diff --git a/openssl/aes_test.go b/openssl/aes_test.go
index c06fcc5..8428c39 100644
--- a/openssl/aes_test.go
+++ b/openssl/aes_test.go
@@ -9,7 +9,6 @@ package openssl
 import (
 	"bytes"
 	"crypto/cipher"
-	"math"
 	"testing"
 )
 
@@ -36,6 +35,7 @@ func TestNewGCMNonce(t *testing.T) {
 	if err != nil {
 		t.Errorf("expected no error for standard tag / nonce size, got: %#v", err)
 	}
+	c.finalize()
 }
 
 func TestSealAndOpen(t *testing.T) {
@@ -60,6 +60,7 @@ func TestSealAndOpen(t *testing.T) {
 	if !bytes.Equal(decrypted, plainText) {
 		t.Errorf("unexpected decrypted result\ngot: %#v\nexp: %#v", decrypted, plainText)
 	}
+	c.finalize()
 }
 
 func TestSealAndOpenAuthenticationError(t *testing.T) {
@@ -81,6 +82,7 @@ func TestSealAndOpenAuthenticationError(t *testing.T) {
 	if err != errOpen {
 		t.Errorf("expected authentication error, got: %#v", err)
 	}
+	c.finalize()
 }
 
 func assertPanic(t *testing.T, f func()) {
@@ -107,7 +109,11 @@ func TestSealPanic(t *testing.T) {
 		gcm.Seal(nil, make([]byte, gcmStandardNonceSize-1), []byte{0x01, 0x02, 0x03}, nil)
 	})
 	assertPanic(t, func() {
-		gcm.Seal(nil, make([]byte, gcmStandardNonceSize), make([]byte, math.MaxInt), nil)
+		// maxInt is implemented as math.MaxInt, but this constant
+		// is only available since go1.17.
+		// TODO: use math.MaxInt once go1.16 is no longer supported.
+		maxInt := int((^uint(0)) >> 1)
+		gcm.Seal(nil, make([]byte, gcmStandardNonceSize), make([]byte, maxInt), nil)
 	})
 }
 
diff --git a/openssl/ecdsa_test.go b/openssl/ecdsa_test.go
new file mode 100644
index 0000000..132a488
--- /dev/null
+++ b/openssl/ecdsa_test.go
@@ -0,0 +1,88 @@
+// Copyright (c) Microsoft Corporation.
+// Licensed under the MIT License.
+
+//go:build linux && !android
+// +build linux,!android
+
+package openssl
+
+import (
+	"crypto/ecdsa"
+	"crypto/elliptic"
+	"testing"
+)
+
+func testAllCurves(t *testing.T, f func(*testing.T, elliptic.Curve)) {
+	tests := []struct {
+		name  string
+		curve elliptic.Curve
+	}{
+		{"P256", elliptic.P256()},
+		{"P224", elliptic.P224()},
+		{"P384", elliptic.P384()},
+		{"P521", elliptic.P521()},
+	}
+	for _, test := range tests {
+		curve := test.curve
+		t.Run(test.name, func(t *testing.T) {
+			t.Parallel()
+			f(t, curve)
+		})
+	}
+}
+
+func TestECDSAKeyGeneration(t *testing.T) {
+	testAllCurves(t, testECDSAKeyGeneration)
+}
+
+func testECDSAKeyGeneration(t *testing.T, c elliptic.Curve) {
+	priv, err := generateKeycurve(c)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !c.IsOnCurve(priv.PublicKey.X, priv.PublicKey.Y) {
+		t.Errorf("public key invalid: %s", err)
+	}
+}
+
+func TestECDSASignAndVerify(t *testing.T) {
+	testAllCurves(t, testECDSASignAndVerify)
+}
+
+func testECDSASignAndVerify(t *testing.T, c elliptic.Curve) {
+	key, err := generateKeycurve(c)
+	if err != nil {
+		t.Fatal(err)
+	}
+
+	priv, err := NewPrivateKeyECDSA(key.Params().Name, key.X, key.Y, key.D)
+	if err != nil {
+		t.Fatal(err)
+	}
+	hashed := []byte("testing")
+	r, s, err := SignECDSA(priv, hashed)
+	if err != nil {
+		t.Errorf("error signing: %s", err)
+		return
+	}
+
+	pub, err := NewPublicKeyECDSA(key.Params().Name, key.X, key.Y)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !VerifyECDSA(pub, hashed, r, s) {
+		t.Errorf("Verify failed")
+	}
+	hashed[0] ^= 0xff
+	if VerifyECDSA(pub, hashed, r, s) {
+		t.Errorf("Verify succeeded despite intentionally invalid hash!")
+	}
+}
+
+func generateKeycurve(c elliptic.Curve) (*ecdsa.PrivateKey, error) {
+	x, y, d, err := GenerateKeyECDSA(c.Params().Name)
+	if err != nil {
+		return nil, err
+	}
+	return &ecdsa.PrivateKey{PublicKey: ecdsa.PublicKey{Curve: c, X: x, Y: y}, D: d}, nil
+}
diff --git a/openssl/hmac_test.go b/openssl/hmac_test.go
index 097b584..1b6a6e8 100644
--- a/openssl/hmac_test.go
+++ b/openssl/hmac_test.go
@@ -7,14 +7,58 @@
 package openssl
 
 import (
+	"bytes"
+	"hash"
 	"testing"
 )
 
-// Just tests that we can create an HMAC instance.
-// Previously would cause panic because of incorrect
-// stack allocation of opaque OpenSSL type.
-func TestNewHMAC(t *testing.T) {
-	mac := NewHMAC(NewSHA256, nil)
-	mac.Write([]byte("foo"))
-	t.Logf("%x\n", mac.Sum(nil))
+func TestHMAC(t *testing.T) {
+	var tests = []struct {
+		name string
+		fn   func() hash.Hash
+	}{
+		{"sha1", NewSHA1},
+		{"sha224", NewSHA224},
+		{"sha256", NewSHA256},
+		{"sha384", NewSHA384},
+		{"sha512", NewSHA512},
+	}
+	for _, tt := range tests {
+		t.Run(tt.name, func(t *testing.T) {
+			t.Parallel()
+			h := NewHMAC(tt.fn, nil)
+			h.Write([]byte("hello"))
+			sumHello := h.Sum(nil)
+
+			h = NewHMAC(tt.fn, nil)
+			h.Write([]byte("hello world"))
+			sumHelloWorld := h.Sum(nil)
+
+			// Test that Sum has no effect on future Sum or Write operations.
+			// This is a bit unusual as far as usage, but it's allowed
+			// by the definition of Go hash.Hash, and some clients expect it to work.
+			h = NewHMAC(tt.fn, nil)
+			h.Write([]byte("hello"))
+			if sum := h.Sum(nil); !bytes.Equal(sum, sumHello) {
+				t.Fatalf("1st Sum after hello = %x, want %x", sum, sumHello)
+			}
+			if sum := h.Sum(nil); !bytes.Equal(sum, sumHello) {
+				t.Fatalf("2nd Sum after hello = %x, want %x", sum, sumHello)
+			}
+
+			h.Write([]byte(" world"))
+			if sum := h.Sum(nil); !bytes.Equal(sum, sumHelloWorld) {
+				t.Fatalf("1st Sum after hello world = %x, want %x", sum, sumHelloWorld)
+			}
+			if sum := h.Sum(nil); !bytes.Equal(sum, sumHelloWorld) {
+				t.Fatalf("2nd Sum after hello world = %x, want %x", sum, sumHelloWorld)
+			}
+
+			h.Reset()
+			h.Write([]byte("hello"))
+			if sum := h.Sum(nil); !bytes.Equal(sum, sumHello) {
+				t.Fatalf("Sum after Reset + hello = %x, want %x", sum, sumHello)
+			}
+		})
+	}
 }
diff --git a/openssl/rand_test.go b/openssl/rand_test.go
new file mode 100644
index 0000000..ee8bcf0
--- /dev/null
+++ b/openssl/rand_test.go
@@ -0,0 +1,18 @@
+// Copyright (c) Microsoft Corporation.
+// Licensed under the MIT License.
+
+//go:build linux && !android
+// +build linux,!android
+
+package openssl
+
+import (
+	"testing"
+)
+
+func TestRand(t *testing.T) {
+	_, err := RandReader.Read(make([]byte, 5))
+	if err != nil {
+		t.Fatal(err)
+	}
+}
diff --git a/openssl/rsa_test.go b/openssl/rsa_test.go
new file mode 100644
index 0000000..a9849f8
--- /dev/null
+++ b/openssl/rsa_test.go
@@ -0,0 +1,146 @@
+// Copyright (c) Microsoft Corporation.
+// Licensed under the MIT License.
+
+//go:build linux && !android
+// +build linux,!android
+
+package openssl
+
+import (
+	"bytes"
+	"crypto"
+	"strconv"
+	"testing"
+)
+
+func TestRSAKeyGeneration(t *testing.T) {
+	for _, size := range []int{2048, 3072} {
+		t.Run(strconv.Itoa(size), func(t *testing.T) {
+			t.Parallel()
+			priv, pub := newRSAKey(t, size)
+			msg := []byte("hi!")
+			enc, err := EncryptRSAPKCS1(pub, msg)
+			if err != nil {
+				t.Fatalf("EncryptPKCS1v15: %v", err)
+			}
+			dec, err := DecryptRSAPKCS1(priv, enc)
+			if err != nil {
+				t.Fatalf("DecryptPKCS1v15: %v", err)
+			}
+			if !bytes.Equal(dec, msg) {
+				t.Fatalf("got:%x want:%x", dec, msg)
+			}
+		})
+	}
+}
+
+func TestEncryptDecryptOAEP(t *testing.T) {
+	sha256 := NewSHA256()
+	msg := []byte("hi!")
+	label := []byte("ho!")
+	priv, pub := newRSAKey(t, 2048)
+	enc, err := EncryptRSAOAEP(sha256, pub, msg, label)
+	if err != nil {
+		t.Fatal(err)
+	}
+	dec, err := DecryptRSAOAEP(sha256, priv, enc, label)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !bytes.Equal(dec, msg) {
+		t.Errorf("got:%x want:%x", dec, msg)
+	}
+}
+
+func TestEncryptDecryptOAEP_WrongLabel(t *testing.T) {
+	sha256 := NewSHA256()
+	msg := []byte("hi!")
+	priv, pub := newRSAKey(t, 2048)
+	enc, err := EncryptRSAOAEP(sha256, pub, msg, []byte("ho!"))
+	if err != nil {
+		t.Fatal(err)
+	}
+	dec, err := DecryptRSAOAEP(sha256, priv, enc, []byte("wrong!"))
+	if err == nil {
+		t.Errorf("error expected")
+	}
+	if dec != nil {
+		t.Errorf("got:%x want: nil", dec)
+	}
+}
+
+func TestSignVerifyPKCS1v15(t *testing.T) {
+	sha256 := NewSHA256()
+	priv, pub := newRSAKey(t, 2048)
+	sha256.Write([]byte("hi!"))
+	hashed := sha256.Sum(nil)
+	signed, err := SignRSAPKCS1v15(priv, crypto.SHA256, hashed)
+	if err != nil {
+		t.Fatal(err)
+	}
+	err = VerifyRSAPKCS1v15(pub, crypto.SHA256, hashed, signed)
+	if err != nil {
+		t.Fatal(err)
+	}
+}
+
+func TestSignVerifyPKCS1v15_Unhashed(t *testing.T) {
+	msg := []byte("hi!")
+	priv, pub := newRSAKey(t, 2048)
+	signed, err := SignRSAPKCS1v15(priv, 0, msg)
+	if err != nil {
+		t.Fatal(err)
+	}
+	err = VerifyRSAPKCS1v15(pub, 0, msg, signed)
+	if err != nil {
+		t.Fatal(err)
+	}
+}
+
+func TestSignVerifyPKCS1v15_Invalid(t *testing.T) {
+	sha256 := NewSHA256()
+	msg := []byte("hi!")
+	priv, pub := newRSAKey(t, 2048)
+	sha256.Write(msg)
+	hashed := sha256.Sum(nil)
+	signed, err := SignRSAPKCS1v15(priv, crypto.SHA256, hashed)
+	if err != nil {
+		t.Fatal(err)
+	}
+	err = VerifyRSAPKCS1v15(pub, crypto.SHA256, msg, signed)
+	if err == nil {
+		t.Fatal("error expected")
+	}
+}
+
+func TestSignVerifyRSAPSS(t *testing.T) {
+	sha1 := NewSHA1()
+	priv, pub := newRSAKey(t, 2048)
+	sha1.Write([]byte("testing"))
+	hashed := sha1.Sum(nil)
+	signed, err := SignRSAPSS(priv, crypto.SHA1, hashed, 0)
+	if err != nil {
+		t.Fatal(err)
+	}
+	err = VerifyRSAPSS(pub, crypto.SHA1, hashed, signed, 0)
+	if err != nil {
+		t.Fatal(err)
+	}
+}
+
+func newRSAKey(t *testing.T, size int) (*PrivateKeyRSA, *PublicKeyRSA) {
+	t.Helper()
+	N, E, D, P, Q, Dp, Dq, Qinv, err := GenerateKeyRSA(size)
+	if err != nil {
+		t.Fatalf("GenerateKeyRSA(%d): %v", size, err)
+	}
+	priv, err := NewPrivateKeyRSA(N, E, D, P, Q, Dp, Dq, Qinv)
+	if err != nil {
+		t.Fatalf("NewPrivateKeyRSA(%d): %v", size, err)
+	}
+	pub, err := NewPublicKeyRSA(N, E)
+	if err != nil {
+		t.Fatalf("NewPublicKeyRSA(%d): %v", size, err)
+	}
+	return priv, pub
+}
diff --git a/openssl/sha_test.go b/openssl/sha_test.go
new file mode 100644
index 0000000..ef43037
--- /dev/null
+++ b/openssl/sha_test.go
@@ -0,0 +1,67 @@
+// Copyright (c) Microsoft Corporation.
+// Licensed under the MIT License.
+
+//go:build linux && !android
+// +build linux,!android
+
+package openssl
+
+import (
+	"bytes"
+	"encoding"
+	"hash"
+	"testing"
+)
+
+func TestSha(t *testing.T) {
+	msg := []byte("testig")
+	var tests = []struct {
+		name string
+		fn   func() hash.Hash
+	}{
+		{"sha1", NewSHA1},
+		{"sha224", NewSHA224},
+		{"sha256", NewSHA256},
+		{"sha384", NewSHA384},
+		{"sha512", NewSHA512},
+	}
+	for _, tt := range tests {
+		t.Run(tt.name, func(t *testing.T) {
+			t.Parallel()
+			h := tt.fn()
+			initSum := h.Sum(nil)
+			n, err := h.Write(msg)
+			if err != nil {
+				t.Fatal(err)
+			}
+			if n != len(msg) {
+				t.Errorf("got: %d, want: %d", n, len(msg))
+			}
+			sum := h.Sum(nil)
+			if size := h.Size(); len(sum) != size {
+				t.Errorf("got: %d, want: %d", len(sum), size)
+			}
+			if bytes.Equal(sum, initSum) {
+				t.Error("Write didn't change internal hash state")
+			}
+
+			state, err := h.(encoding.BinaryMarshaler).MarshalBinary()
+			if err != nil {
+				t.Errorf("could not marshal: %v", err)
+			}
+			h2 := tt.fn()
+			if err := h2.(encoding.BinaryUnmarshaler).UnmarshalBinary(state); err != nil {
+				t.Errorf("could not unmarshal: %v", err)
+			}
+			if actual, actual2 := h.Sum(nil), h2.Sum(nil); !bytes.Equal(actual, actual2) {
+				t.Errorf("0x%x != marshaled 0x%x", actual, actual2)
+			}
+
+			h.Reset()
+			sum = h.Sum(nil)
+			if !bytes.Equal(sum, initSum) {
+				t.Errorf("got:%x want:%x", sum, initSum)
+			}
+		})
+	}
+}