@Test
  public void dblPrecomputedTableContainsExpectedGroupElements() {
    // Arrange:
    GroupElement g = ed25519.getB();
    GroupElement h = MathUtils.addGroupElements(g, g);

    // Act + Assert:
    for (int i = 0; i < 8; i++) {
      Assert.assertThat(
          MathUtils.toRepresentation(g, GroupElement.Representation.PRECOMP),
          IsEqual.equalTo(ed25519.getB().dblPrecmp[i]));
      g = MathUtils.addGroupElements(g, h);
    }
  }
  @Test
  public void precomputedTableContainsExpectedGroupElements() {
    // Arrange:
    GroupElement g = ed25519.getB();

    // Act + Assert:
    for (int i = 0; i < 32; i++) {
      GroupElement h = g;
      for (int j = 0; j < 8; j++) {
        Assert.assertThat(
            MathUtils.toRepresentation(h, GroupElement.Representation.PRECOMP),
            IsEqual.equalTo(ed25519.getB().precmp[i][j]));
        h = MathUtils.addGroupElements(h, g);
      }
      for (int k = 0; k < 8; k++) {
        g = MathUtils.addGroupElements(g, g);
      }
    }
  }
  @Test
  public void scalarMultiplyBasePointWithOneReturnsBasePoint() {
    // Arrange:
    final GroupElement basePoint = ed25519.getB();

    // Act:
    final GroupElement g = basePoint.scalarMultiply(curve.getField().ONE.toByteArray());

    // Assert:
    Assert.assertThat(basePoint, IsEqual.equalTo(g));
  }
  @Test
  public void scalarMultiplyBasePointWithZeroReturnsNeutralElement() {
    // Arrange:
    final GroupElement basePoint = ed25519.getB();

    // Act:
    final GroupElement g = basePoint.scalarMultiply(curve.getField().ZERO.toByteArray());

    // Assert:
    Assert.assertThat(curve.getZero(GroupElement.Representation.P3), IsEqual.equalTo(g));
  }
  /**
   * Test method for {@link GroupElement#scalarMultiply(byte[])}. Test values generated with Python
   * Ed25519 implementation.
   */
  @Test
  public void testScalarMultiplyByteArray() {
    // Little-endian
    byte[] zero =
        Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
    byte[] one =
        Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000000");
    byte[] two =
        Utils.hexToBytes("0200000000000000000000000000000000000000000000000000000000000000");
    byte[] a = Utils.hexToBytes("d072f8dd9c07fa7bc8d22a4b325d26301ee9202f6db89aa7c3731529e37e437c");
    GroupElement A =
        new GroupElement(
            curve,
            Utils.hexToBytes("d4cf8595571830644bd14af416954d09ab7159751ad9e0f7a6cbd92379e71a66"));

    assertThat(
        "scalarMultiply(0) failed",
        ed25519.getB().scalarMultiply(zero),
        is(equalTo(curve.getZero(GroupElement.Representation.P3))));
    assertThat(
        "scalarMultiply(1) failed",
        ed25519.getB().scalarMultiply(one),
        is(equalTo(ed25519.getB())));
    assertThat(
        "scalarMultiply(2) failed",
        ed25519.getB().scalarMultiply(two),
        is(equalTo(ed25519.getB().dbl())));

    assertThat("scalarMultiply(a) failed", ed25519.getB().scalarMultiply(a), is(equalTo(A)));
  }
  // This test is slow (~6s) due to math utils using an inferior algorithm to calculate the result.
  @Test
  public void scalarMultiplyBasePointReturnsExpectedResult() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement basePoint = ed25519.getB();
      final FieldElement f = MathUtils.getRandomFieldElement();

      // Act:
      final GroupElement g = basePoint.scalarMultiply(f.toByteArray());
      final GroupElement h = MathUtils.scalarMultiplyGroupElement(basePoint, f);

      // Assert:
      Assert.assertThat(g, IsEqual.equalTo(h));
    }
  }
  /** Test method for {@link GroupElement#toP2()}. */
  @Test
  public void testToP2() {
    GroupElement p3zero = curve.getZero(GroupElement.Representation.P3);
    GroupElement t = p3zero.toP2();
    assertThat(t.repr, is(GroupElement.Representation.P2));
    assertThat(t.X, is(p3zero.X));
    assertThat(t.Y, is(p3zero.Y));
    assertThat(t.Z, is(p3zero.Z));
    assertThat(t.T, is((FieldElement) null));

    GroupElement B = ed25519.getB();
    t = B.toP2();
    assertThat(t.repr, is(GroupElement.Representation.P2));
    assertThat(t.X, is(B.X));
    assertThat(t.Y, is(B.Y));
    assertThat(t.Z, is(B.Z));
    assertThat(t.T, is((FieldElement) null));
  }
  // This test is slow (~6s) due to math utils using an inferior algorithm to calculate the result.
  @Test
  public void doubleScalarMultiplyVariableTimeReturnsExpectedResult() {
    for (int i = 0; i < 50; i++) {
      // Arrange:
      final GroupElement basePoint = ed25519.getB();
      final GroupElement g = MathUtils.getRandomGroupElement();
      g.precompute(false);
      final FieldElement f1 = MathUtils.getRandomFieldElement();
      final FieldElement f2 = MathUtils.getRandomFieldElement();

      // Act:
      final GroupElement h1 =
          basePoint.doubleScalarMultiplyVariableTime(g, f2.toByteArray(), f1.toByteArray());
      final GroupElement h2 = MathUtils.doubleScalarMultiplyGroupElements(basePoint, f1, g, f2);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
    }
  }
  @Test
  public void testDoubleScalarMultiplyVariableTime() {
    // Little-endian
    byte[] zero =
        Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
    byte[] one =
        Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000000");
    byte[] two =
        Utils.hexToBytes("0200000000000000000000000000000000000000000000000000000000000000");
    byte[] a = Utils.hexToBytes("d072f8dd9c07fa7bc8d22a4b325d26301ee9202f6db89aa7c3731529e37e437c");
    GroupElement A =
        new GroupElement(
            curve,
            Utils.hexToBytes("d4cf8595571830644bd14af416954d09ab7159751ad9e0f7a6cbd92379e71a66"));
    GroupElement B = ed25519.getB();
    GroupElement geZero = curve.getZero(GroupElement.Representation.P3);
    geZero.precompute(false);

    // 0 * GE(0) + 0 * GE(0) = GE(0)
    assertThat(geZero.doubleScalarMultiplyVariableTime(geZero, zero, zero), is(equalTo(geZero)));
    // 0 * GE(0) + 0 * B = GE(0)
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, zero, zero), is(equalTo(geZero)));
    // 1 * GE(0) + 0 * B = GE(0)
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, one, zero), is(equalTo(geZero)));
    // 1 * GE(0) + 1 * B = B
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, one, one), is(equalTo(B)));
    // 1 * B + 1 * B = 2 * B
    assertThat(B.doubleScalarMultiplyVariableTime(B, one, one), is(equalTo(B.dbl())));
    // 1 * B + 2 * B = 3 * B
    assertThat(
        B.doubleScalarMultiplyVariableTime(B, one, two),
        is(equalTo(B.dbl().toP3().add(B.toCached()))));
    // 2 * B + 2 * B = 4 * B
    assertThat(B.doubleScalarMultiplyVariableTime(B, two, two), is(equalTo(B.dbl().toP3().dbl())));

    // 0 * B + a * B = A
    assertThat(B.doubleScalarMultiplyVariableTime(B, zero, a), is(equalTo(A)));
    // a * B + 0 * B = A
    assertThat(B.doubleScalarMultiplyVariableTime(B, a, zero), is(equalTo(A)));
    // a * B + a * B = 2 * A
    assertThat(B.doubleScalarMultiplyVariableTime(B, a, a), is(equalTo(A.dbl())));
  }
Exemplo n.º 10
0
 /** Test method for {@link GroupElement#select(int, int)}. */
 @Test
 public void testSelect() {
   GroupElement B = ed25519.getB();
   for (int i = 0; i < 32; i++) {
     // 16^i 0 B
     assertThat(
         i + ",0", B.select(i, 0), is(equalTo(GroupElement.precomp(curve, ONE, ONE, ZERO))));
     for (int j = 1; j < 8; j++) {
       // 16^i r_i B
       GroupElement t = B.select(i, j);
       assertThat(i + "," + j, t, is(equalTo(B.precmp[i][j - 1])));
       // -16^i r_i B
       t = B.select(i, -j);
       GroupElement neg =
           GroupElement.precomp(
               curve, B.precmp[i][j - 1].Y, B.precmp[i][j - 1].X, B.precmp[i][j - 1].Z.negate());
       assertThat(i + "," + -j, t, is(equalTo(neg)));
     }
   }
 }
Exemplo n.º 11
0
 /** Test method for {@link GroupElement#dbl()}. */
 @Test
 public void testDbl() {
   GroupElement B = ed25519.getB();
   // 2 * B = B + B
   assertThat(B.dbl(), is(equalTo(B.add(B.toCached()))));
 }
Exemplo n.º 12
0
 /** Test method for {@link GroupElement#precompute(boolean)}. */
 @Test
 public void testPrecompute() {
   GroupElement B = ed25519.getB();
   assertThat(B.precmp, is(equalTo(PrecomputationTestVectors.testPrecmp)));
   assertThat(B.dblPrecmp, is(equalTo(PrecomputationTestVectors.testDblPrecmp)));
 }
Exemplo n.º 13
0
/** @author str4d Additional tests by NEM project team. */
public class GroupElementTest {
  static final byte[] BYTES_ZEROZERO =
      Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
  static final byte[] BYTES_ONEONE =
      Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000080");
  static final byte[] BYTES_TENZERO =
      Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
  static final byte[] BYTES_ONETEN =
      Utils.hexToBytes("0a00000000000000000000000000000000000000000000000000000000000080");

  static final EdDSANamedCurveSpec ed25519 = EdDSANamedCurveTable.getByName("ed25519-sha-512");
  static final Curve curve = ed25519.getCurve();

  static final FieldElement ZERO = curve.getField().ZERO;
  static final FieldElement ONE = curve.getField().ONE;
  static final FieldElement TWO = curve.getField().TWO;
  static final FieldElement TEN =
      curve
          .getField()
          .fromByteArray(
              Utils.hexToBytes("0a00000000000000000000000000000000000000000000000000000000000000"));

  static final GroupElement P2_ZERO = GroupElement.p2(curve, ZERO, ONE, ONE);

  static final FieldElement[] PKR =
      new FieldElement[] {
        curve
            .getField()
            .fromByteArray(
                Utils.hexToBytes(
                    "5849722e338aced7b50c7f0e9328f9a10c847b08e40af5c5b0577b0fd8984f15")),
        curve
            .getField()
            .fromByteArray(
                Utils.hexToBytes(
                    "3b6a27bcceb6a42d62a3a8d02a6f0d73653215771de243a63ac048a18b59da29"))
      };
  static final byte[] BYTES_PKR =
      Utils.hexToBytes("3b6a27bcceb6a42d62a3a8d02a6f0d73653215771de243a63ac048a18b59da29");

  @Rule public ExpectedException exception = ExpectedException.none();

  /** Test method for {@link GroupElement#p2(Curve, FieldElement, FieldElement, FieldElement)}. */
  @Test
  public void testP2() {
    final GroupElement t = GroupElement.p2(curve, ZERO, ONE, ONE);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.P2));
    assertThat(t.X, is(ZERO));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ONE));
    assertThat(t.T, is((FieldElement) null));
  }

  /**
   * Test method for {@link GroupElement#p3(Curve, FieldElement, FieldElement, FieldElement,
   * FieldElement)}.
   */
  @Test
  public void testP3() {
    final GroupElement t = GroupElement.p3(curve, ZERO, ONE, ONE, ZERO);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.P3));
    assertThat(t.X, is(ZERO));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ONE));
    assertThat(t.T, is(ZERO));
  }

  /**
   * Test method for {@link GroupElement#p1p1(Curve, FieldElement, FieldElement, FieldElement,
   * FieldElement)}.
   */
  @Test
  public void testP1p1() {
    final GroupElement t = GroupElement.p1p1(curve, ZERO, ONE, ONE, ONE);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.P1P1));
    assertThat(t.X, is(ZERO));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ONE));
    assertThat(t.T, is(ONE));
  }

  /**
   * Test method for {@link GroupElement#precomp(Curve, FieldElement, FieldElement, FieldElement)}.
   */
  @Test
  public void testPrecomp() {
    final GroupElement t = GroupElement.precomp(curve, ONE, ONE, ZERO);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.PRECOMP));
    assertThat(t.X, is(ONE));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ZERO));
    assertThat(t.T, is((FieldElement) null));
  }

  /**
   * Test method for {@link GroupElement#cached(Curve, FieldElement, FieldElement, FieldElement,
   * FieldElement)}.
   */
  @Test
  public void testCached() {
    final GroupElement t = GroupElement.cached(curve, ONE, ONE, ONE, ZERO);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.CACHED));
    assertThat(t.X, is(ONE));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ONE));
    assertThat(t.T, is(ZERO));
  }

  /**
   * Test method for {@link GroupElement#GroupElement(Curve, GroupElement.Representation,
   * FieldElement, FieldElement, FieldElement, FieldElement)}.
   */
  @Test
  public void
      testGroupElementCurveRepresentationFieldElementFieldElementFieldElementFieldElement() {
    final GroupElement t =
        new GroupElement(curve, GroupElement.Representation.P3, ZERO, ONE, ONE, ZERO);
    assertThat(t.curve, is(equalTo(curve)));
    assertThat(t.repr, is(GroupElement.Representation.P3));
    assertThat(t.X, is(ZERO));
    assertThat(t.Y, is(ONE));
    assertThat(t.Z, is(ONE));
    assertThat(t.T, is(ZERO));
  }

  /**
   * Tests {@link GroupElement#GroupElement(Curve, byte[])} and {@link GroupElement#toByteArray()}
   * against valid public keys.
   */
  @Test
  public void testToAndFromByteArray() {
    GroupElement t;
    for (Ed25519TestVectors.TestTuple testCase : Ed25519TestVectors.testCases) {
      t = new GroupElement(curve, testCase.pk);
      assertThat(
          "Test case " + testCase.caseNum + " failed", t.toByteArray(), is(equalTo(testCase.pk)));
    }
  }

  /** Test method for {@link GroupElement#GroupElement(Curve, byte[])}. */
  @Test
  public void testGroupElementByteArray() {
    final GroupElement t = new GroupElement(curve, BYTES_PKR);
    final GroupElement s = GroupElement.p3(curve, PKR[0], PKR[1], ONE, PKR[0].multiply(PKR[1]));
    assertThat(t, is(equalTo(s)));
  }

  @Test
  public void constructorUsingByteArrayReturnsExpectedResult() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();
      final byte[] bytes = g.toByteArray();

      // Act:
      final GroupElement h1 = new GroupElement(curve, bytes);
      final GroupElement h2 = MathUtils.toGroupElement(bytes);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
    }
  }

  /**
   * Test method for {@link GroupElement#toByteArray()}. TODO 20141001 BR: why test with points
   * which are not on the curve?
   */
  @Test
  public void testToByteArray() {
    byte[] zerozero = GroupElement.p2(curve, ZERO, ZERO, ONE).toByteArray();
    assertThat(zerozero.length, is(equalTo(BYTES_ZEROZERO.length)));
    assertThat(zerozero, is(equalTo(BYTES_ZEROZERO)));

    byte[] oneone = GroupElement.p2(curve, ONE, ONE, ONE).toByteArray();
    assertThat(oneone.length, is(equalTo(BYTES_ONEONE.length)));
    assertThat(oneone, is(equalTo(BYTES_ONEONE)));

    byte[] tenzero = GroupElement.p2(curve, TEN, ZERO, ONE).toByteArray();
    assertThat(tenzero.length, is(equalTo(BYTES_TENZERO.length)));
    assertThat(tenzero, is(equalTo(BYTES_TENZERO)));

    byte[] oneten = GroupElement.p2(curve, ONE, TEN, ONE).toByteArray();
    assertThat(oneten.length, is(equalTo(BYTES_ONETEN.length)));
    assertThat(oneten, is(equalTo(BYTES_ONETEN)));

    byte[] pkr = GroupElement.p2(curve, PKR[0], PKR[1], ONE).toByteArray();
    assertThat(pkr.length, is(equalTo(BYTES_PKR.length)));
    assertThat(pkr, is(equalTo(BYTES_PKR)));
  }

  @Test
  public void toByteArrayReturnsExpectedResult() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final byte[] gBytes = g.toByteArray();
      final byte[] bytes = MathUtils.toByteArray(MathUtils.toBigInteger(g.getY()));
      if (MathUtils.toBigInteger(g.getX()).mod(new BigInteger("2")).equals(BigInteger.ONE)) {
        bytes[31] |= 0x80;
      }

      // Assert:
      Assert.assertThat(Arrays.equals(gBytes, bytes), IsEqual.equalTo(true));
    }
  }

  // region toX where X is the representation

  /** Test method for {@link GroupElement#toP2()}. */
  @Test
  public void testToP2() {
    GroupElement p3zero = curve.getZero(GroupElement.Representation.P3);
    GroupElement t = p3zero.toP2();
    assertThat(t.repr, is(GroupElement.Representation.P2));
    assertThat(t.X, is(p3zero.X));
    assertThat(t.Y, is(p3zero.Y));
    assertThat(t.Z, is(p3zero.Z));
    assertThat(t.T, is((FieldElement) null));

    GroupElement B = ed25519.getB();
    t = B.toP2();
    assertThat(t.repr, is(GroupElement.Representation.P2));
    assertThat(t.X, is(B.X));
    assertThat(t.Y, is(B.Y));
    assertThat(t.Z, is(B.Z));
    assertThat(t.T, is((FieldElement) null));
  }

  @Test(expected = IllegalArgumentException.class)
  public void toP2ThrowsIfGroupElementHasPrecompRepresentation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.PRECOMP);

    // Assert:
    g.toP2();
  }

  @Test(expected = IllegalArgumentException.class)
  public void toP2ThrowsIfGroupElementHasCachedRepresentation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.CACHED);

    // Assert:
    g.toP2();
  }

  @Test
  public void toP2ReturnsExpectedResultIfGroupElementHasP2Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g =
          MathUtils.toRepresentation(
              MathUtils.getRandomGroupElement(), GroupElement.Representation.P2);

      // Act:
      final GroupElement h = g.toP2();

      // Assert:
      Assert.assertThat(h, IsEqual.equalTo(g));
      Assert.assertThat(h.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.P2));
      Assert.assertThat(h.getX(), IsEqual.equalTo(g.getX()));
      Assert.assertThat(h.getY(), IsEqual.equalTo(g.getY()));
      Assert.assertThat(h.getZ(), IsEqual.equalTo(g.getZ()));
      Assert.assertThat(h.getT(), IsEqual.equalTo(null));
    }
  }

  @Test
  public void toP2ReturnsExpectedResultIfGroupElementHasP3Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g.toP2();
      final GroupElement h2 = MathUtils.toRepresentation(g, GroupElement.Representation.P2);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
      Assert.assertThat(h1.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.P2));
      Assert.assertThat(h1.getX(), IsEqual.equalTo(g.getX()));
      Assert.assertThat(h1.getY(), IsEqual.equalTo(g.getY()));
      Assert.assertThat(h1.getZ(), IsEqual.equalTo(g.getZ()));
      Assert.assertThat(h1.getT(), IsEqual.equalTo(null));
    }
  }

  @Test
  public void toP2ReturnsExpectedResultIfGroupElementHasP1P1Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g =
          MathUtils.toRepresentation(
              MathUtils.getRandomGroupElement(), GroupElement.Representation.P1P1);

      // Act:
      final GroupElement h1 = g.toP2();
      final GroupElement h2 = MathUtils.toRepresentation(g, GroupElement.Representation.P2);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
      Assert.assertThat(h1.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.P2));
      Assert.assertThat(h1.getX(), IsEqual.equalTo(g.getX().multiply(g.getT())));
      Assert.assertThat(h1.getY(), IsEqual.equalTo(g.getY().multiply(g.getZ())));
      Assert.assertThat(h1.getZ(), IsEqual.equalTo(g.getZ().multiply(g.getT())));
      Assert.assertThat(h1.getT(), IsEqual.equalTo(null));
    }
  }

  @Test(expected = IllegalArgumentException.class)
  public void toP3ThrowsIfGroupElementHasP2Representation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.P2);

    // Assert:
    g.toP3();
  }

  @Test(expected = IllegalArgumentException.class)
  public void toP3ThrowsIfGroupElementHasPrecompRepresentation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.PRECOMP);

    // Assert:
    g.toP3();
  }

  @Test(expected = IllegalArgumentException.class)
  public void toP3ThrowsIfGroupElementHasCachedRepresentation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.CACHED);

    // Assert:
    g.toP3();
  }

  @Test
  public void toP3ReturnsExpectedResultIfGroupElementHasP1P1Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g =
          MathUtils.toRepresentation(
              MathUtils.getRandomGroupElement(), GroupElement.Representation.P1P1);

      // Act:
      final GroupElement h1 = g.toP3();
      final GroupElement h2 = MathUtils.toRepresentation(g, GroupElement.Representation.P3);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
      Assert.assertThat(h1.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.P3));
      Assert.assertThat(h1.getX(), IsEqual.equalTo(g.getX().multiply(g.getT())));
      Assert.assertThat(h1.getY(), IsEqual.equalTo(g.getY().multiply(g.getZ())));
      Assert.assertThat(h1.getZ(), IsEqual.equalTo(g.getZ().multiply(g.getT())));
      Assert.assertThat(h1.getT(), IsEqual.equalTo(g.getX().multiply(g.getY())));
    }
  }

  @Test
  public void toP3ReturnsExpectedResultIfGroupElementHasP3Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h = g.toP3();

      // Assert:
      Assert.assertThat(h, IsEqual.equalTo(g));
      Assert.assertThat(h.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.P3));
      Assert.assertThat(h, IsEqual.equalTo(g));
      Assert.assertThat(h.getX(), IsEqual.equalTo(g.getX()));
      Assert.assertThat(h.getY(), IsEqual.equalTo(g.getY()));
      Assert.assertThat(h.getZ(), IsEqual.equalTo(g.getZ()));
      Assert.assertThat(h.getT(), IsEqual.equalTo(g.getT()));
    }
  }

  @Test(expected = IllegalArgumentException.class)
  public void toCachedThrowsIfGroupElementHasP2Representation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.P2);

    // Assert:
    g.toCached();
  }

  @Test(expected = IllegalArgumentException.class)
  public void toCachedThrowsIfGroupElementHasPrecompRepresentation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.PRECOMP);

    // Assert:
    g.toCached();
  }

  @Test(expected = IllegalArgumentException.class)
  public void toCachedThrowsIfGroupElementHasP1P1Representation() {
    // Arrange:
    final GroupElement g =
        MathUtils.toRepresentation(
            MathUtils.getRandomGroupElement(), GroupElement.Representation.P1P1);

    // Assert:
    g.toCached();
  }

  @Test
  public void toCachedReturnsExpectedResultIfGroupElementHasCachedRepresentation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g =
          MathUtils.toRepresentation(
              MathUtils.getRandomGroupElement(), GroupElement.Representation.CACHED);

      // Act:
      final GroupElement h = g.toCached();

      // Assert:
      Assert.assertThat(h, IsEqual.equalTo(g));
      Assert.assertThat(h.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.CACHED));
      Assert.assertThat(h, IsEqual.equalTo(g));
      Assert.assertThat(h.getX(), IsEqual.equalTo(g.getX()));
      Assert.assertThat(h.getY(), IsEqual.equalTo(g.getY()));
      Assert.assertThat(h.getZ(), IsEqual.equalTo(g.getZ()));
      Assert.assertThat(h.getT(), IsEqual.equalTo(g.getT()));
    }
  }

  @Test
  public void toCachedReturnsExpectedResultIfGroupElementHasP3Representation() {
    for (int i = 0; i < 10; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g.toCached();
      final GroupElement h2 = MathUtils.toRepresentation(g, GroupElement.Representation.CACHED);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
      Assert.assertThat(
          h1.getRepresentation(), IsEqual.equalTo(GroupElement.Representation.CACHED));
      Assert.assertThat(h1, IsEqual.equalTo(g));
      Assert.assertThat(h1.getX(), IsEqual.equalTo(g.getY().add(g.getX())));
      Assert.assertThat(h1.getY(), IsEqual.equalTo(g.getY().subtract(g.getX())));
      Assert.assertThat(h1.getZ(), IsEqual.equalTo(g.getZ()));
      Assert.assertThat(h1.getT(), IsEqual.equalTo(g.getT().multiply(curve.get2D())));
    }
  }

  // endregion

  /** Test method for {@link GroupElement#precompute(boolean)}. */
  @Test
  public void testPrecompute() {
    GroupElement B = ed25519.getB();
    assertThat(B.precmp, is(equalTo(PrecomputationTestVectors.testPrecmp)));
    assertThat(B.dblPrecmp, is(equalTo(PrecomputationTestVectors.testDblPrecmp)));
  }

  @Test
  public void precomputedTableContainsExpectedGroupElements() {
    // Arrange:
    GroupElement g = ed25519.getB();

    // Act + Assert:
    for (int i = 0; i < 32; i++) {
      GroupElement h = g;
      for (int j = 0; j < 8; j++) {
        Assert.assertThat(
            MathUtils.toRepresentation(h, GroupElement.Representation.PRECOMP),
            IsEqual.equalTo(ed25519.getB().precmp[i][j]));
        h = MathUtils.addGroupElements(h, g);
      }
      for (int k = 0; k < 8; k++) {
        g = MathUtils.addGroupElements(g, g);
      }
    }
  }

  @Test
  public void dblPrecomputedTableContainsExpectedGroupElements() {
    // Arrange:
    GroupElement g = ed25519.getB();
    GroupElement h = MathUtils.addGroupElements(g, g);

    // Act + Assert:
    for (int i = 0; i < 8; i++) {
      Assert.assertThat(
          MathUtils.toRepresentation(g, GroupElement.Representation.PRECOMP),
          IsEqual.equalTo(ed25519.getB().dblPrecmp[i]));
      g = MathUtils.addGroupElements(g, h);
    }
  }

  /** Test method for {@link GroupElement#dbl()}. */
  @Test
  public void testDbl() {
    GroupElement B = ed25519.getB();
    // 2 * B = B + B
    assertThat(B.dbl(), is(equalTo(B.add(B.toCached()))));
  }

  @Test
  public void dblReturnsExpectedResult() {
    for (int i = 0; i < 1000; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g.dbl();
      final GroupElement h2 = MathUtils.doubleGroupElement(g);

      // Assert:
      Assert.assertThat(h2, IsEqual.equalTo(h1));
    }
  }

  @Test
  public void addingNeutralGroupElementDoesNotChangeGroupElement() {
    final GroupElement neutral =
        GroupElement.p3(
            curve,
            curve.getField().ZERO,
            curve.getField().ONE,
            curve.getField().ONE,
            curve.getField().ZERO);
    for (int i = 0; i < 1000; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g.add(neutral.toCached());
      final GroupElement h2 = neutral.add(g.toCached());

      // Assert:
      Assert.assertThat(g, IsEqual.equalTo(h1));
      Assert.assertThat(g, IsEqual.equalTo(h2));
    }
  }

  @Test
  public void addReturnsExpectedResult() {
    for (int i = 0; i < 1000; i++) {
      // Arrange:
      final GroupElement g1 = MathUtils.getRandomGroupElement();
      final GroupElement g2 = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g1.add(g2.toCached());
      final GroupElement h2 = MathUtils.addGroupElements(g1, g2);

      // Assert:
      Assert.assertThat(h2, IsEqual.equalTo(h1));
    }
  }

  @Test
  public void subReturnsExpectedResult() {
    for (int i = 0; i < 1000; i++) {
      // Arrange:
      final GroupElement g1 = MathUtils.getRandomGroupElement();
      final GroupElement g2 = MathUtils.getRandomGroupElement();

      // Act:
      final GroupElement h1 = g1.sub(g2.toCached());
      final GroupElement h2 = MathUtils.addGroupElements(g1, MathUtils.negateGroupElement(g2));

      // Assert:
      Assert.assertThat(h2, IsEqual.equalTo(h1));
    }
  }

  // region hashCode / equals
  /** Test method for {@link GroupElement#equals(java.lang.Object)}. */
  @Test
  public void testEqualsObject() {
    assertThat(GroupElement.p2(curve, ZERO, ONE, ONE), is(equalTo(P2_ZERO)));
  }

  @Test
  public void equalsOnlyReturnsTrueForEquivalentObjects() {
    // Arrange:
    final GroupElement g1 = MathUtils.getRandomGroupElement();
    final GroupElement g2 = MathUtils.toRepresentation(g1, GroupElement.Representation.P2);
    final GroupElement g3 = MathUtils.toRepresentation(g1, GroupElement.Representation.CACHED);
    final GroupElement g4 = MathUtils.toRepresentation(g1, GroupElement.Representation.P1P1);
    final GroupElement g5 = MathUtils.getRandomGroupElement();

    // Assert
    Assert.assertThat(g2, IsEqual.equalTo(g1));
    Assert.assertThat(g3, IsEqual.equalTo(g1));
    Assert.assertThat(g1, IsEqual.equalTo(g4));
    Assert.assertThat(g1, IsNot.not(IsEqual.equalTo(g5)));
    Assert.assertThat(g2, IsNot.not(IsEqual.equalTo(g5)));
    Assert.assertThat(g3, IsNot.not(IsEqual.equalTo(g5)));
    Assert.assertThat(g5, IsNot.not(IsEqual.equalTo(g4)));
  }

  @Test
  public void hashCodesAreEqualForEquivalentObjects() {
    // Arrange:
    final GroupElement g1 = MathUtils.getRandomGroupElement();
    final GroupElement g2 = MathUtils.toRepresentation(g1, GroupElement.Representation.P2);
    final GroupElement g3 = MathUtils.toRepresentation(g1, GroupElement.Representation.P1P1);
    final GroupElement g4 = MathUtils.getRandomGroupElement();

    // Assert
    Assert.assertThat(g2.hashCode(), IsEqual.equalTo(g1.hashCode()));
    Assert.assertThat(g3.hashCode(), IsEqual.equalTo(g1.hashCode()));
    Assert.assertThat(g1.hashCode(), IsNot.not(IsEqual.equalTo(g4.hashCode())));
    Assert.assertThat(g2.hashCode(), IsNot.not(IsEqual.equalTo(g4.hashCode())));
    Assert.assertThat(g3.hashCode(), IsNot.not(IsEqual.equalTo(g4.hashCode())));
  }

  // endregion

  static final byte[] BYTES_ZERO =
      Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
  static final byte[] BYTES_ONE =
      Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000000");
  static final byte[] BYTES_42 =
      Utils.hexToBytes("2A00000000000000000000000000000000000000000000000000000000000000");
  static final byte[] BYTES_1234567890 =
      Utils.hexToBytes("D202964900000000000000000000000000000000000000000000000000000000");

  static final byte[] RADIX16_ZERO =
      Utils.hexToBytes(
          "00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000");
  static final byte[] RADIX16_ONE =
      Utils.hexToBytes(
          "01000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000");
  static final byte[] RADIX16_42 =
      Utils.hexToBytes(
          "FA030000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000");

  /** Test method for {@link GroupElement#toRadix16(byte[])}. */
  @Test
  public void testToRadix16() {
    assertThat(GroupElement.toRadix16(BYTES_ZERO), is(RADIX16_ZERO));
    assertThat(GroupElement.toRadix16(BYTES_ONE), is(RADIX16_ONE));
    assertThat(GroupElement.toRadix16(BYTES_42), is(RADIX16_42));

    byte[] from1234567890 = GroupElement.toRadix16(BYTES_1234567890);
    int total = 0;
    for (int i = 0; i < from1234567890.length; i++) {
      assertThat(from1234567890[i], is(greaterThanOrEqualTo((byte) -8)));
      assertThat(from1234567890[i], is(lessThanOrEqualTo((byte) 8)));
      total += from1234567890[i] * Math.pow(16, i);
    }
    assertThat(total, is(1234567890));

    byte[] pkrR16 = GroupElement.toRadix16(BYTES_PKR);
    for (int i = 0; i < pkrR16.length; i++) {
      assertThat(pkrR16[i], is(greaterThanOrEqualTo((byte) -8)));
      assertThat(pkrR16[i], is(lessThanOrEqualTo((byte) 8)));
    }
  }

  /** Test method for {@link GroupElement#cmov(GroupElement, int)}. */
  @Test
  public void testCmov() {
    GroupElement a = curve.getZero(GroupElement.Representation.PRECOMP);
    GroupElement b = GroupElement.precomp(curve, TWO, ZERO, TEN);
    assertThat(a.cmov(b, 0), is(equalTo(a)));
    assertThat(a.cmov(b, 1), is(equalTo(b)));
  }

  /** Test method for {@link GroupElement#select(int, int)}. */
  @Test
  public void testSelect() {
    GroupElement B = ed25519.getB();
    for (int i = 0; i < 32; i++) {
      // 16^i 0 B
      assertThat(
          i + ",0", B.select(i, 0), is(equalTo(GroupElement.precomp(curve, ONE, ONE, ZERO))));
      for (int j = 1; j < 8; j++) {
        // 16^i r_i B
        GroupElement t = B.select(i, j);
        assertThat(i + "," + j, t, is(equalTo(B.precmp[i][j - 1])));
        // -16^i r_i B
        t = B.select(i, -j);
        GroupElement neg =
            GroupElement.precomp(
                curve, B.precmp[i][j - 1].Y, B.precmp[i][j - 1].X, B.precmp[i][j - 1].Z.negate());
        assertThat(i + "," + -j, t, is(equalTo(neg)));
      }
    }
  }

  // region scalar multiplication
  /**
   * Test method for {@link GroupElement#scalarMultiply(byte[])}. Test values generated with Python
   * Ed25519 implementation.
   */
  @Test
  public void testScalarMultiplyByteArray() {
    // Little-endian
    byte[] zero =
        Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
    byte[] one =
        Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000000");
    byte[] two =
        Utils.hexToBytes("0200000000000000000000000000000000000000000000000000000000000000");
    byte[] a = Utils.hexToBytes("d072f8dd9c07fa7bc8d22a4b325d26301ee9202f6db89aa7c3731529e37e437c");
    GroupElement A =
        new GroupElement(
            curve,
            Utils.hexToBytes("d4cf8595571830644bd14af416954d09ab7159751ad9e0f7a6cbd92379e71a66"));

    assertThat(
        "scalarMultiply(0) failed",
        ed25519.getB().scalarMultiply(zero),
        is(equalTo(curve.getZero(GroupElement.Representation.P3))));
    assertThat(
        "scalarMultiply(1) failed",
        ed25519.getB().scalarMultiply(one),
        is(equalTo(ed25519.getB())));
    assertThat(
        "scalarMultiply(2) failed",
        ed25519.getB().scalarMultiply(two),
        is(equalTo(ed25519.getB().dbl())));

    assertThat("scalarMultiply(a) failed", ed25519.getB().scalarMultiply(a), is(equalTo(A)));
  }

  @Test
  public void scalarMultiplyBasePointWithZeroReturnsNeutralElement() {
    // Arrange:
    final GroupElement basePoint = ed25519.getB();

    // Act:
    final GroupElement g = basePoint.scalarMultiply(curve.getField().ZERO.toByteArray());

    // Assert:
    Assert.assertThat(curve.getZero(GroupElement.Representation.P3), IsEqual.equalTo(g));
  }

  @Test
  public void scalarMultiplyBasePointWithOneReturnsBasePoint() {
    // Arrange:
    final GroupElement basePoint = ed25519.getB();

    // Act:
    final GroupElement g = basePoint.scalarMultiply(curve.getField().ONE.toByteArray());

    // Assert:
    Assert.assertThat(basePoint, IsEqual.equalTo(g));
  }

  // This test is slow (~6s) due to math utils using an inferior algorithm to calculate the result.
  @Test
  public void scalarMultiplyBasePointReturnsExpectedResult() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement basePoint = ed25519.getB();
      final FieldElement f = MathUtils.getRandomFieldElement();

      // Act:
      final GroupElement g = basePoint.scalarMultiply(f.toByteArray());
      final GroupElement h = MathUtils.scalarMultiplyGroupElement(basePoint, f);

      // Assert:
      Assert.assertThat(g, IsEqual.equalTo(h));
    }
  }

  @Test
  public void testDoubleScalarMultiplyVariableTime() {
    // Little-endian
    byte[] zero =
        Utils.hexToBytes("0000000000000000000000000000000000000000000000000000000000000000");
    byte[] one =
        Utils.hexToBytes("0100000000000000000000000000000000000000000000000000000000000000");
    byte[] two =
        Utils.hexToBytes("0200000000000000000000000000000000000000000000000000000000000000");
    byte[] a = Utils.hexToBytes("d072f8dd9c07fa7bc8d22a4b325d26301ee9202f6db89aa7c3731529e37e437c");
    GroupElement A =
        new GroupElement(
            curve,
            Utils.hexToBytes("d4cf8595571830644bd14af416954d09ab7159751ad9e0f7a6cbd92379e71a66"));
    GroupElement B = ed25519.getB();
    GroupElement geZero = curve.getZero(GroupElement.Representation.P3);
    geZero.precompute(false);

    // 0 * GE(0) + 0 * GE(0) = GE(0)
    assertThat(geZero.doubleScalarMultiplyVariableTime(geZero, zero, zero), is(equalTo(geZero)));
    // 0 * GE(0) + 0 * B = GE(0)
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, zero, zero), is(equalTo(geZero)));
    // 1 * GE(0) + 0 * B = GE(0)
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, one, zero), is(equalTo(geZero)));
    // 1 * GE(0) + 1 * B = B
    assertThat(B.doubleScalarMultiplyVariableTime(geZero, one, one), is(equalTo(B)));
    // 1 * B + 1 * B = 2 * B
    assertThat(B.doubleScalarMultiplyVariableTime(B, one, one), is(equalTo(B.dbl())));
    // 1 * B + 2 * B = 3 * B
    assertThat(
        B.doubleScalarMultiplyVariableTime(B, one, two),
        is(equalTo(B.dbl().toP3().add(B.toCached()))));
    // 2 * B + 2 * B = 4 * B
    assertThat(B.doubleScalarMultiplyVariableTime(B, two, two), is(equalTo(B.dbl().toP3().dbl())));

    // 0 * B + a * B = A
    assertThat(B.doubleScalarMultiplyVariableTime(B, zero, a), is(equalTo(A)));
    // a * B + 0 * B = A
    assertThat(B.doubleScalarMultiplyVariableTime(B, a, zero), is(equalTo(A)));
    // a * B + a * B = 2 * A
    assertThat(B.doubleScalarMultiplyVariableTime(B, a, a), is(equalTo(A.dbl())));
  }

  // This test is slow (~6s) due to math utils using an inferior algorithm to calculate the result.
  @Test
  public void doubleScalarMultiplyVariableTimeReturnsExpectedResult() {
    for (int i = 0; i < 50; i++) {
      // Arrange:
      final GroupElement basePoint = ed25519.getB();
      final GroupElement g = MathUtils.getRandomGroupElement();
      g.precompute(false);
      final FieldElement f1 = MathUtils.getRandomFieldElement();
      final FieldElement f2 = MathUtils.getRandomFieldElement();

      // Act:
      final GroupElement h1 =
          basePoint.doubleScalarMultiplyVariableTime(g, f2.toByteArray(), f1.toByteArray());
      final GroupElement h2 = MathUtils.doubleScalarMultiplyGroupElements(basePoint, f1, g, f2);

      // Assert:
      Assert.assertThat(h1, IsEqual.equalTo(h2));
    }
  }

  // endregion

  /** Test method for {@link GroupElement#isOnCurve(Curve)}. */
  @Test
  public void testIsOnCurve() {
    assertThat(P2_ZERO.isOnCurve(curve), is(true));
    assertThat(GroupElement.p2(curve, ZERO, ZERO, ONE).isOnCurve(curve), is(false));
    assertThat(GroupElement.p2(curve, ONE, ONE, ONE).isOnCurve(curve), is(false));
    assertThat(GroupElement.p2(curve, TEN, ZERO, ONE).isOnCurve(curve), is(false));
    assertThat(GroupElement.p2(curve, ONE, TEN, ONE).isOnCurve(curve), is(false));
    assertThat(GroupElement.p2(curve, PKR[0], PKR[1], ONE).isOnCurve(curve), is(true));
  }

  @Test
  public void isOnCurveReturnsTrueForPointsOnTheCurve() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();

      // Assert:
      Assert.assertThat(g.isOnCurve(), IsEqual.equalTo(true));
    }
  }

  @Test
  public void isOnCurveReturnsFalseForPointsNotOnTheCurve() {
    for (int i = 0; i < 100; i++) {
      // Arrange:
      final GroupElement g = MathUtils.getRandomGroupElement();
      final GroupElement h =
          GroupElement.p2(curve, g.getX(), g.getY(), g.getZ().multiply(curve.getField().TWO));

      // Assert (can only fail for 5*Z^2=1):
      Assert.assertThat(h.isOnCurve(), IsEqual.equalTo(false));
    }
  }
}