diff --git a/crypto/rsa/key_test.go b/crypto/rsa/key_test.go
index bccbfeb..e862481 100644
--- a/crypto/rsa/key_test.go
+++ b/crypto/rsa/key_test.go
@@ -1,6 +1,7 @@
 package rsa
 
 import (
+	"crypto/x509"
 	"encoding/base64"
 	"testing"
 
@@ -165,3 +166,101 @@ MCB+kOgWk51uJwuiuHlffGMBPxku/t+skxI7Bw==`
 
 	require.True(t, pub.VerifyASN1([]byte("message"), sig))
 }
+
+func TestPublicKeyPKCS1RoundTrip(t *testing.T) {
+	pub, _, err := GenerateKeyPair(2048)
+	require.NoError(t, err)
+
+	der := x509.MarshalPKCS1PublicKey(pub.Unwrap())
+	rt, err := PublicKeyFromPKCS1DER(der)
+	require.NoError(t, err)
+	require.True(t, pub.Equal(rt))
+}
+
+func TestPublicKeyFromNERoundTrip(t *testing.T) {
+	pub, _, err := GenerateKeyPair(2048)
+	require.NoError(t, err)
+
+	rt, err := PublicKeyFromNE(pub.NBytes(), pub.EBytes())
+	require.NoError(t, err)
+	require.True(t, pub.Equal(rt))
+}
+
+func TestPrivateKeyFromNEDPQRoundTrip(t *testing.T) {
+	pub, priv, err := GenerateKeyPair(2048)
+	require.NoError(t, err)
+
+	rt, err := PrivateKeyFromNEDPQ(pub.NBytes(), pub.EBytes(), priv.DBytes(), priv.PBytes(), priv.QBytes())
+	require.NoError(t, err)
+	require.True(t, priv.Equal(rt))
+	require.True(t, pub.Equal(rt.Public()))
+}
+
+func TestRejectWeirdPublicKeyInputs(t *testing.T) {
+	// Reuse a known-good 2048-bit modulus so each case changes only the input under test.
+	validModulus, err := base64.RawURLEncoding.DecodeString("sbX82NTV6IylxCh7MfV4hlyvaniCajuP97GyOqSvTmoEdBOflFvZ06kR_9D6ctt45Fk6hskfnag2GG69NALVH2o4RCR6tQiLRpKcMRtDYE_thEmfBvDzm_VVkOIYfxu-Ipuo9J_S5XDNDjczx2v-3oDh5-CIHkU46hvFeCvpUS-L8TJSbgX0kjVk_m4eIb9wh63rtmD6Uz_KBtCo5mmR4TEtcLZKYdqMp3wCjN-TlgHiz_4oVXWbHUefCEe8rFnX1iQnpDHU49_SaXQoud1jCaexFn25n-Aa8f8bc5Vm-5SeRwidHa6ErvEhTvf1dz6GoNPp2iRvm-wJ1gxwWJEYPQ")
+	require.NoError(t, err)
+
+	evenModulus := append([]byte{}, validModulus...)
+	evenModulus[len(evenModulus)-1] &^= 1
+
+	// Build a 8193-bit odd modulus to exceed the package's 8192-bit upper bound.
+	tooLargeModulus := make([]byte, 1025)
+	tooLargeModulus[0] = 0x80
+	tooLargeModulus[len(tooLargeModulus)-1] = 0x01
+
+	// This is 2^63, which does not fit in a signed int64 and must be rejected.
+	exponentTooLarge := []byte{0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}
+
+	for _, tc := range []struct {
+		name string
+		n    []byte
+		e    []byte
+	}{
+		{
+			name: "empty modulus",
+			n:    nil,
+			e:    []byte{0x03},
+		},
+		{
+			name: "too small modulus",
+			n:    validModulus[:len(validModulus)-1],
+			e:    []byte{0x03},
+		},
+		{
+			name: "too large modulus",
+			n:    tooLargeModulus,
+			e:    []byte{0x03},
+		},
+		{
+			name: "even modulus",
+			n:    evenModulus,
+			e:    []byte{0x03},
+		},
+		{
+			name: "empty exponent",
+			n:    validModulus,
+			e:    nil,
+		},
+		{
+			name: "exponent one",
+			n:    validModulus,
+			e:    []byte{0x01},
+		},
+		{
+			name: "even exponent",
+			n:    validModulus,
+			e:    []byte{0x02},
+		},
+		{
+			name: "exponent too large",
+			n:    validModulus,
+			e:    exponentTooLarge,
+		},
+	} {
+		t.Run(tc.name, func(t *testing.T) {
+			_, err := PublicKeyFromNE(tc.n, tc.e)
+			require.Error(t, err)
+		})
+	}
+}
diff --git a/crypto/rsa/public.go b/crypto/rsa/public.go
index d89b2e6..e093c5c 100644
--- a/crypto/rsa/public.go
+++ b/crypto/rsa/public.go
@@ -53,6 +53,9 @@ func PublicKeyFromNE(n, e []byte) (*PublicKey, error) {
 	if eBInt.Sign() <= 0 {
 		return nil, fmt.Errorf("exponent must be positive")
 	}
+	if eBInt.Cmp(big.NewInt(2)) < 0 {
+		return nil, fmt.Errorf("exponent too small")
+	}
 	if eBInt.Bit(0) == 0 {
 		return nil, fmt.Errorf("exponent must be odd")
 	}