exit;
End;
If Q.Infinity Then
Begin
ECPointCopy(P, Sum);
exit;
End;
If FGIntCompareAbs(P.XCoordinate, Q.XCoordinate) = Eq Then
Begin
If (FGIntCompareAbs(P.YCoordinate, Q.YCoordinate) = Eq) Then
Begin
ECDoublePoint(P, Prime, a, Sum);
Exit;
End;
Base2StringToFGInt('0', Sum.XCoordinate);
Base2StringToFGInt('0', Sum.YCoordinate);
Sum.Infinity := true;
End
Else
Begin
FGIntCopy(P.YCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(Q.YCoordinate, temp2, Prime, temp3);
FGIntDestroy(temp2);
FGIntCopy(P.XCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(Q.XCoordinate, temp2, Prime, temp1);
FGIntDestroy(temp2);
FGIntModInv(temp1, Prime, temp2);
FGIntDestroy(temp1);
FGIntMulMod(temp3, temp2, Prime, temp1);
FGIntSquareMod(temp1, Prime, temp4);
FGIntDestroy(temp2);
FGIntDestroy(temp3);
FGIntCopy(P.XCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(temp4, temp2, Prime, temp3);
FGIntDestroy(temp2);
FGIntCopy(Q.XCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(temp3, temp2, Prime, Sum.XCoordinate);
FGIntDestroy(temp2);
FGIntDestroy(temp3);
FGIntDestroy(temp4);
FGIntCopy(Sum.XCoordinate, temp2);
FGIntChangeSign(temp2);
FGIntAddMod(P.XCoordinate, temp2, Prime, temp3);
FGIntDestroy(temp2);
FGIntMulMod(temp1, temp3, Prime, temp2);
FGIntDestroy(temp1);
FGIntDestroy(temp3);
FGIntCopy(P.YCoordinate, temp3);
FGIntChangeSign(temp3);
FGIntAddMod(temp2, temp3, Prime, Sum.YCoordinate);
FGIntDestroy(temp2);
FGIntDestroy(temp3);
Sum.Infinity := False;
End
End;
// Compute the k-multiple of a point P on an Elliptic Curve
// y^2 = x^3 + a*x + b, over the finite prime field with "Prime" elements
// k * P = Multiple
Procedure ECPointKMultiple(P : TECPoint; Prime, a, k : TFGInt; Out Multiple : TECPoint);
Var
temp : String;
i : longint;
temp1 : TECPoint;
Begin
FGIntToBase2String(k, temp);
Multiple.Infinity := True;
Base2StringToFGInt('0', Multiple.XCoordinate);
Base2StringToFGInt('0', Multiple.YCoordinate);
For i := 1 To Length(temp) Do
Begin
If temp[i] = '1' Then
Begin
ECAddPoints(Multiple, P, Prime, a, temp1);
ECPointDestroy(Multiple);
ECPointCopy(temp1, Multiple);
ECPointDestroy(temp1);
End;
If i < Length(temp) Then
Begin
ECDoublePoint(Multiple, Prime, a, temp1);
ECPointDestroy(Multiple);
ECPointCopy(temp1, Multiple);
ECPointDestroy(temp1);
End;
End;
End;
// compute the inverse of an ECPoint P on an Elliptic curve over
// the finite field GF(Prime), P + Q = O, where O is the point
// at infnity
Procedure ECPointInverse(P : TECPoint; Prime : TFGInt; Var Inverse : TECPoint);
Begin
If P.Infinity Then
Begin
ECPointCopy(P, Inverse);
exit;
End;
Inverse.Infinity := false;
FGIntCopy(P.XCoordinate, Inverse.XCoordinate);
FGIntCopy(Prime, Inverse.YCoordinate);
FGIntSubBis(Inverse.YCoordinate, P.YCoordinate);
End;
// Inbed a string, InString, on an Elliptic Curve, y^2 = x^3 + a*x + b,
// (length(InString) + 2)*8 must be less than the size of Prime in bits
Procedure ECInbedStringOnEC(InString : RawByteString; Prime, a, b : TFGInt; Out P : TECPoint; Out DidItWork : Boolean);
Var
temp : RawByteString;
pad : byte;
i, L : integer;
one, limit, counter, temp1, temp2, temp3, YSquare : TFGInt;
Begin
DidItWork := false;
FGIntToBase256String(Prime, temp);
If length(temp) < (length(InString) + 3) Then exit;
pad := min(255, length(temp) - length(InString) - 2);
Base2StringToFGInt('1', one);
Base2StringToFGInt('1', limit);
temp := chr(pad) + InString;
For i := 1 To pad Do temp := temp + chr(0);
Base256StringToFGInt(temp, P.XCoordinate);
temp := '1';
For i := 1 To pad Do temp := temp + '00000000';
Base2StringToFGInt(temp, Limit);
Base2StringToFGInt('0', counter);
While (Not DidItWork) And (FGIntCompareAbs(counter, Limit) = St) Do
Begin
FGIntAddBis(counter, one);
FGIntSquareMod(P.XCoordinate, Prime, temp1);
FGIntMulMod(P.XCoordinate, temp1, Prime, temp2);
FGIntDestroy(temp1);
FGIntMulMod(a, P.XCoordinate, Prime, temp1);
FGIntAddMod(temp1, temp2, Prime, temp3);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
FGIntAddMod(temp3, b, Prime, YSquare);
FGIntDestroy(temp3);
FGIntLegendreSymbol(YSquare, Prime, L);
If L = 1 Then DidItWork := true
Else
Begin
FGIntAddBis(P.XCoordinate, one);
FGIntDestroy(YSquare);
End;
End;
If DidItWork Then
Begin
FGIntSquareRootModP(YSquare, Prime, P.YCoordinate);
FGIntDestroy(YSquare);
P.Infinity := false;
End;
FGIntDestroy(counter);
FGIntDestroy(Limit);
FGIntDestroy(one);
End;
// Extract an inbedded string which is inbedded with the procedure above
Procedure ECExtractInbeddedString(P : TECPoint; Var InBeddedString : RawByteString);
Var
b : byte;
Begin
FGIntToBase256String(P.XCoordinate, InBeddedString);
b := ord(InBeddedString[1]);
delete(InBeddedString, 1, 1);
delete(InBeddedString, Length(InBeddedString) - b + 1, b);
End;
// Find a prime using a standard method but with nrRMtests Rabin-Miller tests
Procedure ECPrimeSearch(Var GInt : TFGInt; nrRMtests : byte);
Var
two : TFGInt;
ok : Boolean;
Begin
If (GInt.Number[1] Mod 2) = 0 Then GInt.Number[1] := GInt.Number[1] + 1;
Base10StringToFGInt('2', two);
ok := false;
While Not ok Do
Begin
FGIntAddBis(GInt, two);
FGIntPrimeTest(GInt, nrRMtests, ok);
End;
FGIntDestroy(two);
End;
// Find a (next) point on the EC by incrementing the X coordinate
Procedure ECFindNextPointOnEC(Var Prime, a, b : TFGInt; Var P : TECPoint);
Var
L : integer;
one, temp1, temp2, temp3, YSquare : TFGInt;
DidItWork : boolean;
Begin
DidItWork := false;
FGIntDestroy(P.YCoordinate);
Base2StringToFGInt('1', one);
While (Not DidItWork) Do
Begin
FGIntAddBis(P.XCoordinate, one);
FGIntSquareMod(P.XCoordinate, Prime, temp1);
FGIntMulMod(P.XCoordinate, temp1, Prime, temp2);
FGIntDestroy(temp1);
FGIntMulMod(a, P.XCoordinate, Prime, temp1);
FGIntAddMod(temp1, temp2, Prime, temp3);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
FGIntAddMod(temp3, b, Prime, YSquare);
FGIntDestroy(temp3);
FGIntLegendreSymbol(YSquare, Prime, L);
If L = 1 Then DidItWork := true Else FGIntDestroy(YSquare);
End;
FGIntSquareRootModP(YSquare, Prime, P.YCoordinate);
FGIntDestroy(YSquare);
P.Infinity := false;
FGIntDestroy(one);
End;
// Construct an EC with the Complex Multiplication Discriminant D (P1363A)
Procedure ConstructCurveWithCMD(D : byte; Var a0, b0 : TFGInt);
Begin
Case D Of
1 : Begin
Base2StringToFGInt('1', a0);
Base2StringToFGInt('0', b0);
End;
2 : Begin
Base10StringToFGInt('-30', a0);
Base10StringToFGInt('56', b0);
End;
3 : Begin
Base2StringToFGInt('0', a0);
Base2StringToFGInt('1', b0);
End;
7 : Begin
Base10StringToFGInt('-35', a0);
Base10StringToFGInt('98', b0);
End;
11 : Begin
Base10StringToFGInt('-264', a0);
Base10StringToFGInt('1694', b0);
End;
19 : Begin
Base10StringToFGInt('-152', a0);
Base10StringToFGInt('722', b0);
End;
43 : Begin
Base10StringToFGInt('-3440', a0);
Base10StringToFGInt('77658', b0);
End;
67 : Begin
Base10StringToFGInt('-29480', a0);
Base10StringToFGInt('1948226', b0);
End;
163 : Begin
Base10StringToFGInt('-8697680', a0);
Base10StringToFGInt('9873093538', b0);
End;
End;
End;
// Test wether D is a CM Discriminant or not (P1363A)
// If so compute W and if necessary V so that 4*p = W^2 + D*V^2
Procedure IsCMD(Const D, p : TFGInt; Out W, V : TFGInt; Out IsCMD : boolean);
Var
temp, tempd, temp1, temp2, A, B, C, U1, U2, S11, S12, S21, S22 : TFGInt;
l : integer;
Begin
FGIntSub(p, D, temp);
FGIntLegendreSymbol(temp, p, l);
IsCMD := false;
If l = -1 Then
Begin
FGIntDestroy(temp);
exit;
End;
FGIntSquareRootModP(temp, p, B);
FGIntDestroy(temp);
FGIntCopy(p, A);
FGIntSquare(B, temp);
FGIntAdd(temp, D, C);
FGIntDestroy(temp);
FGIntDiv(C, p, temp);
FGIntDestroy(C);
FGIntCopy(temp, C);
FGIntDestroy(temp);
FGIntCopy(A, S11);
FGIntCopy(B, S12);
FGIntCopy(B, S21);
FGIntCopy(C, S22);
Base2StringToFGInt('1', U1);
Base2StringToFGInt('0', U2);
FGIntAdd(B, B, temp);
While (FGIntCompareAbs(temp, A) = Lt) Or (FGIntCompareAbs(A, C) = Lt) Do
Begin
FGIntAdd(temp, C, temp1);
FGIntAdd(C, C, temp2);
FGIntDiv(temp1, temp2, tempd);
FGIntDestroy(temp);
FGIntDestroy(temp1);
FGIntDestroy(temp2);
FGIntMul(tempd, U1, temp1);
FGIntAdd(temp1, U2, temp2);
FGIntDestroy(temp1);
FGIntDestroy(U2);
FGIntChangeSign(U1);
FGIntCopy(U1, U2);
FGIntDestroy(U1);
FGIntCopy(temp2, U1);
FGIntDestroy(temp2);
FGIntDestroy(S11);
FGIntCopy(C, S11);
FGIntMul(tempd, C, temp1);
FGIntSub(temp1, B, temp2);
FGIntDestroy(temp1);
FGIntDestroy(S12);
FGIntDestroy(S21);
FGIntCopy(temp2, S12);
FGIntCopy(temp2, S21);
FGIntDestroy(temp2);
FGIntMul(tempd, S12, temp2);
FGIntMul(B, tempd, temp1);
FGIntSub(temp2, temp1, temp);
FGIntDestroy(temp2);
FGIntDestroy(temp1);
FGIntDestroy(S22);
FGIntAdd(A, temp, S22);
FGIntDestroy(temp);
FGIntDestroy(A);
FGIntDestroy(B);
FGIntDestroy(C);
FGIntCopy(S11, A);
FGIntCopy(S21, B);
FGIntCopy(S22, C);
FGIntAdd(B, B, temp);
FGIntDestroy(tempd);
End;
FGIntDestroy(temp);
FGIntDestroy(S11);
FGIntDestroy(S22);
FGIntDestroy(S21);
FGIntDestroy(S12);
If D.Number[0] = 1 Then
If D.Number[1] = 11 Then
If A.Sign = positive Then
If A.Number[0] = 1 Then
If A.Number[1] = 3 Then
Begin
FGIntCopy(U2, temp);
FGIntDestroy(U2);
FGIntCopy(U1, U2);
FGIntDestroy(U1);
FGIntCopy(temp, U1);
FGIntDestroy(temp);
FGIntChangeSign(U2);
FGIntChangeSign(B);
FGIntCopy(A, temp);
FGIntDestroy(A);
FGIntCopy(C, A);
FGIntDestroy(C);
FGIntCopy(temp, C);
FGIntDestroy(temp);
End;
Base2StringToFGInt('1', temp1);
FGIntCopy(temp1, temp2);
temp2.Number[1] := 3;
If (FGIntCompareAbs(D, temp1) = Eq) Or (FGIntCompareAbs(D, temp2) = Eq) Then
Begin
IsCMD := true;
FGIntAdd(U1, U1, W);
FGIntAdd(U2, U2, V);
End
Else If (FGIntCompareAbs(A, temp1) = Eq) Then
Begin
IsCMD := true;
FGIntAdd(U1, U1, W);
End;
FGIntAddBis(temp1, temp2);
FGIntDestroy(temp2);
If (FGIntCompareAbs(A, temp1) = Eq) Then
Begin
IsCMD := true;
FGIntMulByIntBis(U1, 4);
FGIntMul(B, U2, temp2);
FGIntAdd(temp2, U1, W);
FGIntDestroy(temp2);
End;
FGIntDestroy(temp1);
FGIntDestroy(U1);
FGIntDestroy(U2);
FGIntDestroy(A);
FGIntDestroy(B);
FGIntDestroy(C);
End;
// Speaks for itself (P1363A)
Procedure FindNextCMCandidate(Const p : TFGInt; Var D : byte; Out Found : boolean);
Const
candidates1 : Array[0..9] Of byte = (9, 1, 2, 3, 7, 11, 19, 43, 67, 163);
candidates3 : Array[0..8] Of byte = (8, 2, 3, 7, 11, 19, 43, 67, 163);
candidates5 : Array[0..8] Of byte = (8, 1, 3, 7, 11, 19, 43, 67, 163);
candidates7 : Array[0..7] Of byte = (7, 3, 7, 11, 19, 43, 67, 163);
Var
i : integer;
Begin
Found := false;
Case (p.Number[1] Mod 8) Of
1 : For i := 1 To candidates1[0] Do If candidates1[i] > D Then
Begin
D := candidates1[i];
Found := true;
break;
End;
3 : For i := 1 To candidates3[0] Do If candidates3[i] > D Then
Begin
D := candidates3[i];
Found := true;
break;
End;
5 : For i := 1 To candidates5[0] Do If candidates5[i] > D Then
Begin
D := candidates5[i];
Found := true;
break;
End;
7 : For i := 1 To candidates7[0] Do If candidates7[i] > D Then
Begin
D := candidates7[i];
Found := true;
break;
End;
End;
End;
// Given a prime p, a CMD D, find possible orders for a curve with CMD D,
// Found = false if no orders are found
Procedure FindOrders(Const p : TFGInt; Out D : byte; Out Orders : TOrderList; Out Found : boolean);
Var
DGInt, temp, W, V : TFGInt;
l : integer;
Begin
Found := false;
D := 0;
While Not Found Do
Begin
FindNextCMCandidate(p, D, Found);
If Not Found Then exit;
Base256StringToFGInt(chr(D), DGInt);
FGIntChangeSign(DGInt);
FGIntLegendreSymbol(DGInt, p, l);
FGIntChangeSign(DGInt);
If l = -1 Then Found := false Else
Begin
If D > 2 Then FGIntLegendreSymbol(p, DGInt, l) Else l := 1;
If l = -1 Then Found := false Else
Begin
IsCMD(DGInt, p, W, V, Found);
If Found Then
Begin
FGIntCopy(p, Orders.Order1);
Base2StringToFGInt('1', temp);
FGIntAddBis(Orders.Order1, temp);
FGIntCopy(Orders.Order1, Orders.Order2);
FGIntDestroy(temp);
If (D = 1) Then
Begin
FGIntCopy(Orders.Order1, Orders.Order3);
FGIntCopy(Orders.Order1, Orders.Order4);
FGIntAddBis(Orders.Order3, V);
FGIntSubBis(Orders.Order4, V);
FGIntDestroy(V);
End;
If (D = 3) Then
Begin
FGIntCopy(Orders.Order1, Orders.Order3);
FGIntCopy(Orders.Order1, Orders.Order4);
FGIntCopy(Orders.Order1, Orders.Order5);
FGIntCopy(Orders.Order1, Orders.Order6);
FGIntCopy(W, temp);
FGIntShiftRight(temp);
FGIntAddBis(Orders.Order3, temp);
FGIntSubBis(Orders.Order4, temp);
FGIntAddBis(Orders.Order5, temp);
FGIntSubBis(Orders.Order6, temp);
FGIntDestroy(temp);
FGIntCopy(V, temp);
FGIntShiftRight(temp);
FGIntMulByIntBis(temp, 3);
FGIntAddBis(Orders.Order3, temp);
FGIntSubBis(Orders.Order4, temp);
FGIntSubBis(Orders.Order5, temp);
FGIntAddBis(Orders.Order6, temp);
FGIntDestroy(temp);
FGIntDestroy(V);
End;
FGIntAddBis(Orders.Order1, W);
FGIntSubBis(Orders.Order2, W);
FGIntDestroy(W);
End;
End;
End;
FGIntDestroy(DGInt);
End;
End;
// check if n has a prime factor r of at least leastsize bits, n = k*r
// INP = false if no large primefactor is found
Procedure IsNearlyPrime(Var n : TFGInt; leastsize : longint; Var k, r : TFGInt; Var INP : boolean);
Var
i : integer;
m : longword;
S : String;
Begin
FGIntCopy(n, r);
While (r.Number[1] Mod 2) = 0 Do FGIntShiftRight(r);
For i := 1 To 1228 Do
Begin
FGIntModByInt(r, primes[i], m);
While m = 0 Do
Begin
FGIntDivByIntBis(r, primes[i], m);
FGIntModByInt(r, primes[i], m);
End;
End;
FGIntToBase2String(r, S);
If length(S) < leastsize Then
Begin
INP := false;
FGIntDestroy(r);
End
Else
Begin
FGIntRabinMiller(r, 4, INP);
If INP Then FGIntDiv(n, r, k) Else FGIntDestroy(r);
End;
End;
// Given a prime p, k, r and CMD D, find a curve of order k*r and a point G of order r
// a, b are provided by the procedure ConstructCurveWithCMD
Procedure ConstructCurveAndPointWithGoodORder(Var p, k, r : TFGInt; D : byte; Var a, b : TFGInt; Var G : TECPoint);
Var
one, tempa, tempb, x, temp, tempx : TFGInt;
DidItWork : boolean;
tempG : TECPoint;
Begin
Base2StringToFGInt('1', one);
FGIntCopy(one, x);
DidItWork := false;
While Not DidItWork Do
Begin
If D = 1 Then
Begin
Base2StringToFGInt('0', tempb);
FGIntMulMod(a, x, p, tempa);
End;
If D = 3 Then
Begin
Base2StringToFGInt('0', tempa);
FGIntMulMod(b, x, p, tempb);
End;
If (D <> 1) And (D <> 3) Then
Begin
FGIntSquareMod(x, p, temp);
FGIntMulMod(temp, a, p, tempa);
FGIntMulMod(temp, b, p, tempx);
FGIntMulMod(tempx, x, p, tempb);
FGIntDestroy(temp);
FGIntDestroy(tempx);
End;
FGIntCopy(one, tempG.XCoordinate);
FGIntCopy(one, tempG.YCoordinate);
tempG.Infinity := false;
ECFindNextPointOnEC(p, tempa, tempb, tempG);
ECPointKMultiple(tempG, p, tempa, k, G);
While G.Infinity Do
Begin
ECFindNextPointOnEC(p, tempa, tempb, tempG);
ECPointDestroy(G);
ECPointKMultiple(tempG, p, tempa, k, G);
End;
ECpointDestroy(tempG);
ECPointKMultiple(G, p, tempa, r, tempG);
If tempG.Infinity Then
Begin
DidItwork := true;
FGIntDestroy(a);
FGIntCopy(tempa, a);
FGIntDestroy(b);
FGIntCopy(tempb, b);
End
Else
Begin
DidItWork := false;
ECPointDestroy(G);
End;
FGIntDestroy(tempa);
FGIntDestroy(tempb);
ECPointDestroy(tempG);
FGIntAddBis(x, one);
End;
FGIntDestroy(one);
FGIntDestroy(x);
End;
// the name says it all (P1363A)
// provide a prime p and a leastsize longint for the least number of bits of the point order
Procedure ConstructCurveAndPoint(Var p : TFGINt; leastsize : longint; Var a, b, k, r : TFGInt; Var G : TECPoint; Var DidItWork : boolean);
Var
Orders : TOrderList;
D : byte;
Begin
FindOrders(p, D, Orders, DidItWork);
If Not DidItWork Then exit;
If DidItWork Then
Begin
IsNearlyPrime(Orders.Order1, leastsize, k, r, DidItWork);
If DidItWork Then
Begin
ConstructCurveWithCMD(D, a, b);
ConstructCurveAndPointWithGoodORder(p, k, r, D, a, b, G);
End;
End;
If (Not DidItWork) Then
Begin
IsNearlyPrime(Orders.Order2, leastsize, k, r, DidItWork);
If DidItWork Then
Begin
ConstructCurveWithCMD(D, a, b);
ConstructCurveAndPointWithGoodORder(p, k, r, D, a, b, G);
End;
End;
If (Not DidItWork) And ((D = 3) Or (D = 1)) Then
Begin
IsNearlyPrime(Orders.Order3, leastsize, k, r, DidItWork);
If DidItWork Then
Begin
ConstructCurveWithCMD(D, a, b);
ConstructCurveAndPointWithGoodORder(p, k, r, D, a, b, G);
End;
End;
If (Not DidItWork) And ((D = 3) Or (D = 1)) Then
Begin
IsNearlyPrime(Orders.Order4, leastsize, k, r, DidItWork);
If DidItWork Then
Begin
ConstructCurveWithCMD(D, a, b);
ConstructCurveAndPointWithGoodORder(p, k, r, D, a, b, G);
End;
End;
If (Not DidItWork) And (D = 3) Then
Begin
IsNearlyPrime(Orders.Order5, leastsize, k, r, DidItWork);
If DidItWork Then
Begin
ConstructCurveWithCMD(D, a, b);
ConstructCurveAndPointWithGoodORder(p, k, r, D, a, b, G);
End;
End;