diff --git a/src/crypto/elliptic/p224.go b/src/crypto/elliptic/p224.go
index 2ea63f3..8c76021 100644
--- a/src/crypto/elliptic/p224.go
+++ b/src/crypto/elliptic/p224.go
@@ -386,10 +386,11 @@
 // p224Contract converts a FieldElement to its unique, minimal form.
 //
 // On entry, in[i] < 2**29
-// On exit, in[i] < 2**28
+// On exit, out[i] < 2**28 and out < p
 func p224Contract(out, in *p224FieldElement) {
 	copy(out[:], in[:])
 
+	// First, carry the bits above 28 to the higher limb.
 	for i := 0; i < 7; i++ {
 		out[i+1] += out[i] >> 28
 		out[i] &= bottom28Bits
@@ -397,10 +398,13 @@
 	top := out[7] >> 28
 	out[7] &= bottom28Bits
 
+	// Use the reduction identity to carry the overflow.
+	//
+	//   a + top * 2²²⁴ = a + top * 2⁹⁶ - top
 	out[0] -= top
 	out[3] += top << 12
 
-	// We may just have made out[i] negative. So we carry down. If we made
+	// We may just have made out[0] negative. So we carry down. If we made
 	// out[0] negative then we know that out[3] is sufficiently positive
 	// because we just added to it.
 	for i := 0; i < 3; i++ {
@@ -425,13 +429,12 @@
 	// There are two cases to consider for out[3]:
 	//   1) The first time that we eliminated top, we didn't push out[3] over
 	//      2**28. In this case, the partial carry chain didn't change any values
-	//      and top is zero.
+	//      and top is now zero.
 	//   2) We did push out[3] over 2**28 the first time that we eliminated top.
-	//      The first value of top was in [0..16), therefore, prior to eliminating
-	//      the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after
-	//      overflowing and being reduced by the second carry chain, out[3] <=
-	//      0xf000. Thus it cannot have overflowed when we eliminated top for the
-	//      second time.
+	//      The first value of top was in [0..2], therefore, after overflowing
+	//      and being reduced by the second carry chain, out[3] <= 2<<12 - 1.
+	// In both cases, out[3] cannot have overflowed when we eliminated top for
+	// the second time.
 
 	// Again, we may just have made out[0] negative, so do the same carry down.
 	// As before, if we made out[0] negative then we know that out[3] is
@@ -470,12 +473,11 @@
 	bottom3NonZero |= bottom3NonZero >> 1
 	bottom3NonZero = uint32(int32(bottom3NonZero<<31) >> 31)
 
-	// Everything depends on the value of out[3].
-	//    If it's > 0xffff000 and top4AllOnes != 0 then the whole value is >= p
-	//    If it's = 0xffff000 and top4AllOnes != 0 and bottom3NonZero != 0,
-	//      then the whole value is >= p
+	// Assuming top4AllOnes != 0, everything depends on the value of out[3].
+	//    If it's > 0xffff000 then the whole value is > p
+	//    If it's = 0xffff000 and bottom3NonZero != 0, then the whole value is >= p
 	//    If it's < 0xffff000, then the whole value is < p
-	n := out[3] - 0xffff000
+	n := 0xffff000 - out[3]
 	out3Equal := n
 	out3Equal |= out3Equal >> 16
 	out3Equal |= out3Equal >> 8
@@ -484,8 +486,8 @@
 	out3Equal |= out3Equal >> 1
 	out3Equal = ^uint32(int32(out3Equal<<31) >> 31)
 
-	// If out[3] > 0xffff000 then n's MSB will be zero.
-	out3GT := ^uint32(int32(n) >> 31)
+	// If out[3] > 0xffff000 then n's MSB will be one.
+	out3GT := uint32(int32(n) >> 31)
 
 	mask := top4AllOnes & ((out3Equal & bottom3NonZero) | out3GT)
 	out[0] -= 1 & mask
@@ -494,6 +496,15 @@
 	out[5] -= 0xfffffff & mask
 	out[6] -= 0xfffffff & mask
 	out[7] -= 0xfffffff & mask
+
+	// Do one final carry down, in case we made out[0] negative. One of
+	// out[0..3] needs to be positive and able to absorb the -1 or the value
+	// would have been < p, and the subtraction wouldn't have happened.
+	for i := 0; i < 3; i++ {
+		mask := uint32(int32(out[i]) >> 31)
+		out[i] += (1 << 28) & mask
+		out[i+1] -= 1 & mask
+	}
 }
 
 // Group element functions.
diff --git a/src/crypto/elliptic/p224_test.go b/src/crypto/elliptic/p224_test.go
index 8b4fa04..c3141b6 100644
--- a/src/crypto/elliptic/p224_test.go
+++ b/src/crypto/elliptic/p224_test.go
@@ -6,7 +6,11 @@
 
 import (
 	"math/big"
+	"math/bits"
+	"math/rand"
+	"reflect"
 	"testing"
+	"testing/quick"
 )
 
 var toFromBigTests = []string{
@@ -21,16 +25,16 @@
 	ret := new(big.Int)
 	tmp := new(big.Int)
 
-	for i := uint(0); i < 8; i++ {
+	for i := len(in) - 1; i >= 0; i-- {
+		ret.Lsh(ret, 28)
 		tmp.SetInt64(int64(in[i]))
-		tmp.Lsh(tmp, 28*i)
 		ret.Add(ret, tmp)
 	}
-	ret.Mod(ret, p224.P)
+	ret.Mod(ret, P224().Params().P)
 	return ret
 }
 
-func TestToFromBig(t *testing.T) {
+func TestP224ToFromBig(t *testing.T) {
 	for i, test := range toFromBigTests {
 		n, _ := new(big.Int).SetString(test, 16)
 		var x p224FieldElement
@@ -41,7 +45,270 @@
 		}
 		q := p224AlternativeToBig(&x)
 		if n.Cmp(q) != 0 {
-			t.Errorf("#%d: %x != %x (alternative)", i, n, m)
+			t.Errorf("#%d: %x != %x (alternative)", i, n, q)
 		}
 	}
 }
+
+// quickCheckConfig32 will make each quickcheck test run (32 * -quickchecks)
+// times. The default value of -quickchecks is 100.
+var quickCheckConfig32 = &quick.Config{MaxCountScale: 32}
+
+// weirdLimbs can be combined to generate a range of edge-case field elements.
+var weirdLimbs = [...]uint32{
+	0, 1, (1 << 29) - 1,
+	(1 << 12), (1 << 12) - 1,
+	(1 << 28), (1 << 28) - 1,
+}
+
+func generateLimb(rand *rand.Rand) uint32 {
+	const bottom29Bits = 0x1fffffff
+	n := rand.Intn(len(weirdLimbs) + 3)
+	switch n {
+	case len(weirdLimbs):
+		// Random value.
+		return uint32(rand.Int31n(1 << 29))
+	case len(weirdLimbs) + 1:
+		// Sum of two values.
+		k := generateLimb(rand) + generateLimb(rand)
+		return k & bottom29Bits
+	case len(weirdLimbs) + 2:
+		// Difference of two values.
+		k := generateLimb(rand) - generateLimb(rand)
+		return k & bottom29Bits
+	default:
+		return weirdLimbs[n]
+	}
+}
+
+func (p224FieldElement) Generate(rand *rand.Rand, size int) reflect.Value {
+	return reflect.ValueOf(p224FieldElement{
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+		generateLimb(rand),
+	})
+}
+
+func isInBounds(x *p224FieldElement) bool {
+	return bits.Len32(x[0]) <= 29 &&
+		bits.Len32(x[1]) <= 29 &&
+		bits.Len32(x[2]) <= 29 &&
+		bits.Len32(x[3]) <= 29 &&
+		bits.Len32(x[4]) <= 29 &&
+		bits.Len32(x[5]) <= 29 &&
+		bits.Len32(x[6]) <= 29 &&
+		bits.Len32(x[7]) <= 29
+}
+
+func TestP224Mul(t *testing.T) {
+	mulMatchesBigInt := func(a, b, out p224FieldElement) bool {
+		var tmp p224LargeFieldElement
+		p224Mul(&out, &a, &b, &tmp)
+
+		exp := new(big.Int).Mul(p224AlternativeToBig(&a), p224AlternativeToBig(&b))
+		exp.Mod(exp, P224().Params().P)
+		got := p224AlternativeToBig(&out)
+		if exp.Cmp(got) != 0 || !isInBounds(&out) {
+			t.Logf("a = %x", a)
+			t.Logf("b = %x", b)
+			t.Logf("p224Mul(a, b) = %x = %v", out, got)
+			t.Logf("a * b = %v", exp)
+			return false
+		}
+
+		return true
+	}
+
+	a := p224FieldElement{0xfffffff, 0xfffffff, 0xf00ffff, 0x20f, 0x0, 0x0, 0x0, 0x0}
+	b := p224FieldElement{1, 0, 0, 0, 0, 0, 0, 0}
+	if !mulMatchesBigInt(a, b, p224FieldElement{}) {
+		t.Fail()
+	}
+
+	if err := quick.Check(mulMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224Square(t *testing.T) {
+	squareMatchesBigInt := func(a, out p224FieldElement) bool {
+		var tmp p224LargeFieldElement
+		p224Square(&out, &a, &tmp)
+
+		exp := p224AlternativeToBig(&a)
+		exp.Mul(exp, exp)
+		exp.Mod(exp, P224().Params().P)
+		got := p224AlternativeToBig(&out)
+		if exp.Cmp(got) != 0 || !isInBounds(&out) {
+			t.Logf("a = %x", a)
+			t.Logf("p224Square(a, b) = %x = %v", out, got)
+			t.Logf("a * a = %v", exp)
+			return false
+		}
+
+		return true
+	}
+
+	if err := quick.Check(squareMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224Add(t *testing.T) {
+	addMatchesBigInt := func(a, b, out p224FieldElement) bool {
+		p224Add(&out, &a, &b)
+
+		exp := new(big.Int).Add(p224AlternativeToBig(&a), p224AlternativeToBig(&b))
+		exp.Mod(exp, P224().Params().P)
+		got := p224AlternativeToBig(&out)
+		if exp.Cmp(got) != 0 {
+			t.Logf("a = %x", a)
+			t.Logf("b = %x", b)
+			t.Logf("p224Add(a, b) = %x = %v", out, got)
+			t.Logf("a + b = %v", exp)
+			return false
+		}
+
+		return true
+	}
+
+	if err := quick.Check(addMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224Reduce(t *testing.T) {
+	reduceMatchesBigInt := func(a p224FieldElement) bool {
+		out := a
+		// TODO: generate higher values for functions like p224Reduce that are
+		// expected to work with higher input bounds.
+		p224Reduce(&out)
+
+		exp := p224AlternativeToBig(&a)
+		got := p224AlternativeToBig(&out)
+		if exp.Cmp(got) != 0 || !isInBounds(&out) {
+			t.Logf("a = %x = %v", a, exp)
+			t.Logf("p224Reduce(a) = %x = %v", out, got)
+			return false
+		}
+
+		return true
+	}
+
+	if err := quick.Check(reduceMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224Contract(t *testing.T) {
+	contractMatchesBigInt := func(a, out p224FieldElement) bool {
+		p224Contract(&out, &a)
+
+		exp := p224AlternativeToBig(&a)
+		got := p224AlternativeToBig(&out)
+		if exp.Cmp(got) != 0 {
+			t.Logf("a = %x = %v", a, exp)
+			t.Logf("p224Contract(a) = %x = %v", out, got)
+			return false
+		}
+
+		// Check that out < P.
+		for i := range p224P {
+			k := 8 - i - 1
+			if out[k] > p224P[k] {
+				t.Logf("p224Contract(a) = %x", out)
+				return false
+			}
+			if out[k] < p224P[k] {
+				return true
+			}
+		}
+		t.Logf("p224Contract(a) = %x", out)
+		return false
+	}
+
+	if !contractMatchesBigInt(p224P, p224FieldElement{}) {
+		t.Error("p224Contract(p) is broken")
+	}
+	pMinus1 := p224FieldElement{0, 0, 0, 0xffff000, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff}
+	if !contractMatchesBigInt(pMinus1, p224FieldElement{}) {
+		t.Error("p224Contract(p - 1) is broken")
+	}
+	// Check that we can handle input above p, but lowest limb zero.
+	a := p224FieldElement{0, 1, 0, 0xffff000, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff}
+	if !contractMatchesBigInt(a, p224FieldElement{}) {
+		t.Error("p224Contract(p + 2²⁸) is broken")
+	}
+	// Check that we can handle input above p, but lowest three limbs zero.
+	b := p224FieldElement{0, 0, 0, 0xffff001, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff}
+	if !contractMatchesBigInt(b, p224FieldElement{}) {
+		t.Error("p224Contract(p + 2⁸⁴) is broken")
+	}
+
+	if err := quick.Check(contractMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224IsZero(t *testing.T) {
+	if got := p224IsZero(&p224FieldElement{}); got != 1 {
+		t.Errorf("p224IsZero(0) = %d, expected 1", got)
+	}
+	if got := p224IsZero((*p224FieldElement)(&p224P)); got != 1 {
+		t.Errorf("p224IsZero(p) = %d, expected 1", got)
+	}
+	if got := p224IsZero(&p224FieldElement{1}); got != 0 {
+		t.Errorf("p224IsZero(1) = %d, expected 0", got)
+	}
+
+	isZeroMatchesBigInt := func(a p224FieldElement) bool {
+		isZero := p224IsZero(&a)
+
+		big := p224AlternativeToBig(&a)
+		if big.Sign() == 0 && isZero != 1 {
+			return false
+		}
+		if big.Sign() != 0 && isZero != 0 {
+			return false
+		}
+		return true
+	}
+
+	if err := quick.Check(isZeroMatchesBigInt, quickCheckConfig32); err != nil {
+		t.Error(err)
+	}
+}
+
+func TestP224Invert(t *testing.T) {
+	var out p224FieldElement
+
+	p224Invert(&out, &p224FieldElement{})
+	if got := p224IsZero(&out); got != 1 {
+		t.Errorf("p224Invert(0) = %x, expected 0", out)
+	}
+
+	p224Invert(&out, (*p224FieldElement)(&p224P))
+	if got := p224IsZero(&out); got != 1 {
+		t.Errorf("p224Invert(p) = %x, expected 0", out)
+	}
+
+	p224Invert(&out, &p224FieldElement{1})
+	p224Contract(&out, &out)
+	if out != (p224FieldElement{1}) {
+		t.Errorf("p224Invert(1) = %x, expected 1", out)
+	}
+
+	var tmp p224LargeFieldElement
+	a := p224FieldElement{1, 2, 3, 4, 5, 6, 7, 8}
+	p224Invert(&out, &a)
+	p224Mul(&out, &out, &a, &tmp)
+	p224Contract(&out, &out)
+	if out != (p224FieldElement{1}) {
+		t.Errorf("p224Invert(a) * a = %x, expected 1", out)
+	}
+}