src/main/java/org/mariadb/jdbc/plugin/authentication/standard/ed25519/math/ed25519/Ed25519FieldElement.java

package org.mariadb.jdbc.plugin.authentication.standard.ed25519.math.ed25519;

import java.util.Arrays;
import org.mariadb.jdbc.plugin.authentication.standard.ed25519.Utils;
import org.mariadb.jdbc.plugin.authentication.standard.ed25519.math.Field;
import org.mariadb.jdbc.plugin.authentication.standard.ed25519.math.FieldElement;

/**
 * Class to represent a field element of the finite field $p = 2^{255} - 19$ elements.
 *
 * <p>An element $t$, entries $t[0] \dots t[9]$, represents the integer $t[0]+2^{26} t[1]+2^{51}
 * t[2]+2^{77} t[3]+2^{102} t[4]+\dots+2^{230} t[9]$. Bounds on each $t[i]$ vary depending on
 * context.
 *
 * <p>Reviewed/commented by Bloody Rookie (nemproject@gmx.de)
 */
public class Ed25519FieldElement extends FieldElement {
  private static final long serialVersionUID = -2455098303824960263L;
  /** 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:
   *
   * <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:
   *
   * <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:
   *
   * <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:
   *
   * <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:
   *
   * <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:
   *
   * <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:
   *
   * <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:
   *
   * <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;

    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:
   *
   * <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:
   *
   * <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;

    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:
   *
   * <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:
   *
   * <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>
   * $a^p \cong a \mod p$ and therefore $a^{(p-2)} \cong a^{-1} \mod p$
   *
   * @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);
  }

  /**
   * Constant-time conditional move. Well, actually it is a conditional copy. Logic is inspired by
   * the SUPERCOP implementation at:
   * https://github.com/floodyberry/supercop/blob/master/crypto_sign/ed25519/ref10/fe_cmov.c
   *
   * @param val the other field element.
   * @param b must be 0 or 1, otherwise results are undefined.
   * @return a copy of this if $b == 0$, or a copy of val if $b == 1$.
   */
  @Override
  public FieldElement cmov(FieldElement val, int b) {
    Ed25519FieldElement that = (Ed25519FieldElement) val;
    b = -b;
    int[] result = new int[10];
    for (int i = 0; i < 10; i++) {
      result[i] = this.t[i];
      int x = this.t[i] ^ that.t[i];
      x &= b;
      result[i] ^= x;
    }
    return new Ed25519FieldElement(this.f, result);
  }

  @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()) + "]";
  }
}