/// <summary>
/// Calculates f*f. Can overlap h with f.
/// </summary>
/// <returns></returns>
public FieldElement Square()
{
int f0 = _elements[0];
int f1 = _elements[1];
int f2 = _elements[2];
int f3 = _elements[3];
int f4 = _elements[4];
int f5 = _elements[5];
int f6 = _elements[6];
int f7 = _elements[7];
int f8 = _elements[8];
int f9 = _elements[9];
int f0_2 = 2 * f0;
int f1_2 = 2 * f1;
int f2_2 = 2 * f2;
int f3_2 = 2 * f3;
int f4_2 = 2 * f4;
int f5_2 = 2 * f5;
int f6_2 = 2 * f6;
int f7_2 = 2 * f7;
int f5_38 = 38 * f5; // 1.31*2^30
int f6_19 = 19 * f6; // 1.31*2^30
int f7_38 = 38 * f7; // 1.31*2^30
int f8_19 = 19 * f8; // 1.31*2^30
int f9_38 = 38 * f9; // 1.31*2^30
long f0f0 = (long)(f0) * (long)(f0);
long f0f1_2 = (long)(f0_2) * (long)(f1);
long f0f2_2 = (long)(f0_2) * (long)(f2);
long f0f3_2 = (long)(f0_2) * (long)(f3);
long f0f4_2 = (long)(f0_2) * (long)(f4);
long f0f5_2 = (long)(f0_2) * (long)(f5);
long f0f6_2 = (long)(f0_2) * (long)(f6);
long f0f7_2 = (long)(f0_2) * (long)(f7);
long f0f8_2 = (long)(f0_2) * (long)(f8);
long f0f9_2 = (long)(f0_2) * (long)(f9);
long f1f1_2 = (long)(f1_2) * (long)(f1);
long f1f2_2 = (long)(f1_2) * (long)(f2);
long f1f3_4 = (long)(f1_2) * (long)(f3_2);
long f1f4_2 = (long)(f1_2) * (long)(f4);
long f1f5_4 = (long)(f1_2) * (long)(f5_2);
long f1f6_2 = (long)(f1_2) * (long)(f6);
long f1f7_4 = (long)(f1_2) * (long)(f7_2);
long f1f8_2 = (long)(f1_2) * (long)(f8);
long f1f9_76 = (long)(f1_2) * (long)(f9_38);
long f2f2 = (long)(f2) * (long)(f2);
long f2f3_2 = (long)(f2_2) * (long)(f3);
long f2f4_2 = (long)(f2_2) * (long)(f4);
long f2f5_2 = (long)(f2_2) * (long)(f5);
long f2f6_2 = (long)(f2_2) * (long)(f6);
long f2f7_2 = (long)(f2_2) * (long)(f7);
long f2f8_38 = (long)(f2_2) * (long)(f8_19);
long f2f9_38 = (long)(f2) * (long)(f9_38);
long f3f3_2 = (long)(f3_2) * (long)(f3);
long f3f4_2 = (long)(f3_2) * (long)(f4);
long f3f5_4 = (long)(f3_2) * (long)(f5_2);
long f3f6_2 = (long)(f3_2) * (long)(f6);
long f3f7_76 = (long)(f3_2) * (long)(f7_38);
long f3f8_38 = (long)(f3_2) * (long)(f8_19);
long f3f9_76 = (long)(f3_2) * (long)(f9_38);
long f4f4 = (long)(f4) * (long)(f4);
long f4f5_2 = (long)(f4_2) * (long)(f5);
long f4f6_38 = (long)(f4_2) * (long)(f6_19);
long f4f7_38 = (long)(f4) * (long)(f7_38);
long f4f8_38 = (long)(f4_2) * (long)(f8_19);
long f4f9_38 = (long)(f4) * (long)(f9_38);
long f5f5_38 = (long)(f5) * (long)(f5_38);
long f5f6_38 = (long)(f5_2) * (long)(f6_19);
long f5f7_76 = (long)(f5_2) * (long)(f7_38);
long f5f8_38 = (long)(f5_2) * (long)(f8_19);
long f5f9_76 = (long)(f5_2) * (long)(f9_38);
long f6f6_19 = (long)(f6) * (long)(f6_19);
long f6f7_38 = (long)(f6) * (long)(f7_38);
long f6f8_38 = (long)(f6_2) * (long)(f8_19);
long f6f9_38 = (long)(f6) * (long)(f9_38);
long f7f7_38 = (long)(f7) * (long)(f7_38);
long f7f8_38 = (long)(f7_2) * (long)(f8_19);
long f7f9_76 = (long)(f7_2) * (long)(f9_38);
long f8f8_19 = (long)(f8) * (long)(f8_19);
long f8f9_38 = (long)(f8) * (long)(f9_38);
long f9f9_38 = (long)(f9) * (long)(f9_38);
long h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
long h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
long h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
long h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
long h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
long h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
long h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
long h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
long h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
long h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
var carry = new long[10];
carry[0] = (h0 + (1 << 25)) >> 26;
h1 += carry[0];
h0 -= carry[0] << 26;
carry[4] = (h4 + (1 << 25)) >> 26;
h5 += carry[4];
h4 -= carry[4] << 26;
carry[1] = (h1 + (1 << 24)) >> 25;
h2 += carry[1];
h1 -= carry[1] << 25;
carry[5] = (h5 + (1 << 24)) >> 25;
h6 += carry[5];
h5 -= carry[5] << 25;
carry[2] = (h2 + (1 << 25)) >> 26;
h3 += carry[2];
h2 -= carry[2] << 26;
carry[6] = (h6 + (1 << 25)) >> 26;
h7 += carry[6];
h6 -= carry[6] << 26;
carry[3] = (h3 + (1 << 24)) >> 25;
h4 += carry[3];
h3 -= carry[3] << 25;
carry[7] = (h7 + (1 << 24)) >> 25;
h8 += carry[7];
h7 -= carry[7] << 25;
carry[4] = (h4 + (1 << 25)) >> 26;
h5 += carry[4];
h4 -= carry[4] << 26;
carry[8] = (h8 + (1 << 25)) >> 26;
h9 += carry[8];
h8 -= carry[8] << 26;
carry[9] = (h9 + (1 << 24)) >> 25;
h0 += carry[9] * 19;
h9 -= carry[9] << 25;
carry[0] = (h0 + (1 << 25)) >> 26;
h1 += carry[0];
h0 -= carry[0] << 26;
var final = new FieldElement();
var h = new int[10];
h[0] = (int)(h0);
h[1] = (int)(h1);
h[2] = (int)(h2);
h[3] = (int)(h3);
h[4] = (int)(h4);
h[5] = (int)(h5);
h[6] = (int)(h6);
h[7] = (int)(h7);
h[8] = (int)(h8);
h[9] = (int)(h9);
final.SetElementsDirect(h);
return(final);
}
/// <summary>
/// Calculates h = f * 121666. Can overlap h with f; I have no clue why this is a thing
/// </summary>
/// <remarks>
/// Preconditions:
/// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
///
/// Postconditions:
/// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
/// </remarks>
/// <returns></returns>
public FieldElement Mul121666()
{
long h0 = (long)(_elements[0]) * 121666;
long h1 = (long)(_elements[1]) * 121666;
long h2 = (long)(_elements[2]) * 121666;
long h3 = (long)(_elements[3]) * 121666;
long h4 = (long)(_elements[4]) * 121666;
long h5 = (long)(_elements[5]) * 121666;
long h6 = (long)(_elements[6]) * 121666;
long h7 = (long)(_elements[7]) * 121666;
long h8 = (long)(_elements[8]) * 121666;
long h9 = (long)(_elements[9]) * 121666;
var carry = new long[10];
carry[9] = (h9 + (1 << 24)) >> 25;
h0 += carry[9] * 19;
h9 -= carry[9] << 25;
carry[1] = (h1 + (1 << 24)) >> 25;
h2 += carry[1];
h1 -= carry[1] << 25;
carry[3] = (h3 + (1 << 24)) >> 25;
h4 += carry[3];
h3 -= carry[3] << 25;
carry[5] = (h5 + (1 << 24)) >> 25;
h6 += carry[5];
h5 -= carry[5] << 25;
carry[7] = (h7 + (1 << 24)) >> 25;
h8 += carry[7];
h7 -= carry[7] << 25;
carry[0] = (h0 + (1 << 25)) >> 26;
h1 += carry[0];
h0 -= carry[0] << 26;
carry[2] = (h2 + (1 << 25)) >> 26;
h3 += carry[2];
h2 -= carry[2] << 26;
carry[4] = (h4 + (1 << 25)) >> 26;
h5 += carry[4];
h4 -= carry[4] << 26;
carry[6] = (h6 + (1 << 25)) >> 26;
h7 += carry[6];
h6 -= carry[6] << 26;
carry[8] = (h8 + (1 << 25)) >> 26;
h9 += carry[8];
h8 -= carry[8] << 26;
var final = new FieldElement();
var h = new int[10];
h[0] = (int)(h0);
h[1] = (int)(h1);
h[2] = (int)(h2);
h[3] = (int)(h3);
h[4] = (int)(h4);
h[5] = (int)(h5);
h[6] = (int)(h6);
h[7] = (int)(h7);
h[8] = (int)(h8);
h[9] = (int)(h9);
final.SetElementsDirect(h);
return(final);
}
/// <summary>
/// Calculates this * g
/// </summary>
/// <param name="g"></param>
/// <remarks>
/// Can overlap h with f or g.
///
/// Preconditions:
/// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
/// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
///
/// Postconditions:
/// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
///
/// Notes on implementation strategy:
///
/// Using schoolbook multiplication.
/// Karatsuba would save a little in some cost models.
///
/// Most multiplications by 2 and 19 are 32-bit precomputations;
/// cheaper than 64-bit postcomputations.
///
/// There is one remaining multiplication by 19 in the carry chain;
/// one *19 precomputation can be merged into this,
/// but the resulting data flow is considerably less clean.
///
/// There are 12 carries below.
/// 10 of them are 2-way parallelizable and vectorizable.
/// Can get away with 11 carries, but then data flow is much deeper.
///
/// With tighter constraints on inputs can squeeze carries into int32.
/// </remarks>
/// <returns></returns>
public FieldElement Multiply(FieldElement g)
{
int f0 = _elements[0];
int f1 = _elements[1];
int f2 = _elements[2];
int f3 = _elements[3];
int f4 = _elements[4];
int f5 = _elements[5];
int f6 = _elements[6];
int f7 = _elements[7];
int f8 = _elements[8];
int f9 = _elements[9];
int g0 = g[0];
int g1 = g[1];
int g2 = g[2];
int g3 = g[3];
int g4 = g[4];
int g5 = g[5];
int g6 = g[6];
int g7 = g[7];
int g8 = g[8];
int g9 = g[9];
int g1_19 = 19 * g1; // 1.4*2^29
int g2_19 = 19 * g2; // 1.4*2^30; still ok
int g3_19 = 19 * g3;
int g4_19 = 19 * g4;
int g5_19 = 19 * g5;
int g6_19 = 19 * g6;
int g7_19 = 19 * g7;
int g8_19 = 19 * g8;
int g9_19 = 19 * g9;
int f1_2 = 2 * f1;
int f3_2 = 2 * f3;
int f5_2 = 2 * f5;
int f7_2 = 2 * f7;
int f9_2 = 2 * f9;
long f0g0 = (long)f0 * g0;
long f0g1 = (long)(f0) * (long)(g1);
long f0g2 = (long)(f0) * (long)(g2);
long f0g3 = (long)(f0) * (long)(g3);
long f0g4 = (long)(f0) * (long)(g4);
long f0g5 = (long)(f0) * (long)(g5);
long f0g6 = (long)(f0) * (long)(g6);
long f0g7 = (long)(f0) * (long)(g7);
long f0g8 = (long)(f0) * (long)(g8);
long f0g9 = (long)(f0) * (long)(g9);
long f1g0 = (long)(f1) * (long)(g0);
long f1g1_2 = (long)(f1_2) * (long)(g1);
long f1g2 = (long)(f1) * (long)(g2);
long f1g3_2 = (long)(f1_2) * (long)(g3);
long f1g4 = (long)(f1) * (long)(g4);
long f1g5_2 = (long)(f1_2) * (long)(g5);
long f1g6 = (long)(f1) * (long)(g6);
long f1g7_2 = (long)(f1_2) * (long)(g7);
long f1g8 = (long)(f1) * (long)(g8);
long f1g9_38 = (long)(f1_2) * (long)(g9_19);
long f2g0 = (long)(f2) * (long)(g0);
long f2g1 = (long)(f2) * (long)(g1);
long f2g2 = (long)(f2) * (long)(g2);
long f2g3 = (long)(f2) * (long)(g3);
long f2g4 = (long)(f2) * (long)(g4);
long f2g5 = (long)(f2) * (long)(g5);
long f2g6 = (long)(f2) * (long)(g6);
long f2g7 = (long)(f2) * (long)(g7);
long f2g8_19 = (long)(f2) * (long)(g8_19);
long f2g9_19 = (long)(f2) * (long)(g9_19);
long f3g0 = (long)(f3) * (long)(g0);
long f3g1_2 = (long)(f3_2) * (long)(g1);
long f3g2 = (long)(f3) * (long)(g2);
long f3g3_2 = (long)(f3_2) * (long)(g3);
long f3g4 = (long)(f3) * (long)(g4);
long f3g5_2 = (long)(f3_2) * (long)(g5);
long f3g6 = (long)(f3) * (long)(g6);
long f3g7_38 = (long)(f3_2) * (long)(g7_19);
long f3g8_19 = (long)(f3) * (long)(g8_19);
long f3g9_38 = (long)(f3_2) * (long)(g9_19);
long f4g0 = (long)(f4) * (long)(g0);
long f4g1 = (long)(f4) * (long)(g1);
long f4g2 = (long)(f4) * (long)(g2);
long f4g3 = (long)(f4) * (long)(g3);
long f4g4 = (long)(f4) * (long)(g4);
long f4g5 = (long)(f4) * (long)(g5);
long f4g6_19 = (long)(f4) * (long)(g6_19);
long f4g7_19 = (long)(f4) * (long)(g7_19);
long f4g8_19 = (long)(f4) * (long)(g8_19);
long f4g9_19 = (long)(f4) * (long)(g9_19);
long f5g0 = (long)(f5) * (long)(g0);
long f5g1_2 = (long)(f5_2) * (long)(g1);
long f5g2 = (long)(f5) * (long)(g2);
long f5g3_2 = (long)(f5_2) * (long)(g3);
long f5g4 = (long)(f5) * (long)(g4);
long f5g5_38 = (long)(f5_2) * (long)(g5_19);
long f5g6_19 = (long)(f5) * (long)(g6_19);
long f5g7_38 = (long)(f5_2) * (long)(g7_19);
long f5g8_19 = (long)(f5) * (long)(g8_19);
long f5g9_38 = (long)(f5_2) * (long)(g9_19);
long f6g0 = (long)(f6) * (long)(g0);
long f6g1 = (long)(f6) * (long)(g1);
long f6g2 = (long)(f6) * (long)(g2);
long f6g3 = (long)(f6) * (long)(g3);
long f6g4_19 = (long)(f6) * (long)(g4_19);
long f6g5_19 = (long)(f6) * (long)(g5_19);
long f6g6_19 = (long)(f6) * (long)(g6_19);
long f6g7_19 = (long)(f6) * (long)(g7_19);
long f6g8_19 = (long)(f6) * (long)(g8_19);
long f6g9_19 = (long)(f6) * (long)(g9_19);
long f7g0 = (long)(f7) * (long)(g0);
long f7g1_2 = (long)(f7_2) * (long)(g1);
long f7g2 = (long)(f7) * (long)(g2);
long f7g3_38 = (long)(f7_2) * (long)(g3_19);
long f7g4_19 = (long)(f7) * (long)(g4_19);
long f7g5_38 = (long)(f7_2) * (long)(g5_19);
long f7g6_19 = (long)(f7) * (long)(g6_19);
long f7g7_38 = (long)(f7_2) * (long)(g7_19);
long f7g8_19 = (long)(f7) * (long)(g8_19);
long f7g9_38 = (long)(f7_2) * (long)(g9_19);
long f8g0 = (long)(f8) * (long)(g0);
long f8g1 = (long)(f8) * (long)(g1);
long f8g2_19 = (long)(f8) * (long)(g2_19);
long f8g3_19 = (long)(f8) * (long)(g3_19);
long f8g4_19 = (long)(f8) * (long)(g4_19);
long f8g5_19 = (long)(f8) * (long)(g5_19);
long f8g6_19 = (long)(f8) * (long)(g6_19);
long f8g7_19 = (long)(f8) * (long)(g7_19);
long f8g8_19 = (long)(f8) * (long)(g8_19);
long f8g9_19 = (long)(f8) * (long)(g9_19);
long f9g0 = (long)(f9) * (long)(g0);
long f9g1_38 = (long)(f9_2) * (long)(g1_19);
long f9g2_19 = (long)(f9) * (long)(g2_19);
long f9g3_38 = (long)(f9_2) * (long)(g3_19);
long f9g4_19 = (long)(f9) * (long)(g4_19);
long f9g5_38 = (long)(f9_2) * (long)(g5_19);
long f9g6_19 = (long)(f9) * (long)(g6_19);
long f9g7_38 = (long)(f9_2) * (long)(g7_19);
long f9g8_19 = (long)(f9) * (long)(g8_19);
long f9g9_38 = (long)(f9_2) * (long)(g9_19);
long h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
long h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
long h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
long h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
long h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
long h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
long h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
long h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
long h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
long h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0;
var carry = new long[10];
// |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
// i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
// |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
// i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9
carry[0] = (h0 + (1 << 25)) >> 26;
h1 += carry[0];
h0 -= carry[0] << 26;
carry[4] = (h4 + (1 << 25)) >> 26;
h5 += carry[4];
h4 -= carry[4] << 26;
// |h0| <= 2^25
// |h4| <= 2^25
// |h1| <= 1.51*2^58
// |h5| <= 1.51*2^58
carry[1] = (h1 + (1 << 24)) >> 25;
h2 += carry[1];
h1 -= carry[1] << 25;
carry[5] = (h5 + (1 << 24)) >> 25;
h6 += carry[5];
h5 -= carry[5] << 25;
// |h1| <= 2^24; from now on fits into int32
// |h5| <= 2^24; from now on fits into int32
// |h2| <= 1.21*2^59
// |h6| <= 1.21*2^59
carry[2] = (h2 + (1 << 25)) >> 26;
h3 += carry[2];
h2 -= carry[2] << 26;
carry[6] = (h6 + (1 << 25)) >> 26;
h7 += carry[6];
h6 -= carry[6] << 26;
// |h2| <= 2^25; from now on fits into int32 unchanged
// |h6| <= 2^25; from now on fits into int32 unchanged
// |h3| <= 1.51*2^58
// |h7| <= 1.51*2^58
carry[3] = (h3 + (1 << 24)) >> 25;
h4 += carry[3];
h3 -= carry[3] << 25;
carry[7] = (h7 + (1 << 24)) >> 25;
h8 += carry[7];
h7 -= carry[7] << 25;
// |h3| <= 2^24; from now on fits into int32 unchanged
// |h7| <= 2^24; from now on fits into int32 unchanged
// |h4| <= 1.52*2^33
// |h8| <= 1.52*2^33
carry[4] = (h4 + (1 << 25)) >> 26;
h5 += carry[4];
h4 -= carry[4] << 26;
carry[8] = (h8 + (1 << 25)) >> 26;
h9 += carry[8];
h8 -= carry[8] << 26;
// |h4| <= 2^25; from now on fits into int32 unchanged
// |h8| <= 2^25; from now on fits into int32 unchanged
// |h5| <= 1.01*2^24
// |h9| <= 1.51*2^58
carry[9] = (h9 + (1 << 24)) >> 25;
h0 += carry[9] * 19;
h9 -= carry[9] << 25;
// |h9| <= 2^24; from now on fits into int32 unchanged
// |h0| <= 1.8*2^37
carry[0] = (h0 + (1 << 25)) >> 26;
h1 += carry[0];
h0 -= carry[0] << 26;
// |h0| <= 2^25; from now on fits into int32 unchanged
// |h1| <= 1.01*2^24
int[] h = new int[10];
h[0] = (int)(h0);
h[1] = (int)(h1);
h[2] = (int)(h2);
h[3] = (int)(h3);
h[4] = (int)(h4);
h[5] = (int)(h5);
h[6] = (int)(h6);
h[7] = (int)(h7);
h[8] = (int)(h8);
h[9] = (int)(h9);
var final = new FieldElement();
final.SetElementsDirect(h);
return(final);
}