diff --git a/rsa.go b/rsa.go
index 564ccdcf..4e45b02d 100644
--- a/rsa.go
+++ b/rsa.go
@@ -396,15 +396,24 @@ func newRSAKey3(isPriv bool, n, e, d, p, q, dp, dq, qinv BigInt) (C.GO_EVP_PKEY_
 	}
 	comps = append(comps, required[:]...)
 
-	// OpenSSL 3.0 and 3.1 required all the precomputed values if
-	// P and Q are present. See:
-	// https://github.com/openssl/openssl/pull/22334
-	if p != nil && q != nil && dp != nil && dq != nil && qinv != nil {
-		precomputed := [...]bigIntParam{
-			{OSSL_PKEY_PARAM_RSA_FACTOR1, p}, {OSSL_PKEY_PARAM_RSA_FACTOR2, q},
-			{OSSL_PKEY_PARAM_RSA_EXPONENT1, dp}, {OSSL_PKEY_PARAM_RSA_EXPONENT2, dq}, {OSSL_PKEY_PARAM_RSA_COEFFICIENT1, qinv},
+	if p != nil && q != nil {
+		allPrecomputedExists := dp != nil && dq != nil && qinv != nil
+		// The precomputed values should only be passed if P and Q are present
+		// and every precomputed value is present. (If any precomputed value is
+		// missing, don't pass any of them.)
+		//
+		// In OpenSSL 3.0 and 3.1, we must also omit P and Q if any precomputed
+		// value is missing. See https://github.com/openssl/openssl/pull/22334
+		if vMinor >= 2 || allPrecomputedExists {
+			comps = append(comps, bigIntParam{OSSL_PKEY_PARAM_RSA_FACTOR1, p}, bigIntParam{OSSL_PKEY_PARAM_RSA_FACTOR2, q})
+		}
+		if allPrecomputedExists {
+			comps = append(comps,
+				bigIntParam{OSSL_PKEY_PARAM_RSA_EXPONENT1, dp},
+				bigIntParam{OSSL_PKEY_PARAM_RSA_EXPONENT2, dq},
+				bigIntParam{OSSL_PKEY_PARAM_RSA_COEFFICIENT1, qinv},
+			)
 		}
-		comps = append(comps, precomputed[:]...)
 	}
 
 	for _, comp := range comps {
diff --git a/rsa_test.go b/rsa_test.go
index 35de951a..94775332 100644
--- a/rsa_test.go
+++ b/rsa_test.go
@@ -4,6 +4,7 @@ import (
 	"bytes"
 	"crypto"
 	"crypto/rsa"
+	"fmt"
 	"math/big"
 	"strconv"
 	"testing"
@@ -13,11 +14,80 @@ import (
 )
 
 func TestRSAKeyGeneration(t *testing.T) {
+	for _, size := range []int{2048, 3072} {
+		size := size
+		t.Run(strconv.Itoa(size), func(t *testing.T) {
+			t.Parallel()
+			_, _, _, _, _, _, _, _, err := openssl.GenerateKeyRSA(size)
+			if err != nil {
+				t.Fatal(err)
+			}
+		})
+	}
+}
+
+func testRSAEncryptDecryptPKCS1(t *testing.T, priv *openssl.PrivateKeyRSA, pub *openssl.PublicKeyRSA) {
+	msg := []byte("hi!")
+	enc, err := openssl.EncryptRSAPKCS1(pub, msg)
+	if err != nil {
+		t.Fatalf("EncryptPKCS1v15: %v", err)
+	}
+	dec, err := openssl.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 TestRSAEncryptDecryptPKCS1(t *testing.T) {
 	for _, size := range []int{2048, 3072} {
 		size := size
 		t.Run(strconv.Itoa(size), func(t *testing.T) {
 			t.Parallel()
 			priv, pub := newRSAKey(t, size)
+			testRSAEncryptDecryptPKCS1(t, priv, pub)
+		})
+	}
+}
+
+func TestRSAEncryptDecryptPKCS1_MissingPrecomputedValues(t *testing.T) {
+	n, e, d, p, q, dp, dq, qinv, err := openssl.GenerateKeyRSA(2048)
+	if err != nil {
+		t.Fatalf("GenerateKeyRSA: %v", err)
+	}
+	tt := []struct {
+		withDp   bool
+		withDq   bool
+		withQinv bool
+	}{
+		{true, true, false},
+		{true, false, true},
+		{false, true, true},
+		{false, false, false},
+		{false, false, true},
+		{false, true, false},
+		{true, false, false},
+		{true, true, true},
+	}
+	for _, tt := range tt {
+		tt := tt
+		t.Run(fmt.Sprintf("dp=%v,dq=%v,qinv=%v", tt.withDp, tt.withDq, tt.withQinv), func(t *testing.T) {
+			t.Parallel()
+			dp1, dq1, qinv1 := dp, dq, qinv
+			if !tt.withDp {
+				dp1 = nil
+			}
+			if !tt.withDq {
+				dq1 = nil
+			}
+			if !tt.withQinv {
+				qinv1 = nil
+			}
+
+			priv, pub := newRSAKeyFromParams(t, n, e, d, p, q, dp1, dq1, qinv1)
+			testRSAEncryptDecryptPKCS1(t, priv, pub)
 			msg := []byte("hi!")
 			enc, err := openssl.EncryptRSAPKCS1(pub, msg)
 			if err != nil {
@@ -34,7 +104,7 @@ func TestRSAKeyGeneration(t *testing.T) {
 	}
 }
 
-func TestEncryptDecryptOAEP(t *testing.T) {
+func TestRSAEncryptDecryptOAEP(t *testing.T) {
 	sha256 := openssl.NewSHA256()
 	msg := []byte("hi!")
 	label := []byte("ho!")
@@ -57,7 +127,7 @@ func TestEncryptDecryptOAEP(t *testing.T) {
 	}
 }
 
-func TestEncryptDecryptOAEP_EmptyLabel(t *testing.T) {
+func TestRSAEncryptDecryptOAEP_EmptyLabel(t *testing.T) {
 	sha256 := openssl.NewSHA256()
 	msg := []byte("hi!")
 	label := []byte("")
@@ -80,7 +150,7 @@ func TestEncryptDecryptOAEP_EmptyLabel(t *testing.T) {
 	}
 }
 
-func TestEncryptDecryptOAEP_WithMGF1Hash(t *testing.T) {
+func TestRSAEncryptDecryptOAEP_WithMGF1Hash(t *testing.T) {
 	if openssl.SymCryptProviderAvailable() {
 		t.Skip("SymCrypt provider does not support MGF1 hash")
 	}
@@ -107,7 +177,7 @@ func TestEncryptDecryptOAEP_WithMGF1Hash(t *testing.T) {
 	}
 }
 
-func TestEncryptDecryptOAEP_WrongLabel(t *testing.T) {
+func TestRSAEncryptDecryptOAEP_WrongLabel(t *testing.T) {
 	sha256 := openssl.NewSHA256()
 	msg := []byte("hi!")
 	priv, pub := newRSAKey(t, 2048)
@@ -124,7 +194,7 @@ func TestEncryptDecryptOAEP_WrongLabel(t *testing.T) {
 	}
 }
 
-func TestSignVerifyPKCS1v15(t *testing.T) {
+func TestRSASignVerifyPKCS1v15(t *testing.T) {
 	sha256 := openssl.NewSHA256()
 	priv, pub := newRSAKey(t, 2048)
 	msg := []byte("hi!")
@@ -151,7 +221,7 @@ func TestSignVerifyPKCS1v15(t *testing.T) {
 	}
 }
 
-func TestSignVerifyPKCS1v15_Unhashed(t *testing.T) {
+func TestRSASignVerifyPKCS1v15_Unhashed(t *testing.T) {
 	if openssl.SymCryptProviderAvailable() {
 		t.Skip("SymCrypt provider does not support unhashed PKCS1v15")
 	}
@@ -168,7 +238,7 @@ func TestSignVerifyPKCS1v15_Unhashed(t *testing.T) {
 	}
 }
 
-func TestSignVerifyPKCS1v15_Invalid(t *testing.T) {
+func TestRSASignVerifyPKCS1v15_Invalid(t *testing.T) {
 	sha256 := openssl.NewSHA256()
 	msg := []byte("hi!")
 	priv, pub := newRSAKey(t, 2048)
@@ -184,7 +254,7 @@ func TestSignVerifyPKCS1v15_Invalid(t *testing.T) {
 	}
 }
 
-func TestSignVerifyRSAPSS(t *testing.T) {
+func TestRSASignVerifyRSAPSS(t *testing.T) {
 	// Test cases taken from
 	// https://github.com/golang/go/blob/54182ff54a687272dd7632c3a963e036ce03cb7c/src/crypto/rsa/pss_test.go#L200.
 	const keyBits = 2048
@@ -225,15 +295,18 @@ func newRSAKey(t *testing.T, size int) (*openssl.PrivateKeyRSA, *openssl.PublicK
 	if err != nil {
 		t.Fatalf("GenerateKeyRSA(%d): %v", size, err)
 	}
-	// Exercise omission of precomputed value
-	Dp = nil
+	return newRSAKeyFromParams(t, N, E, D, P, Q, Dp, Dq, Qinv)
+}
+
+func newRSAKeyFromParams(t *testing.T, N, E, D, P, Q, Dp, Dq, Qinv openssl.BigInt) (*openssl.PrivateKeyRSA, *openssl.PublicKeyRSA) {
+	t.Helper()
 	priv, err := openssl.NewPrivateKeyRSA(N, E, D, P, Q, Dp, Dq, Qinv)
 	if err != nil {
-		t.Fatalf("NewPrivateKeyRSA(%d): %v", size, err)
+		t.Fatalf("NewPrivateKeyRSA: %v", err)
 	}
 	pub, err := openssl.NewPublicKeyRSA(N, E)
 	if err != nil {
-		t.Fatalf("NewPublicKeyRSA(%d): %v", size, err)
+		t.Fatalf("NewPublicKeyRSA: %v", err)
 	}
 	return priv, pub
 }