package net.i2p.crypto.eddsa.math.ed25519;
import net.i2p.crypto.eddsa.Utils;
import net.i2p.crypto.eddsa.math.*;
import java.util.Arrays;
/**
* Class to represent a field element of the finite field p=2^255-19 elements.
* <p>
* An element t, entries t[0]...t[9], represents the integer
* t[0]+2^26 t[1]+2^51 t[2]+2^77 t[3]+2^102 t[4]+...+2^230 t[9].
* Bounds on each t[i] vary depending on context.
* <p>
* Reviewed/commented by Bloody Rookie (nemproject@gmx.de)
*/
public class Ed25519FieldElement extends FieldElement {
/**
* Variable is package private for encoding.
*/
final int[] t;
/**
* Creates a field element.
*
* @param f The underlying field, must be the finite field with p = 2^255 - 19 elements
* @param t The 2^25.5 bit representation of the field element.
*/
public Ed25519FieldElement(Field f, int[] t) {
super(f);
if (t.length != 10)
throw new IllegalArgumentException("Invalid radix-2^51 representation");
this.t = t;
}
private static final byte[] ZERO = new byte[32];
/**
* Gets a value indicating whether or not the field element is non-zero.
*
* @return 1 if it is non-zero, 0 otherwise.
*/
public boolean isNonZero() {
final byte[] s = toByteArray();
return Utils.equal(s, ZERO) == 0;
}
/**
* h = f + g
* <p>
* TODO-CR BR: h is allocated via new, probably not a good idea. Do we need the copying into temp variables if we do that?
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* <li>|g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
* </ul>
*
* @param val The field element to add.
* @return The field element this + val.
*/
public FieldElement add(FieldElement val) {
int[] g = ((Ed25519FieldElement)val).t;
int[] h = new int[10];
for (int i = 0; i < 10; i++) {
h[i] = t[i] + g[i];
}
return new Ed25519FieldElement(f, h);
}
/**
* h = f - g
* <p>
* Can overlap h with f or g.
* <p>
* TODO-CR BR: See above.
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* <li>|g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
* </ul>
*
* @param val The field element to subtract.
* @return The field element this - val.
**/
public FieldElement subtract(FieldElement val) {
int[] g = ((Ed25519FieldElement)val).t;
int[] h = new int[10];
for (int i = 0; i < 10; i++) {
h[i] = t[i] - g[i];
}
return new Ed25519FieldElement(f, h);
}
/**
* h = -f
* <p>
* TODO-CR BR: see above.
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
* </ul>
*
* @return The field element (-1) * this.
*/
public FieldElement negate() {
int[] h = new int[10];
for (int i = 0; i < 10; i++) {
h[i] = - t[i];
}
return new Ed25519FieldElement(f, h);
}
/**
* h = f * g
* <p>
* Can overlap h with f or g.
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by
* 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
* <li>|g| bounded by
* 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by
* 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
* </ul><p>
* Notes on implementation strategy:
* <p>
* Using schoolbook multiplication. Karatsuba would save a little in some
* cost models.
* <p>
* Most multiplications by 2 and 19 are 32-bit precomputations; cheaper than
* 64-bit postcomputations.
* <p>
* 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.
* <p>
* 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.
* <p>
* With tighter constraints on inputs can squeeze carries into int32.
*
* @param val The field element to multiply.
* @return The (reasonably reduced) field element this * val.
*/
public FieldElement multiply(FieldElement val) {
int[] g = ((Ed25519FieldElement)val).t;
int g1_19 = 19 * g[1]; /* 1.959375*2^29 */
int g2_19 = 19 * g[2]; /* 1.959375*2^30; still ok */
int g3_19 = 19 * g[3];
int g4_19 = 19 * g[4];
int g5_19 = 19 * g[5];
int g6_19 = 19 * g[6];
int g7_19 = 19 * g[7];
int g8_19 = 19 * g[8];
int g9_19 = 19 * g[9];
int f1_2 = 2 * t[1];
int f3_2 = 2 * t[3];
int f5_2 = 2 * t[5];
int f7_2 = 2 * t[7];
int f9_2 = 2 * t[9];
long f0g0 = t[0] * (long) g[0];
long f0g1 = t[0] * (long) g[1];
long f0g2 = t[0] * (long) g[2];
long f0g3 = t[0] * (long) g[3];
long f0g4 = t[0] * (long) g[4];
long f0g5 = t[0] * (long) g[5];
long f0g6 = t[0] * (long) g[6];
long f0g7 = t[0] * (long) g[7];
long f0g8 = t[0] * (long) g[8];
long f0g9 = t[0] * (long) g[9];
long f1g0 = t[1] * (long) g[0];
long f1g1_2 = f1_2 * (long) g[1];
long f1g2 = t[1] * (long) g[2];
long f1g3_2 = f1_2 * (long) g[3];
long f1g4 = t[1] * (long) g[4];
long f1g5_2 = f1_2 * (long) g[5];
long f1g6 = t[1] * (long) g[6];
long f1g7_2 = f1_2 * (long) g[7];
long f1g8 = t[1] * (long) g[8];
long f1g9_38 = f1_2 * (long) g9_19;
long f2g0 = t[2] * (long) g[0];
long f2g1 = t[2] * (long) g[1];
long f2g2 = t[2] * (long) g[2];
long f2g3 = t[2] * (long) g[3];
long f2g4 = t[2] * (long) g[4];
long f2g5 = t[2] * (long) g[5];
long f2g6 = t[2] * (long) g[6];
long f2g7 = t[2] * (long) g[7];
long f2g8_19 = t[2] * (long) g8_19;
long f2g9_19 = t[2] * (long) g9_19;
long f3g0 = t[3] * (long) g[0];
long f3g1_2 = f3_2 * (long) g[1];
long f3g2 = t[3] * (long) g[2];
long f3g3_2 = f3_2 * (long) g[3];
long f3g4 = t[3] * (long) g[4];
long f3g5_2 = f3_2 * (long) g[5];
long f3g6 = t[3] * (long) g[6];
long f3g7_38 = f3_2 * (long) g7_19;
long f3g8_19 = t[3] * (long) g8_19;
long f3g9_38 = f3_2 * (long) g9_19;
long f4g0 = t[4] * (long) g[0];
long f4g1 = t[4] * (long) g[1];
long f4g2 = t[4] * (long) g[2];
long f4g3 = t[4] * (long) g[3];
long f4g4 = t[4] * (long) g[4];
long f4g5 = t[4] * (long) g[5];
long f4g6_19 = t[4] * (long) g6_19;
long f4g7_19 = t[4] * (long) g7_19;
long f4g8_19 = t[4] * (long) g8_19;
long f4g9_19 = t[4] * (long) g9_19;
long f5g0 = t[5] * (long) g[0];
long f5g1_2 = f5_2 * (long) g[1];
long f5g2 = t[5] * (long) g[2];
long f5g3_2 = f5_2 * (long) g[3];
long f5g4 = t[5] * (long) g[4];
long f5g5_38 = f5_2 * (long) g5_19;
long f5g6_19 = t[5] * (long) g6_19;
long f5g7_38 = f5_2 * (long) g7_19;
long f5g8_19 = t[5] * (long) g8_19;
long f5g9_38 = f5_2 * (long) g9_19;
long f6g0 = t[6] * (long) g[0];
long f6g1 = t[6] * (long) g[1];
long f6g2 = t[6] * (long) g[2];
long f6g3 = t[6] * (long) g[3];
long f6g4_19 = t[6] * (long) g4_19;
long f6g5_19 = t[6] * (long) g5_19;
long f6g6_19 = t[6] * (long) g6_19;
long f6g7_19 = t[6] * (long) g7_19;
long f6g8_19 = t[6] * (long) g8_19;
long f6g9_19 = t[6] * (long) g9_19;
long f7g0 = t[7] * (long) g[0];
long f7g1_2 = f7_2 * (long) g[1];
long f7g2 = t[7] * (long) g[2];
long f7g3_38 = f7_2 * (long) g3_19;
long f7g4_19 = t[7] * (long) g4_19;
long f7g5_38 = f7_2 * (long) g5_19;
long f7g6_19 = t[7] * (long) g6_19;
long f7g7_38 = f7_2 * (long) g7_19;
long f7g8_19 = t[7] * (long) g8_19;
long f7g9_38 = f7_2 * (long) g9_19;
long f8g0 = t[8] * (long) g[0];
long f8g1 = t[8] * (long) g[1];
long f8g2_19 = t[8] * (long) g2_19;
long f8g3_19 = t[8] * (long) g3_19;
long f8g4_19 = t[8] * (long) g4_19;
long f8g5_19 = t[8] * (long) g5_19;
long f8g6_19 = t[8] * (long) g6_19;
long f8g7_19 = t[8] * (long) g7_19;
long f8g8_19 = t[8] * (long) g8_19;
long f8g9_19 = t[8] * (long) g9_19;
long f9g0 = t[9] * (long) g[0];
long f9g1_38 = f9_2 * (long) g1_19;
long f9g2_19 = t[9] * (long) g2_19;
long f9g3_38 = f9_2 * (long) g3_19;
long f9g4_19 = t[9] * (long) g4_19;
long f9g5_38 = f9_2 * (long) g5_19;
long f9g6_19 = t[9] * (long) g6_19;
long f9g7_38 = f9_2 * (long) g7_19;
long f9g8_19 = t[9] * (long) g8_19;
long f9g9_38 = f9_2 * (long) g9_19;
/**
* Remember: 2^255 congruent 19 modulo p.
* h = h0 * 2^0 + h1 * 2^26 + h2 * 2^(26+25) + h3 * 2^(26+25+26) + ... + h9 * 2^(5*26+5*25).
* So to get the real number we would have to multiply the coefficients with the corresponding powers of 2.
* To get an idea what is going on below, look at the calculation of h0:
* h0 is the coefficient to the power 2^0 so it collects (sums) all products that have the power 2^0.
* f0 * g0 really is f0 * 2^0 * g0 * 2^0 = (f0 * g0) * 2^0.
* f1 * g9 really is f1 * 2^26 * g9 * 2^230 = f1 * g9 * 2^256 = 2 * f1 * g9 * 2^255 congruent 2 * 19 * f1 * g9 * 2^0 modulo p.
* f2 * g8 really is f2 * 2^51 * g8 * 2^204 = f2 * g8 * 2^255 congruent 19 * f2 * g8 * 2^0 modulo p.
* and so on...
*/
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;
long carry0;
long carry1;
long carry2;
long carry3;
long carry4;
long carry5;
long carry6;
long carry7;
long carry8;
long carry9;
/*
|h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
|h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9
*/
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
/* |h0| <= 2^25 */
/* |h4| <= 2^25 */
/* |h1| <= 1.71*2^59 */
/* |h5| <= 1.71*2^59 */
carry1 = (h1 + (long) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
carry5 = (h5 + (long) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
/* |h1| <= 2^24; from now on fits into int32 */
/* |h5| <= 2^24; from now on fits into int32 */
/* |h2| <= 1.41*2^60 */
/* |h6| <= 1.41*2^60 */
carry2 = (h2 + (long) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
carry6 = (h6 + (long) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
/* |h2| <= 2^25; from now on fits into int32 unchanged */
/* |h6| <= 2^25; from now on fits into int32 unchanged */
/* |h3| <= 1.71*2^59 */
/* |h7| <= 1.71*2^59 */
carry3 = (h3 + (long) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
carry7 = (h7 + (long) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
/* |h3| <= 2^24; from now on fits into int32 unchanged */
/* |h7| <= 2^24; from now on fits into int32 unchanged */
/* |h4| <= 1.72*2^34 */
/* |h8| <= 1.41*2^60 */
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry8 = (h8 + (long) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 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| <= 1.71*2^59 */
carry9 = (h9 + (long) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
/* |h9| <= 2^24; from now on fits into int32 unchanged */
/* |h0| <= 1.1*2^39 */
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 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;
return new Ed25519FieldElement(f, h);
}
/**
* h = f * f
* <p>
* Can overlap h with f.
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
* </ul><p>
* See {@link #multiply(FieldElement)} for discussion
* of implementation strategy.
*
* @return The (reasonably reduced) square of this field element.
*/
public FieldElement square() {
int f0 = t[0];
int f1 = t[1];
int f2 = t[2];
int f3 = t[3];
int f4 = t[4];
int f5 = t[5];
int f6 = t[6];
int f7 = t[7];
int f8 = t[8];
int f9 = t[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; /* 1.959375*2^30 */
int f6_19 = 19 * f6; /* 1.959375*2^30 */
int f7_38 = 38 * f7; /* 1.959375*2^30 */
int f8_19 = 19 * f8; /* 1.959375*2^30 */
int f9_38 = 38 * f9; /* 1.959375*2^30 */
long f0f0 = f0 * (long) f0;
long f0f1_2 = f0_2 * (long) f1;
long f0f2_2 = f0_2 * (long) f2;
long f0f3_2 = f0_2 * (long) f3;
long f0f4_2 = f0_2 * (long) f4;
long f0f5_2 = f0_2 * (long) f5;
long f0f6_2 = f0_2 * (long) f6;
long f0f7_2 = f0_2 * (long) f7;
long f0f8_2 = f0_2 * (long) f8;
long f0f9_2 = f0_2 * (long) f9;
long f1f1_2 = f1_2 * (long) f1;
long f1f2_2 = f1_2 * (long) f2;
long f1f3_4 = f1_2 * (long) f3_2;
long f1f4_2 = f1_2 * (long) f4;
long f1f5_4 = f1_2 * (long) f5_2;
long f1f6_2 = f1_2 * (long) f6;
long f1f7_4 = f1_2 * (long) f7_2;
long f1f8_2 = f1_2 * (long) f8;
long f1f9_76 = f1_2 * (long) f9_38;
long f2f2 = f2 * (long) f2;
long f2f3_2 = f2_2 * (long) f3;
long f2f4_2 = f2_2 * (long) f4;
long f2f5_2 = f2_2 * (long) f5;
long f2f6_2 = f2_2 * (long) f6;
long f2f7_2 = f2_2 * (long) f7;
long f2f8_38 = f2_2 * (long) f8_19;
long f2f9_38 = f2 * (long) f9_38;
long f3f3_2 = f3_2 * (long) f3;
long f3f4_2 = f3_2 * (long) f4;
long f3f5_4 = f3_2 * (long) f5_2;
long f3f6_2 = f3_2 * (long) f6;
long f3f7_76 = f3_2 * (long) f7_38;
long f3f8_38 = f3_2 * (long) f8_19;
long f3f9_76 = f3_2 * (long) f9_38;
long f4f4 = f4 * (long) f4;
long f4f5_2 = f4_2 * (long) f5;
long f4f6_38 = f4_2 * (long) f6_19;
long f4f7_38 = f4 * (long) f7_38;
long f4f8_38 = f4_2 * (long) f8_19;
long f4f9_38 = f4 * (long) f9_38;
long f5f5_38 = f5 * (long) f5_38;
long f5f6_38 = f5_2 * (long) f6_19;
long f5f7_76 = f5_2 * (long) f7_38;
long f5f8_38 = f5_2 * (long) f8_19;
long f5f9_76 = f5_2 * (long) f9_38;
long f6f6_19 = f6 * (long) f6_19;
long f6f7_38 = f6 * (long) f7_38;
long f6f8_38 = f6_2 * (long) f8_19;
long f6f9_38 = f6 * (long) f9_38;
long f7f7_38 = f7 * (long) f7_38;
long f7f8_38 = f7_2 * (long) f8_19;
long f7f9_76 = f7_2 * (long) f9_38;
long f8f8_19 = f8 * (long) f8_19;
long f8f9_38 = f8 * (long) f9_38;
long f9f9_38 = f9 * (long) f9_38;
/**
* Same procedure as in multiply, but this time we have a higher symmetry leading to less summands.
* e.g. f1f9_76 really stands for f1 * 2^26 * f9 * 2^230 + f9 * 2^230 + f1 * 2^26 congruent 2 * 2 * 19 * f1 * f9 2^0 modulo p.
*/
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;
long carry0;
long carry1;
long carry2;
long carry3;
long carry4;
long carry5;
long carry6;
long carry7;
long carry8;
long carry9;
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry1 = (h1 + (long) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
carry5 = (h5 + (long) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
carry2 = (h2 + (long) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
carry6 = (h6 + (long) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
carry3 = (h3 + (long) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
carry7 = (h7 + (long) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry8 = (h8 + (long) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
carry9 = (h9 + (long) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
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;
return new Ed25519FieldElement(f, h);
}
/**
* h = 2 * f * f
* <p>
* Can overlap h with f.
* <p>
* Preconditions:
* <p><ul>
* <li>|f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
* </ul><p>
* Postconditions:
* <p><ul>
* <li>|h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
* </ul><p>
* See {@link #multiply(FieldElement)} for discussion
* of implementation strategy.
*
* @return The (reasonably reduced) square of this field element times 2.
*/
public FieldElement squareAndDouble() {
int f0 = t[0];
int f1 = t[1];
int f2 = t[2];
int f3 = t[3];
int f4 = t[4];
int f5 = t[5];
int f6 = t[6];
int f7 = t[7];
int f8 = t[8];
int f9 = t[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; /* 1.959375*2^30 */
int f6_19 = 19 * f6; /* 1.959375*2^30 */
int f7_38 = 38 * f7; /* 1.959375*2^30 */
int f8_19 = 19 * f8; /* 1.959375*2^30 */
int f9_38 = 38 * f9; /* 1.959375*2^30 */
long f0f0 = f0 * (long) f0;
long f0f1_2 = f0_2 * (long) f1;
long f0f2_2 = f0_2 * (long) f2;
long f0f3_2 = f0_2 * (long) f3;
long f0f4_2 = f0_2 * (long) f4;
long f0f5_2 = f0_2 * (long) f5;
long f0f6_2 = f0_2 * (long) f6;
long f0f7_2 = f0_2 * (long) f7;
long f0f8_2 = f0_2 * (long) f8;
long f0f9_2 = f0_2 * (long) f9;
long f1f1_2 = f1_2 * (long) f1;
long f1f2_2 = f1_2 * (long) f2;
long f1f3_4 = f1_2 * (long) f3_2;
long f1f4_2 = f1_2 * (long) f4;
long f1f5_4 = f1_2 * (long) f5_2;
long f1f6_2 = f1_2 * (long) f6;
long f1f7_4 = f1_2 * (long) f7_2;
long f1f8_2 = f1_2 * (long) f8;
long f1f9_76 = f1_2 * (long) f9_38;
long f2f2 = f2 * (long) f2;
long f2f3_2 = f2_2 * (long) f3;
long f2f4_2 = f2_2 * (long) f4;
long f2f5_2 = f2_2 * (long) f5;
long f2f6_2 = f2_2 * (long) f6;
long f2f7_2 = f2_2 * (long) f7;
long f2f8_38 = f2_2 * (long) f8_19;
long f2f9_38 = f2 * (long) f9_38;
long f3f3_2 = f3_2 * (long) f3;
long f3f4_2 = f3_2 * (long) f4;
long f3f5_4 = f3_2 * (long) f5_2;
long f3f6_2 = f3_2 * (long) f6;
long f3f7_76 = f3_2 * (long) f7_38;
long f3f8_38 = f3_2 * (long) f8_19;
long f3f9_76 = f3_2 * (long) f9_38;
long f4f4 = f4 * (long) f4;
long f4f5_2 = f4_2 * (long) f5;
long f4f6_38 = f4_2 * (long) f6_19;
long f4f7_38 = f4 * (long) f7_38;
long f4f8_38 = f4_2 * (long) f8_19;
long f4f9_38 = f4 * (long) f9_38;
long f5f5_38 = f5 * (long) f5_38;
long f5f6_38 = f5_2 * (long) f6_19;
long f5f7_76 = f5_2 * (long) f7_38;
long f5f8_38 = f5_2 * (long) f8_19;
long f5f9_76 = f5_2 * (long) f9_38;
long f6f6_19 = f6 * (long) f6_19;
long f6f7_38 = f6 * (long) f7_38;
long f6f8_38 = f6_2 * (long) f8_19;
long f6f9_38 = f6 * (long) f9_38;
long f7f7_38 = f7 * (long) f7_38;
long f7f8_38 = f7_2 * (long) f8_19;
long f7f9_76 = f7_2 * (long) f9_38;
long f8f8_19 = f8 * (long) f8_19;
long f8f9_38 = f8 * (long) f9_38;
long f9f9_38 = 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;
long carry0;
long carry1;
long carry2;
long carry3;
long carry4;
long carry5;
long carry6;
long carry7;
long carry8;
long carry9;
h0 += h0;
h1 += h1;
h2 += h2;
h3 += h3;
h4 += h4;
h5 += h5;
h6 += h6;
h7 += h7;
h8 += h8;
h9 += h9;
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry1 = (h1 + (long) (1<<24)) >> 25; h2 += carry1; h1 -= carry1 << 25;
carry5 = (h5 + (long) (1<<24)) >> 25; h6 += carry5; h5 -= carry5 << 25;
carry2 = (h2 + (long) (1<<25)) >> 26; h3 += carry2; h2 -= carry2 << 26;
carry6 = (h6 + (long) (1<<25)) >> 26; h7 += carry6; h6 -= carry6 << 26;
carry3 = (h3 + (long) (1<<24)) >> 25; h4 += carry3; h3 -= carry3 << 25;
carry7 = (h7 + (long) (1<<24)) >> 25; h8 += carry7; h7 -= carry7 << 25;
carry4 = (h4 + (long) (1<<25)) >> 26; h5 += carry4; h4 -= carry4 << 26;
carry8 = (h8 + (long) (1<<25)) >> 26; h9 += carry8; h8 -= carry8 << 26;
carry9 = (h9 + (long) (1<<24)) >> 25; h0 += carry9 * 19; h9 -= carry9 << 25;
carry0 = (h0 + (long) (1<<25)) >> 26; h1 += carry0; h0 -= carry0 << 26;
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;
return new Ed25519FieldElement(f, h);
}
/**
* Invert this field element.
* <p>
* The inverse is found via Fermat's little theorem:<br>
* a^p congruent a mod p and therefore a^(p-2) congruent a^-1 mod p
*
* @return The inverse of this field element.
*/
public FieldElement invert() {
FieldElement t0, t1, t2, t3;
// 2 == 2 * 1
t0 = square();
// 4 == 2 * 2
t1 = t0.square();
// 8 == 2 * 4
t1 = t1.square();
// 9 == 8 + 1
t1 = multiply(t1);
// 11 == 9 + 2
t0 = t0.multiply(t1);
// 22 == 2 * 11
t2 = t0.square();
// 31 == 22 + 9
t1 = t1.multiply(t2);
// 2^6 - 2^1
t2 = t1.square();
// 2^10 - 2^5
for (int i = 1; i < 5; ++i) {
t2 = t2.square();
}
// 2^10 - 2^0
t1 = t2.multiply(t1);
// 2^11 - 2^1
t2 = t1.square();
// 2^20 - 2^10
for (int i = 1; i < 10; ++i) {
t2 = t2.square();
}
// 2^20 - 2^0
t2 = t2.multiply(t1);
// 2^21 - 2^1
t3 = t2.square();
// 2^40 - 2^20
for (int i = 1; i < 20; ++i) {
t3 = t3.square();
}
// 2^40 - 2^0
t2 = t3.multiply(t2);
// 2^41 - 2^1
t2 = t2.square();
// 2^50 - 2^10
for (int i = 1; i < 10; ++i) {
t2 = t2.square();
}
// 2^50 - 2^0
t1 = t2.multiply(t1);
// 2^51 - 2^1
t2 = t1.square();
// 2^100 - 2^50
for (int i = 1; i < 50; ++i) {
t2 = t2.square();
}
// 2^100 - 2^0
t2 = t2.multiply(t1);
// 2^101 - 2^1
t3 = t2.square();
// 2^200 - 2^100
for (int i = 1; i < 100; ++i) {
t3 = t3.square();
}
// 2^200 - 2^0
t2 = t3.multiply(t2);
// 2^201 - 2^1
t2 = t2.square();
// 2^250 - 2^50
for (int i = 1; i < 50; ++i) {
t2 = t2.square();
}
// 2^250 - 2^0
t1 = t2.multiply(t1);
// 2^251 - 2^1
t1 = t1.square();
// 2^255 - 2^5
for (int i = 1; i < 5; ++i) {
t1 = t1.square();
}
// 2^255 - 21
return t1.multiply(t0);
}
/**
* Gets this field element to the power of (2^252 - 3).
* This is a helper function for calculating the square root.
* <p>
* TODO-CR BR: I think it makes sense to have a sqrt function.
*
* @return This field element to the power of (2^252 - 3).
*/
public FieldElement pow22523() {
FieldElement t0, t1, t2;
// 2 == 2 * 1
t0 = square();
// 4 == 2 * 2
t1 = t0.square();
// 8 == 2 * 4
t1 = t1.square();
// z9 = z1*z8
t1 = multiply(t1);
// 11 == 9 + 2
t0 = t0.multiply(t1);
// 22 == 2 * 11
t0 = t0.square();
// 31 == 22 + 9
t0 = t1.multiply(t0);
// 2^6 - 2^1
t1 = t0.square();
// 2^10 - 2^5
for (int i = 1; i < 5; ++i) {
t1 = t1.square();
}
// 2^10 - 2^0
t0 = t1.multiply(t0);
// 2^11 - 2^1
t1 = t0.square();
// 2^20 - 2^10
for (int i = 1; i < 10; ++i) {
t1 = t1.square();
}
// 2^20 - 2^0
t1 = t1.multiply(t0);
// 2^21 - 2^1
t2 = t1.square();
// 2^40 - 2^20
for (int i = 1; i < 20; ++i) {
t2 = t2.square();
}
// 2^40 - 2^0
t1 = t2.multiply(t1);
// 2^41 - 2^1
t1 = t1.square();
// 2^50 - 2^10
for (int i = 1; i < 10; ++i) {
t1 = t1.square();
}
// 2^50 - 2^0
t0 = t1.multiply(t0);
// 2^51 - 2^1
t1 = t0.square();
// 2^100 - 2^50
for (int i = 1; i < 50; ++i) {
t1 = t1.square();
}
// 2^100 - 2^0
t1 = t1.multiply(t0);
// 2^101 - 2^1
t2 = t1.square();
// 2^200 - 2^100
for (int i = 1; i < 100; ++i) {
t2 = t2.square();
}
// 2^200 - 2^0
t1 = t2.multiply(t1);
// 2^201 - 2^1
t1 = t1.square();
// 2^250 - 2^50
for (int i = 1; i < 50; ++i) {
t1 = t1.square();
}
// 2^250 - 2^0
t0 = t1.multiply(t0);
// 2^251 - 2^1
t0 = t0.square();
// 2^252 - 2^2
t0 = t0.square();
// 2^252 - 3
return multiply(t0);
}
@Override
public int hashCode() {
return Arrays.hashCode(t);
}
@Override
public boolean equals(Object obj) {
if (!(obj instanceof Ed25519FieldElement))
return false;
Ed25519FieldElement fe = (Ed25519FieldElement) obj;
return 1==Utils.equal(toByteArray(), fe.toByteArray());
}
@Override
public String toString() {
return "[Ed25519FieldElement val="+Utils.bytesToHex(toByteArray())+"]";
}
}