backend specific RSA*Key implementation for OpenSSL

5 files changed, 762 insertions, 538 deletions

diff --git a/cryptography/hazmat/backends/openssl/backend.py b/cryptography/hazmat/backends/openssl/backend.py
index ab3f9471..770b2af4 100644
--- a/cryptography/hazmat/backends/openssl/backend.py
+++ b/cryptography/hazmat/backends/openssl/backend.py
@@ -41,6 +41,9 @@ from cryptography.hazmat.primitives.ciphers.algorithms import (
 from cryptography.hazmat.primitives.ciphers.modes import (
     CBC, CFB, CFB8, CTR, ECB, GCM, OFB
 )
+from cryptography.hazmat.primitives.interfaces import (
+    RSAPrivateKeyWithNumbers, RSAPublicKeyWithNumbers
+)
 
 
 _MemoryBIO = collections.namedtuple("_MemoryBIO", ["bio", "char_ptr"])
@@ -374,27 +377,34 @@ class Backend(object):
 
         return self._rsa_cdata_to_private_key(ctx)
 
+    def _generate_rsa_private_key(self, public_exponent, key_size):
+        rsa._verify_rsa_parameters(public_exponent, key_size)
+
+        rsa_cdata = self._lib.RSA_new()
+        assert rsa_cdata != self._ffi.NULL
+        rsa_cdata = self._ffi.gc(rsa_cdata, self._lib.RSA_free)
+
+        bn = self._int_to_bn(public_exponent)
+        bn = self._ffi.gc(bn, self._lib.BN_free)
+
+        res = self._lib.RSA_generate_key_ex(
+            rsa_cdata, key_size, bn, self._ffi.NULL
+        )
+        assert res == 1
+
+        return _RSAPrivateKey(self, rsa_cdata)
+
     def generate_rsa_parameters_supported(self, public_exponent, key_size):
         return (public_exponent >= 3 and public_exponent & 1 != 0 and
                 key_size >= 512)
 
     def load_rsa_private_numbers(self, numbers):
-        return rsa.RSAPrivateKey(
-            p=numbers.p,
-            q=numbers.q,
-            private_exponent=numbers.d,
-            dmp1=numbers.dmp1,
-            dmq1=numbers.dmq1,
-            iqmp=numbers.iqmp,
-            public_exponent=numbers.public_numbers.e,
-            modulus=numbers.public_numbers.n
-        )
+        rsa_cdata = self._rsa_cdata_from_private_numbers(numbers)
+        return _RSAPrivateKey(self, rsa_cdata)
 
     def load_rsa_public_numbers(self, numbers):
-        return rsa.RSAPublicKey(
-            public_exponent=numbers.e,
-            modulus=numbers.n
-        )
+        rsa_cdata = self._rsa_cdata_from_public_numbers(numbers)
+        return _RSAPublicKey(self, rsa_cdata)
 
     def _new_evp_pkey(self):
         evp_pkey = self._lib.EVP_PKEY_new()
@@ -445,8 +455,7 @@ class Backend(object):
         if type == self._lib.EVP_PKEY_RSA:
             rsa_cdata = self._lib.EVP_PKEY_get1_RSA(evp_pkey)
             assert rsa_cdata != self._ffi.NULL
-            rsa_cdata = self._ffi.gc(rsa_cdata, self._lib.RSA_free)
-            return self._rsa_cdata_to_private_key(rsa_cdata)
+            return _RSAPrivateKey(self, rsa_cdata)
         elif type == self._lib.EVP_PKEY_DSA:
             dsa_cdata = self._lib.EVP_PKEY_get1_DSA(evp_pkey)
             assert dsa_cdata != self._ffi.NULL
@@ -538,12 +547,47 @@ class Backend(object):
 
         return ctx
 
+    def _rsa_cdata_from_private_numbers(self, private_numbers):
+        # Does not GC the RSA cdata. You *must* make sure it's freed
+        # correctly yourself!
+        ctx = self._lib.RSA_new()
+        assert ctx != self._ffi.NULL
+        ctx.p = self._int_to_bn(private_numbers.p)
+        ctx.q = self._int_to_bn(private_numbers.q)
+        ctx.d = self._int_to_bn(private_numbers.d)
+        ctx.dmp1 = self._int_to_bn(private_numbers.dmp1)
+        ctx.dmq1 = self._int_to_bn(private_numbers.dmq1)
+        ctx.iqmp = self._int_to_bn(private_numbers.iqmp)
+        ctx.e = self._int_to_bn(private_numbers.public_numbers.e)
+        ctx.n = self._int_to_bn(private_numbers.public_numbers.n)
+        res = self._lib.RSA_blinding_on(ctx, self._ffi.NULL)
+        assert res == 1
+
+        return ctx
+
+    def _rsa_cdata_from_public_numbers(self, public_numbers):
+        # Does not GC the RSA cdata. You *must* make sure it's freed
+        # correctly yourself!
+
+        ctx = self._lib.RSA_new()
+        assert ctx != self._ffi.NULL
+        ctx.e = self._int_to_bn(public_numbers.e)
+        ctx.n = self._int_to_bn(public_numbers.n)
+        res = self._lib.RSA_blinding_on(ctx, self._ffi.NULL)
+        assert res == 1
+
+        return ctx
+
     def create_rsa_signature_ctx(self, private_key, padding, algorithm):
-        return _RSASignatureContext(self, private_key, padding, algorithm)
+        rsa_cdata = self._rsa_cdata_from_private_key(private_key)
+        key = _RSAPrivateKey(self, rsa_cdata)
+        return _RSASignatureContext(self, key, padding, algorithm)
 
     def create_rsa_verification_ctx(self, public_key, signature, padding,
                                     algorithm):
-        return _RSAVerificationContext(self, public_key, signature, padding,
+        rsa_cdata = self._rsa_cdata_from_public_key(public_key)
+        key = _RSAPublicKey(self, rsa_cdata)
+        return _RSAVerificationContext(self, key, signature, padding,
                                        algorithm)
 
     def mgf1_hash_supported(self, algorithm):
@@ -704,13 +748,18 @@ class Backend(object):
 
     def _enc_dec_rsa_pkey_ctx(self, key, data, padding_enum):
         if isinstance(key, rsa.RSAPublicKey):
+            evp_pkey = self._rsa_public_key_to_evp_pkey(key)
+        elif isinstance(key, rsa.RSAPrivateKey):
+            evp_pkey = self._rsa_private_key_to_evp_pkey(key)
+        else:
+            evp_pkey = key._evp_pkey
+
+        if isinstance(key, (rsa.RSAPublicKey, _RSAPublicKey)):
             init = self._lib.EVP_PKEY_encrypt_init
             crypt = self._lib.Cryptography_EVP_PKEY_encrypt
-            evp_pkey = self._rsa_public_key_to_evp_pkey(key)
         else:
             init = self._lib.EVP_PKEY_decrypt_init
             crypt = self._lib.Cryptography_EVP_PKEY_decrypt
-            evp_pkey = self._rsa_private_key_to_evp_pkey(key)
 
         pkey_ctx = self._lib.EVP_PKEY_CTX_new(
             evp_pkey, self._ffi.NULL
@@ -740,13 +789,19 @@ class Backend(object):
 
     def _enc_dec_rsa_098(self, key, data, padding_enum):
         if isinstance(key, rsa.RSAPublicKey):
-            crypt = self._lib.RSA_public_encrypt
             rsa_cdata = self._rsa_cdata_from_public_key(key)
+            rsa_cdata = self._ffi.gc(rsa_cdata, self._lib.RSA_free)
+        elif isinstance(key, rsa.RSAPrivateKey):
+            rsa_cdata = self._rsa_cdata_from_private_key(key)
+            rsa_cdata = self._ffi.gc(rsa_cdata, self._lib.RSA_free)
+        else:
+            rsa_cdata = key._rsa_cdata
+
+        if isinstance(key, (rsa.RSAPublicKey, _RSAPublicKey)):
+            crypt = self._lib.RSA_public_encrypt
         else:
             crypt = self._lib.RSA_private_decrypt
-            rsa_cdata = self._rsa_cdata_from_private_key(key)
 
-        rsa_cdata = self._ffi.gc(rsa_cdata, self._lib.RSA_free)
         key_size = self._lib.RSA_size(rsa_cdata)
         assert key_size > 0
         buf = self._ffi.new("unsigned char[]", key_size)
@@ -766,7 +821,7 @@ class Backend(object):
         errors = self._consume_errors()
         assert errors
         assert errors[0].lib == self._lib.ERR_LIB_RSA
-        if isinstance(key, rsa.RSAPublicKey):
+        if isinstance(key, (_RSAPublicKey, rsa.RSAPublicKey)):
             assert (errors[0].reason ==
                     self._lib.RSA_R_DATA_TOO_LARGE_FOR_KEY_SIZE)
             raise ValueError(
@@ -1468,6 +1523,10 @@ class _RSASignatureContext(object):
             raise TypeError(
                 "Expected provider of interfaces.AsymmetricPadding.")
 
+        self.pkey_size = self._backend._lib.EVP_PKEY_size(
+            self._private_key._evp_pkey
+        )
+
         if isinstance(padding, PKCS1v15):
             if self._backend._lib.Cryptography_HAS_PKEY_CTX:
                 self._finalize_method = self._finalize_pkey_ctx
@@ -1483,8 +1542,8 @@ class _RSASignatureContext(object):
 
             # Size of key in bytes - 2 is the maximum
             # PSS signature length (salt length is checked later)
-            key_size_bytes = int(math.ceil(private_key.key_size / 8.0))
-            if key_size_bytes - algorithm.digest_size - 2 < 0:
+            assert self.pkey_size > 0
+            if self.pkey_size - algorithm.digest_size - 2 < 0:
                 raise ValueError("Digest too large for key size. Use a larger "
                                  "key.")
 
@@ -1514,20 +1573,15 @@ class _RSASignatureContext(object):
         self._hash_ctx.update(data)
 
     def finalize(self):
-        evp_pkey = self._backend._rsa_private_key_to_evp_pkey(
-            self._private_key)
-
         evp_md = self._backend._lib.EVP_get_digestbyname(
             self._algorithm.name.encode("ascii"))
         assert evp_md != self._backend._ffi.NULL
-        pkey_size = self._backend._lib.EVP_PKEY_size(evp_pkey)
-        assert pkey_size > 0
 
-        return self._finalize_method(evp_pkey, pkey_size, evp_md)
+        return self._finalize_method(evp_md)
 
-    def _finalize_pkey_ctx(self, evp_pkey, pkey_size, evp_md):
+    def _finalize_pkey_ctx(self, evp_md):
         pkey_ctx = self._backend._lib.EVP_PKEY_CTX_new(
-            evp_pkey, self._backend._ffi.NULL
+            self._private_key._evp_pkey, self._backend._ffi.NULL
         )
         assert pkey_ctx != self._backend._ffi.NULL
         pkey_ctx = self._backend._ffi.gc(pkey_ctx,
@@ -1590,17 +1644,17 @@ class _RSASignatureContext(object):
 
         return self._backend._ffi.buffer(buf)[:]
 
-    def _finalize_pkcs1(self, evp_pkey, pkey_size, evp_md):
+    def _finalize_pkcs1(self, evp_md):
         if self._hash_ctx._ctx is None:
             raise AlreadyFinalized("Context has already been finalized.")
 
-        sig_buf = self._backend._ffi.new("char[]", pkey_size)
+        sig_buf = self._backend._ffi.new("char[]", self.pkey_size)
         sig_len = self._backend._ffi.new("unsigned int *")
         res = self._backend._lib.EVP_SignFinal(
             self._hash_ctx._ctx._ctx,
             sig_buf,
             sig_len,
-            evp_pkey
+            self._private_key._evp_pkey
         )
         self._hash_ctx.finalize()
         if res == 0:
@@ -1613,15 +1667,11 @@ class _RSASignatureContext(object):
 
         return self._backend._ffi.buffer(sig_buf)[:sig_len[0]]
 
-    def _finalize_pss(self, evp_pkey, pkey_size, evp_md):
+    def _finalize_pss(self, evp_md):
         data_to_sign = self._hash_ctx.finalize()
-        padded = self._backend._ffi.new("unsigned char[]", pkey_size)
-        rsa_cdata = self._backend._lib.EVP_PKEY_get1_RSA(evp_pkey)
-        assert rsa_cdata != self._backend._ffi.NULL
-        rsa_cdata = self._backend._ffi.gc(rsa_cdata,
-                                          self._backend._lib.RSA_free)
+        padded = self._backend._ffi.new("unsigned char[]", self.pkey_size)
         res = self._backend._lib.RSA_padding_add_PKCS1_PSS(
-            rsa_cdata,
+            self._private_key._rsa_cdata,
             padded,
             data_to_sign,
             evp_md,
@@ -1639,12 +1689,12 @@ class _RSASignatureContext(object):
             raise ValueError("Salt length too long for key size. Try using "
                              "MAX_LENGTH instead.")
 
-        sig_buf = self._backend._ffi.new("char[]", pkey_size)
+        sig_buf = self._backend._ffi.new("char[]", self.pkey_size)
         sig_len = self._backend._lib.RSA_private_encrypt(
-            pkey_size,
+            self.pkey_size,
             padded,
             sig_buf,
-            rsa_cdata,
+            self._private_key._rsa_cdata,
             self._backend._lib.RSA_NO_PADDING
         )
         assert sig_len != -1
@@ -1662,6 +1712,10 @@ class _RSAVerificationContext(object):
             raise TypeError(
                 "Expected provider of interfaces.AsymmetricPadding.")
 
+        self.pkey_size = self._backend._lib.EVP_PKEY_size(
+            self._public_key._evp_pkey
+        )
+
         if isinstance(padding, PKCS1v15):
             if self._backend._lib.Cryptography_HAS_PKEY_CTX:
                 self._verify_method = self._verify_pkey_ctx
@@ -1677,8 +1731,8 @@ class _RSAVerificationContext(object):
 
             # Size of key in bytes - 2 is the maximum
             # PSS signature length (salt length is checked later)
-            key_size_bytes = int(math.ceil(public_key.key_size / 8.0))
-            if key_size_bytes - algorithm.digest_size - 2 < 0:
+            assert self.pkey_size > 0
+            if self.pkey_size - algorithm.digest_size - 2 < 0:
                 raise ValueError(
                     "Digest too large for key size. Check that you have the "
                     "correct key and digest algorithm."
@@ -1710,18 +1764,15 @@ class _RSAVerificationContext(object):
         self._hash_ctx.update(data)
 
     def verify(self):
-        evp_pkey = self._backend._rsa_public_key_to_evp_pkey(
-            self._public_key)
-
         evp_md = self._backend._lib.EVP_get_digestbyname(
             self._algorithm.name.encode("ascii"))
         assert evp_md != self._backend._ffi.NULL
 
-        self._verify_method(evp_pkey, evp_md)
+        self._verify_method(evp_md)
 
-    def _verify_pkey_ctx(self, evp_pkey, evp_md):
+    def _verify_pkey_ctx(self, evp_md):
         pkey_ctx = self._backend._lib.EVP_PKEY_CTX_new(
-            evp_pkey, self._backend._ffi.NULL
+            self._public_key._evp_pkey, self._backend._ffi.NULL
         )
         assert pkey_ctx != self._backend._ffi.NULL
         pkey_ctx = self._backend._ffi.gc(pkey_ctx,
@@ -1772,7 +1823,7 @@ class _RSAVerificationContext(object):
             assert errors
             raise InvalidSignature
 
-    def _verify_pkcs1(self, evp_pkey, evp_md):
+    def _verify_pkcs1(self, evp_md):
         if self._hash_ctx._ctx is None:
             raise AlreadyFinalized("Context has already been finalized.")
 
@@ -1780,7 +1831,7 @@ class _RSAVerificationContext(object):
             self._hash_ctx._ctx._ctx,
             self._signature,
             len(self._signature),
-            evp_pkey
+            self._public_key._evp_pkey
         )
         self._hash_ctx.finalize()
         # The previous call can return negative numbers in the event of an
@@ -1792,29 +1843,23 @@ class _RSAVerificationContext(object):
             assert errors
             raise InvalidSignature
 
-    def _verify_pss(self, evp_pkey, evp_md):
-        pkey_size = self._backend._lib.EVP_PKEY_size(evp_pkey)
-        assert pkey_size > 0
-        rsa_cdata = self._backend._lib.EVP_PKEY_get1_RSA(evp_pkey)
-        assert rsa_cdata != self._backend._ffi.NULL
-        rsa_cdata = self._backend._ffi.gc(rsa_cdata,
-                                          self._backend._lib.RSA_free)
-        buf = self._backend._ffi.new("unsigned char[]", pkey_size)
+    def _verify_pss(self, evp_md):
+        buf = self._backend._ffi.new("unsigned char[]", self.pkey_size)
         res = self._backend._lib.RSA_public_decrypt(
             len(self._signature),
             self._signature,
             buf,
-            rsa_cdata,
+            self._public_key._rsa_cdata,
             self._backend._lib.RSA_NO_PADDING
         )
-        if res != pkey_size:
+        if res != self.pkey_size:
             errors = self._backend._consume_errors()
             assert errors
             raise InvalidSignature
 
         data_to_verify = self._hash_ctx.finalize()
         res = self._backend._lib.RSA_verify_PKCS1_PSS(
-            rsa_cdata,
+            self._public_key._rsa_cdata,
             data_to_verify,
             evp_md,
             buf,
@@ -2110,4 +2155,87 @@ class _EllipticCurvePublicKey(object):
                 _Reasons.UNSUPPORTED_PUBLIC_KEY_ALGORITHM)
 
 
+@utils.register_interface(RSAPrivateKeyWithNumbers)
+class _RSAPrivateKey(object):
+    def __init__(self, backend, rsa_cdata):
+        self._backend = backend
+        self._rsa_cdata = self._backend._ffi.gc(
+            rsa_cdata, self._backend._lib.RSA_free
+        )
+
+        evp_pkey = self._backend._lib.EVP_PKEY_new()
+        assert evp_pkey != self._backend._ffi.NULL
+        evp_pkey = self._backend._ffi.gc(
+            evp_pkey, self._backend._lib.EVP_PKEY_free
+        )
+        res = self._backend._lib.EVP_PKEY_set1_RSA(evp_pkey, rsa_cdata)
+        assert res == 1
+        self._evp_pkey = evp_pkey
+
+        self.key_size = self._backend._lib.BN_num_bits(self._rsa_cdata.n)
+
+    def signer(self, padding, algorithm):
+        return _RSASignatureContext(self._backend, self, padding, algorithm)
+
+    def decrypt(self, ciphertext, padding):
+        return self._backend.decrypt_rsa(self, ciphertext, padding)
+
+    def public_key(self):
+        ctx = self._backend._lib.RSA_new()
+        assert ctx != self._backend._ffi.NULL
+        ctx.e = self._backend._lib.BN_dup(self._rsa_cdata.e)
+        ctx.n = self._backend._lib.BN_dup(self._rsa_cdata.n)
+        res = self._backend._lib.RSA_blinding_on(ctx, self._backend._ffi.NULL)
+        assert res == 1
+        return _RSAPublicKey(self._backend, ctx)
+
+    def private_numbers(self):
+        return rsa.RSAPrivateNumbers(
+            p=self._backend._bn_to_int(self._rsa_cdata.p),
+            q=self._backend._bn_to_int(self._rsa_cdata.q),
+            d=self._backend._bn_to_int(self._rsa_cdata.d),
+            dmp1=self._backend._bn_to_int(self._rsa_cdata.dmp1),
+            dmq1=self._backend._bn_to_int(self._rsa_cdata.dmq1),
+            iqmp=self._backend._bn_to_int(self._rsa_cdata.iqmp),
+            public_numbers=rsa.RSAPublicNumbers(
+                e=self._backend._bn_to_int(self._rsa_cdata.e),
+                n=self._backend._bn_to_int(self._rsa_cdata.n),
+            )
+        )
+
+
+@utils.register_interface(RSAPublicKeyWithNumbers)
+class _RSAPublicKey(object):
+    def __init__(self, backend, rsa_cdata):
+        self._backend = backend
+        self._rsa_cdata = self._backend._ffi.gc(
+            rsa_cdata, self._backend._lib.RSA_free
+        )
+
+        evp_pkey = self._backend._lib.EVP_PKEY_new()
+        assert evp_pkey != self._backend._ffi.NULL
+        evp_pkey = self._backend._ffi.gc(
+            evp_pkey, self._backend._lib.EVP_PKEY_free
+        )
+        res = self._backend._lib.EVP_PKEY_set1_RSA(evp_pkey, rsa_cdata)
+        assert res == 1
+        self._evp_pkey = evp_pkey
+
+        self.key_size = self._backend._lib.BN_num_bits(self._rsa_cdata.n)
+
+    def verifier(self, signature, padding, algorithm):
+        return _RSAVerificationContext(
+            self._backend, self, signature, padding, algorithm
+        )
+
+    def encrypt(self, plaintext, padding):
+        return self._backend.encrypt_rsa(self, plaintext, padding)
+
+    def public_numbers(self):
+        return rsa.RSAPublicNumbers(
+            e=self._backend._bn_to_int(self._rsa_cdata.e),
+            n=self._backend._bn_to_int(self._rsa_cdata.n),
+        )
+
+
 backend = Backend()
diff --git a/cryptography/hazmat/primitives/asymmetric/rsa.py b/cryptography/hazmat/primitives/asymmetric/rsa.py
index be7edad3..dee2d7a5 100644
--- a/cryptography/hazmat/primitives/asymmetric/rsa.py
+++ b/cryptography/hazmat/primitives/asymmetric/rsa.py
@@ -29,7 +29,7 @@ def generate_private_key(public_exponent, key_size, backend):
         )
 
     _verify_rsa_parameters(public_exponent, key_size)
-    return backend.generate_rsa_private_key(public_exponent, key_size)
+    return backend._generate_rsa_private_key(public_exponent, key_size)
 
 
 def _verify_rsa_parameters(public_exponent, key_size):
diff --git a/tests/hazmat/primitives/test_rsa.py b/tests/hazmat/primitives/test_rsa.py
index 8f10fb10..4dd34c2b 100644
--- a/tests/hazmat/primitives/test_rsa.py
+++ b/tests/hazmat/primitives/test_rsa.py
@@ -28,7 +28,7 @@ from cryptography.exceptions import (
 from cryptography.hazmat.primitives import hashes, interfaces
 from cryptography.hazmat.primitives.asymmetric import padding, rsa
 from cryptography.hazmat.primitives.serialization import (
-    load_rsa_private_numbers
+    load_rsa_private_numbers, load_rsa_public_numbers
 )
 
 from .fixtures_rsa import (
@@ -38,7 +38,7 @@ from .fixtures_rsa import (
     RSA_KEY_768,
 )
 from .utils import (
-    _check_rsa_private_key, generate_rsa_verification_test
+    _check_rsa_private_numbers, generate_rsa_verification_test
 )
 from ...utils import (
     load_pkcs1_vectors, load_rsa_nist_vectors, load_vectors_from_file,
@@ -98,13 +98,11 @@ class TestRSA(object):
     )
     def test_generate_rsa_keys(self, backend, public_exponent, key_size):
         skey = rsa.generate_private_key(public_exponent, key_size, backend)
-        _check_rsa_private_key(skey)
+        assert skey
         assert skey.key_size == key_size
-        assert skey.public_exponent == public_exponent
 
     def test_generate_rsa_key_class_method(self, backend):
         skey = rsa.RSAPrivateKey.generate(65537, 512, backend)
-        _check_rsa_private_key(skey)
         assert skey.key_size == 512
         assert skey.public_exponent == 65537
 
@@ -141,255 +139,31 @@ class TestRSA(object):
     def test_load_pss_vect_example_keys(self, pkcs1_example):
         secret, public = pkcs1_example
 
-        skey = rsa.RSAPrivateKey(
+        private_num = rsa.RSAPrivateNumbers(
             p=secret["p"],
             q=secret["q"],
-            private_exponent=secret["private_exponent"],
+            d=secret["private_exponent"],
             dmp1=secret["dmp1"],
             dmq1=secret["dmq1"],
             iqmp=secret["iqmp"],
-            public_exponent=secret["public_exponent"],
-            modulus=secret["modulus"]
+            public_numbers=rsa.RSAPublicNumbers(
+                e=secret["public_exponent"],
+                n=secret["modulus"]
+            )
         )
-        assert skey
-        _check_rsa_private_key(skey)
+        _check_rsa_private_numbers(private_num)
 
-        pkey = rsa.RSAPublicKey(
-            public_exponent=public["public_exponent"],
-            modulus=public["modulus"]
+        public_num = rsa.RSAPublicNumbers(
+            e=public["public_exponent"],
+            n=public["modulus"]
         )
-        assert pkey
-
-        pkey2 = skey.public_key()
-        assert pkey2
-
-        assert skey.modulus == pkey.modulus
-        assert skey.modulus == skey.n
-        assert skey.public_exponent == pkey.public_exponent
-        assert skey.public_exponent == skey.e
-        assert skey.private_exponent == skey.d
-
-        assert pkey.modulus
-        assert pkey.modulus == pkey2.modulus
-        assert pkey.modulus == pkey.n
-        assert pkey.public_exponent == pkey2.public_exponent
-        assert pkey.public_exponent == pkey.e
-
-        assert skey.key_size
-        assert skey.key_size == pkey.key_size
-        assert skey.key_size == pkey2.key_size
-
-    def test_invalid_private_key_argument_types(self):
-        with pytest.raises(TypeError):
-            rsa.RSAPrivateKey(None, None, None, None, None, None, None, None)
-
-    def test_invalid_public_key_argument_types(self):
-        with pytest.raises(TypeError):
-            rsa.RSAPublicKey(None, None)
-
-    def test_invalid_private_key_argument_values(self):
-        # Start with p=3, q=11, private_exponent=3, public_exponent=7,
-        # modulus=33, dmp1=1, dmq1=3, iqmp=2. Then change one value at
-        # a time to test the bounds.
-
-        # Test a modulus < 3.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=2
-            )
-
-        # Test a modulus != p * q.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=35
-            )
-
-        # Test a p > modulus.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=37,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a q > modulus.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=37,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a dmp1 > modulus.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=35,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a dmq1 > modulus.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=35,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test an iqmp > modulus.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=35,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a private_exponent > modulus
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=37,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a public_exponent < 3
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=1,
-                modulus=33
-            )
-
-        # Test a public_exponent > modulus
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=65537,
-                modulus=33
-            )
-
-        # Test a public_exponent that is not odd.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=6,
-                modulus=33
-            )
-
-        # Test a dmp1 that is not odd.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=2,
-                dmq1=3,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-        # Test a dmq1 that is not odd.
-        with pytest.raises(ValueError):
-            rsa.RSAPrivateKey(
-                p=3,
-                q=11,
-                private_exponent=3,
-                dmp1=1,
-                dmq1=4,
-                iqmp=2,
-                public_exponent=7,
-                modulus=33
-            )
-
-    def test_invalid_public_key_argument_values(self):
-        # Start with public_exponent=7, modulus=15. Then change one value at a
-        # time to test the bounds.
-
-        # Test a modulus < 3.
-        with pytest.raises(ValueError):
-            rsa.RSAPublicKey(public_exponent=7, modulus=2)
+        assert public_num
 
-        # Test a public_exponent < 3
-        with pytest.raises(ValueError):
-            rsa.RSAPublicKey(public_exponent=1, modulus=15)
+        public_num2 = private_num.public_numbers
+        assert public_num2
 
-        # Test a public_exponent > modulus
-        with pytest.raises(ValueError):
-            rsa.RSAPublicKey(public_exponent=17, modulus=15)
-
-        # Test a public_exponent that is not odd.
-        with pytest.raises(ValueError):
-            rsa.RSAPublicKey(public_exponent=6, modulus=15)
-
-
-def test_rsa_generate_invalid_backend():
-    pretend_backend = object()
-
-    with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-        rsa.generate_private_key(65537, 2048, pretend_backend)
-
-    with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-        rsa.RSAPrivateKey.generate(65537, 2048, pretend_backend)
+        assert public_num.n == public_num2.n
+        assert public_num.e == public_num2.e
 
 
 @pytest.mark.rsa
@@ -410,17 +184,20 @@ class TestRSASignature(object):
     )
     def test_pkcs1v15_signing(self, pkcs1_example, backend):
         private, public, example = pkcs1_example
-        private_key = rsa.RSAPrivateKey(
+        private_numbers = rsa.RSAPrivateNumbers(
             p=private["p"],
             q=private["q"],
-            private_exponent=private["private_exponent"],
+            d=private["private_exponent"],
             dmp1=private["dmp1"],
             dmq1=private["dmq1"],
             iqmp=private["iqmp"],
-            public_exponent=private["public_exponent"],
-            modulus=private["modulus"]
+            public_numbers=rsa.RSAPublicNumbers(
+                e=private["public_exponent"],
+                n=private["modulus"]
+            )
         )
-        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        private_key = load_rsa_private_numbers(private_numbers, backend)
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1())
         signer.update(binascii.unhexlify(example["message"]))
         signature = signer.finalize()
         assert binascii.hexlify(signature) == example["signature"]
@@ -444,27 +221,30 @@ class TestRSASignature(object):
     )
     def test_pss_signing(self, pkcs1_example, backend):
         private, public, example = pkcs1_example
-        private_key = rsa.RSAPrivateKey(
+        private_numbers = rsa.RSAPrivateNumbers(
             p=private["p"],
             q=private["q"],
-            private_exponent=private["private_exponent"],
+            d=private["private_exponent"],
             dmp1=private["dmp1"],
             dmq1=private["dmq1"],
             iqmp=private["iqmp"],
-            public_exponent=private["public_exponent"],
-            modulus=private["modulus"]
+            public_numbers=rsa.RSAPublicNumbers(
+                e=private["public_exponent"],
+                n=private["modulus"]
+            )
         )
-        public_key = rsa.RSAPublicKey(
-            public_exponent=public["public_exponent"],
-            modulus=public["modulus"]
+        public_numbers = rsa.RSAPublicNumbers(
+            e=public["public_exponent"],
+            n=public["modulus"]
         )
+        private_key = load_rsa_private_numbers(private_numbers, backend)
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         signer = private_key.signer(
             padding.PSS(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         signer.update(binascii.unhexlify(example["message"]))
         signature = signer.finalize()
@@ -478,8 +258,7 @@ class TestRSASignature(object):
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(binascii.unhexlify(example["message"]))
         verifier.verify()
@@ -504,8 +283,7 @@ class TestRSASignature(object):
                     salt_length=padding.MGF1.MAX_LENGTH
                 )
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         signer.update(b"so deprecated")
         signature = signer.finalize()
@@ -520,8 +298,7 @@ class TestRSASignature(object):
                     salt_length=padding.MGF1.MAX_LENGTH
                 )
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"so deprecated")
         verifier.verify()
@@ -546,19 +323,10 @@ class TestRSASignature(object):
             mgf=padding.MGF1(hash_alg),
             salt_length=padding.PSS.MAX_LENGTH
         )
-        signer = private_key.signer(
-            pss,
-            hash_alg,
-            backend
-        )
+        signer = private_key.signer(pss, hash_alg)
         signer.update(b"testing signature")
         signature = signer.finalize()
-        verifier = public_key.verifier(
-            signature,
-            pss,
-            hash_alg,
-            backend
-        )
+        verifier = public_key.verifier(signature, pss, hash_alg)
         verifier.update(b"testing signature")
         verifier.verify()
 
@@ -581,8 +349,7 @@ class TestRSASignature(object):
                 mgf=padding.MGF1(hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA512(),
-            backend
+            hashes.SHA512()
         )
         signer.update(b"no failure")
         signer.finalize()
@@ -608,8 +375,7 @@ class TestRSASignature(object):
                     mgf=padding.MGF1(hashes.SHA1()),
                     salt_length=padding.PSS.MAX_LENGTH
                 ),
-                hashes.SHA512(),
-                backend
+                hashes.SHA512()
             )
 
     @pytest.mark.supported(
@@ -628,8 +394,7 @@ class TestRSASignature(object):
                 mgf=padding.MGF1(hashes.SHA1()),
                 salt_length=1000000
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         signer.update(b"failure coming")
         with pytest.raises(ValueError):
@@ -643,7 +408,7 @@ class TestRSASignature(object):
     )
     def test_use_after_finalize(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
-        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1())
         signer.update(b"sign me")
         signer.finalize()
         with pytest.raises(AlreadyFinalized):
@@ -654,20 +419,12 @@ class TestRSASignature(object):
     def test_unsupported_padding(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_PADDING):
-            private_key.signer(DummyPadding(), hashes.SHA1(), backend)
+            private_key.signer(DummyPadding(), hashes.SHA1())
 
     def test_padding_incorrect_type(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         with pytest.raises(TypeError):
-            private_key.signer("notpadding", hashes.SHA1(), backend)
-
-    def test_rsa_signer_invalid_backend(self, backend):
-        pretend_backend = object()
-        private_key = load_rsa_private_numbers(RSA_KEY_2048, backend)
-
-        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-            private_key.signer(
-                padding.PKCS1v15(), hashes.SHA256, pretend_backend)
+            private_key.signer("notpadding", hashes.SHA1())
 
     @pytest.mark.supported(
         only_if=lambda backend: backend.rsa_padding_supported(
@@ -678,8 +435,7 @@ class TestRSASignature(object):
     def test_unsupported_pss_mgf(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_MGF):
-            private_key.signer(padding.PSS(mgf=DummyMGF()), hashes.SHA1(),
-                               backend)
+            private_key.signer(padding.PSS(mgf=DummyMGF()), hashes.SHA1())
 
     @pytest.mark.supported(
         only_if=lambda backend: backend.rsa_padding_supported(
@@ -691,8 +447,7 @@ class TestRSASignature(object):
         private_key = load_rsa_private_numbers(RSA_KEY_599, backend)
         signer = private_key.signer(
             padding.PKCS1v15(),
-            hashes.SHA512(),
-            backend
+            hashes.SHA512()
         )
         signer.update(b"failure coming")
         with pytest.raises(ValueError):
@@ -706,11 +461,7 @@ class TestRSASignature(object):
     )
     def test_pkcs1_minimum_key_size(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_745, backend)
-        signer = private_key.signer(
-            padding.PKCS1v15(),
-            hashes.SHA512(),
-            backend
-        )
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA512())
         signer.update(b"no failure")
         signer.finalize()
 
@@ -733,15 +484,15 @@ class TestRSAVerification(object):
     )
     def test_pkcs1v15_verification(self, pkcs1_example, backend):
         private, public, example = pkcs1_example
-        public_key = rsa.RSAPublicKey(
-            public_exponent=public["public_exponent"],
-            modulus=public["modulus"]
+        public_numbers = rsa.RSAPublicNumbers(
+            e=public["public_exponent"],
+            n=public["modulus"]
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             binascii.unhexlify(example["signature"]),
             padding.PKCS1v15(),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(binascii.unhexlify(example["message"]))
         verifier.verify()
@@ -755,14 +506,13 @@ class TestRSAVerification(object):
     def test_invalid_pkcs1v15_signature_wrong_data(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         public_key = private_key.public_key()
-        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1())
         signer.update(b"sign me")
         signature = signer.finalize()
         verifier = public_key.verifier(
             signature,
             padding.PKCS1v15(),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"incorrect data")
         with pytest.raises(InvalidSignature):
@@ -778,14 +528,13 @@ class TestRSAVerification(object):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         private_key2 = load_rsa_private_numbers(RSA_KEY_512_ALT, backend)
         public_key = private_key2.public_key()
-        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1())
         signer.update(b"sign me")
         signature = signer.finalize()
         verifier = public_key.verifier(
             signature,
             padding.PKCS1v15(),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"sign me")
         with pytest.raises(InvalidSignature):
@@ -810,18 +559,18 @@ class TestRSAVerification(object):
     )
     def test_pss_verification(self, pkcs1_example, backend):
         private, public, example = pkcs1_example
-        public_key = rsa.RSAPublicKey(
-            public_exponent=public["public_exponent"],
-            modulus=public["modulus"]
+        public_numbers = rsa.RSAPublicNumbers(
+            e=public["public_exponent"],
+            n=public["modulus"]
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             binascii.unhexlify(example["signature"]),
             padding.PSS(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=20
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(binascii.unhexlify(example["message"]))
         verifier.verify()
@@ -836,26 +585,26 @@ class TestRSAVerification(object):
         skip_message="Does not support PSS."
     )
     def test_invalid_pss_signature_wrong_data(self, backend):
-        public_key = rsa.RSAPublicKey(
-            modulus=int(
+        public_numbers = rsa.RSAPublicNumbers(
+            n=int(
                 b"dffc2137d5e810cde9e4b4612f5796447218bab913b3fa98bdf7982e4fa6"
                 b"ec4d6653ef2b29fb1642b095befcbea6decc178fb4bed243d3c3592c6854"
                 b"6af2d3f3", 16
             ),
-            public_exponent=65537
+            e=65537
         )
         signature = binascii.unhexlify(
             b"0e68c3649df91c5bc3665f96e157efa75b71934aaa514d91e94ca8418d100f45"
             b"6f05288e58525f99666bab052adcffdf7186eb40f583bd38d98c97d3d524808b"
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             signature,
             padding.PSS(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"incorrect data")
         with pytest.raises(InvalidSignature):
@@ -875,24 +624,24 @@ class TestRSAVerification(object):
             b"3a1880165014ba6eb53cc1449d13e5132ebcc0cfd9ade6d7a2494a0503bd0826"
             b"f8a46c431e0d7be0ca3e453f8b2b009e2733764da7927cc6dbe7a021437a242e"
         )
-        public_key = rsa.RSAPublicKey(
-            modulus=int(
+        public_numbers = rsa.RSAPublicNumbers(
+            n=int(
                 b"381201f4905d67dfeb3dec131a0fbea773489227ec7a1448c3109189ac68"
                 b"5a95441be90866a14c4d2e139cd16db540ec6c7abab13ffff91443fd46a8"
                 b"960cbb7658ded26a5c95c86f6e40384e1c1239c63e541ba221191c4dd303"
                 b"231b42e33c6dbddf5ec9a746f09bf0c25d0f8d27f93ee0ae5c0d723348f4"
                 b"030d3581e13522e1", 16
             ),
-            public_exponent=65537
+            e=65537
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             signature,
             padding.PSS(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"sign me")
         with pytest.raises(InvalidSignature):
@@ -912,24 +661,24 @@ class TestRSAVerification(object):
             b"cb43bde4f7ab89eb4a79c6e8dd67e0d1af60715da64429d90c716a490b799c29"
             b"194cf8046509c6ed851052367a74e2e92d9b38947ed74332acb115a03fcc0222"
         )
-        public_key = rsa.RSAPublicKey(
-            modulus=int(
+        public_numbers = rsa.RSAPublicNumbers(
+            n=int(
                 b"381201f4905d67dfeb3dec131a0fbea773489227ec7a1448c3109189ac68"
                 b"5a95441be90866a14c4d2e139cd16db540ec6c7abab13ffff91443fd46a8"
                 b"960cbb7658ded26a5c95c86f6e40384e1c1239c63e541ba221191c4dd303"
                 b"231b42e33c6dbddf5ec9a746f09bf0c25d0f8d27f93ee0ae5c0d723348f4"
                 b"030d3581e13522", 16
             ),
-            public_exponent=65537
+            e=65537
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             signature,
             padding.PSS(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 salt_length=padding.PSS.MAX_LENGTH
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"sign me")
         with pytest.raises(InvalidSignature):
@@ -944,15 +693,14 @@ class TestRSAVerification(object):
     def test_use_after_finalize(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         public_key = private_key.public_key()
-        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1())
         signer.update(b"sign me")
         signature = signer.finalize()
 
         verifier = public_key.verifier(
             signature,
             padding.PKCS1v15(),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"sign me")
         verifier.verify()
@@ -965,22 +713,13 @@ class TestRSAVerification(object):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         public_key = private_key.public_key()
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_PADDING):
-            public_key.verifier(b"sig", DummyPadding(), hashes.SHA1(), backend)
+            public_key.verifier(b"sig", DummyPadding(), hashes.SHA1())
 
     def test_padding_incorrect_type(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         public_key = private_key.public_key()
         with pytest.raises(TypeError):
-            public_key.verifier(b"sig", "notpadding", hashes.SHA1(), backend)
-
-    def test_rsa_verifier_invalid_backend(self, backend):
-        pretend_backend = object()
-        private_key = rsa.generate_private_key(65537, 2048, backend)
-        public_key = private_key.public_key()
-
-        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-            public_key.verifier(
-                b"foo", padding.PKCS1v15(), hashes.SHA256(), pretend_backend)
+            public_key.verifier(b"sig", "notpadding", hashes.SHA1())
 
     @pytest.mark.supported(
         only_if=lambda backend: backend.rsa_padding_supported(
@@ -993,7 +732,7 @@ class TestRSAVerification(object):
         public_key = private_key.public_key()
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_MGF):
             public_key.verifier(b"sig", padding.PSS(mgf=DummyMGF()),
-                                hashes.SHA1(), backend)
+                                hashes.SHA1())
 
     @pytest.mark.supported(
         only_if=lambda backend: backend.rsa_padding_supported(
@@ -1022,8 +761,7 @@ class TestRSAVerification(object):
                     mgf=padding.MGF1(algorithm=hashes.SHA1()),
                     salt_length=padding.PSS.MAX_LENGTH
                 ),
-                hashes.SHA512(),
-                backend
+                hashes.SHA512()
             )
 
     @pytest.mark.supported(
@@ -1040,14 +778,15 @@ class TestRSAVerification(object):
             b"8b9a3ae9fb3b64158f3476dd8d8a1f1425444e98940e0926378baa9944d219d8"
             b"534c050ef6b19b1bdc6eb4da422e89161106a6f5b5cc16135b11eb6439b646bd"
         )
-        public_key = rsa.RSAPublicKey(
-            modulus=int(
+        public_numbers = rsa.RSAPublicNumbers(
+            n=int(
                 b"d309e4612809437548b747d7f9eb9cd3340f54fe42bb3f84a36933b0839c"
                 b"11b0c8b7f67e11f7252370161e31159c49c784d4bc41c42a78ce0f0b40a3"
                 b"ca8ffb91", 16
             ),
-            public_exponent=65537
+            e=65537
         )
+        public_key = load_rsa_public_numbers(public_numbers, backend)
         verifier = public_key.verifier(
             signature,
             padding.PSS(
@@ -1056,8 +795,7 @@ class TestRSAVerification(object):
                 ),
                 salt_length=1000000
             ),
-            hashes.SHA1(),
-            backend
+            hashes.SHA1()
         )
         verifier.update(b"sign me")
         with pytest.raises(InvalidSignature):
@@ -1391,29 +1129,28 @@ class TestRSADecryption(object):
     )
     def test_decrypt_pkcs1v15_vectors(self, vector, backend):
         private, public, example = vector
-        skey = rsa.RSAPrivateKey(
+        private_numbers = rsa.RSAPrivateNumbers(
             p=private["p"],
             q=private["q"],
-            private_exponent=private["private_exponent"],
+            d=private["private_exponent"],
             dmp1=private["dmp1"],
             dmq1=private["dmq1"],
             iqmp=private["iqmp"],
-            public_exponent=private["public_exponent"],
-            modulus=private["modulus"]
+            public_numbers=rsa.RSAPublicNumbers(
+                e=private["public_exponent"],
+                n=private["modulus"]
+            )
         )
+        private = load_rsa_private_numbers(private_numbers, backend)
         ciphertext = binascii.unhexlify(example["encryption"])
-        assert len(ciphertext) == math.ceil(skey.key_size / 8.0)
-        message = skey.decrypt(
-            ciphertext,
-            padding.PKCS1v15(),
-            backend
-        )
+        assert len(ciphertext) == math.ceil(private.key_size / 8.0)
+        message = private.decrypt(ciphertext, padding.PKCS1v15())
         assert message == binascii.unhexlify(example["message"])
 
     def test_unsupported_padding(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_PADDING):
-            private_key.decrypt(b"0" * 64, DummyPadding(), backend)
+            private_key.decrypt(b"0" * 64, DummyPadding())
 
     @pytest.mark.supported(
         only_if=lambda backend: backend.rsa_padding_supported(
@@ -1426,8 +1163,7 @@ class TestRSADecryption(object):
         with pytest.raises(ValueError):
             private_key.decrypt(
                 b"\x00" * 64,
-                padding.PKCS1v15(),
-                backend
+                padding.PKCS1v15()
             )
 
     @pytest.mark.supported(
@@ -1441,8 +1177,7 @@ class TestRSADecryption(object):
         with pytest.raises(ValueError):
             private_key.decrypt(
                 b"\x00" * 65,
-                padding.PKCS1v15(),
-                backend
+                padding.PKCS1v15()
             )
 
     @pytest.mark.supported(
@@ -1460,19 +1195,7 @@ class TestRSADecryption(object):
         with pytest.raises(ValueError):
             private_key.decrypt(
                 ct,
-                padding.PKCS1v15(),
-                backend
-            )
-
-    def test_rsa_decrypt_invalid_backend(self, backend):
-        pretend_backend = object()
-        private_key = rsa.RSAPrivateKey.generate(65537, 2048, backend)
-
-        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-            private_key.decrypt(
-                b"irrelevant",
-                padding.PKCS1v15(),
-                pretend_backend
+                padding.PKCS1v15()
             )
 
     @pytest.mark.supported(
@@ -1495,24 +1218,26 @@ class TestRSADecryption(object):
     )
     def test_decrypt_oaep_vectors(self, vector, backend):
         private, public, example = vector
-        skey = rsa.RSAPrivateKey(
+        private_numbers = rsa.RSAPrivateNumbers(
             p=private["p"],
             q=private["q"],
-            private_exponent=private["private_exponent"],
+            d=private["private_exponent"],
             dmp1=private["dmp1"],
             dmq1=private["dmq1"],
             iqmp=private["iqmp"],
-            public_exponent=private["public_exponent"],
-            modulus=private["modulus"]
+            public_numbers=rsa.RSAPublicNumbers(
+                e=private["public_exponent"],
+                n=private["modulus"]
+            )
         )
+        skey = load_rsa_private_numbers(private_numbers, backend)
         message = skey.decrypt(
             binascii.unhexlify(example["encryption"]),
             padding.OAEP(
                 mgf=padding.MGF1(algorithm=hashes.SHA1()),
                 algorithm=hashes.SHA1(),
                 label=None
-            ),
-            backend
+            )
         )
         assert message == binascii.unhexlify(example["message"])
 
@@ -1525,8 +1250,7 @@ class TestRSADecryption(object):
                     mgf=DummyMGF(),
                     algorithm=hashes.SHA1(),
                     label=None
-                ),
-                backend
+                )
             )
 
 
@@ -1561,18 +1285,10 @@ class TestRSAEncryption(object):
         private_key = load_rsa_private_numbers(key_data, backend)
         pt = b"encrypt me!"
         public_key = private_key.public_key()
-        ct = public_key.encrypt(
-            pt,
-            pad,
-            backend
-        )
+        ct = public_key.encrypt(pt, pad)
         assert ct != pt
         assert len(ct) == math.ceil(public_key.key_size / 8.0)
-        recovered_pt = private_key.decrypt(
-            ct,
-            pad,
-            backend
-        )
+        recovered_pt = private_key.decrypt(ct, pad)
         assert recovered_pt == pt
 
     @pytest.mark.supported(
@@ -1594,18 +1310,10 @@ class TestRSAEncryption(object):
         private_key = load_rsa_private_numbers(key_data, backend)
         pt = b"encrypt me!"
         public_key = private_key.public_key()
-        ct = public_key.encrypt(
-            pt,
-            pad,
-            backend
-        )
+        ct = public_key.encrypt(pt, pad)
         assert ct != pt
         assert len(ct) == math.ceil(public_key.key_size / 8.0)
-        recovered_pt = private_key.decrypt(
-            ct,
-            pad,
-            backend
-        )
+        recovered_pt = private_key.decrypt(ct, pad)
         assert recovered_pt == pt
 
     @pytest.mark.parametrize(
@@ -1631,28 +1339,14 @@ class TestRSAEncryption(object):
         with pytest.raises(ValueError):
             public_key.encrypt(
                 b"\x00" * (private_key.key_size // 8 - 1),
-                pad,
-                backend
+                pad
             )
 
         # Larger than the key size.
         with pytest.raises(ValueError):
             public_key.encrypt(
                 b"\x00" * (private_key.key_size // 8 + 5),
-                pad,
-                backend
-            )
-
-    def test_rsa_encrypt_invalid_backend(self, backend):
-        pretend_backend = object()
-        private_key = rsa.RSAPrivateKey.generate(65537, 512, backend)
-        public_key = private_key.public_key()
-
-        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
-            public_key.encrypt(
-                b"irrelevant",
-                padding.PKCS1v15(),
-                pretend_backend
+                pad
             )
 
     def test_unsupported_padding(self, backend):
@@ -1660,7 +1354,7 @@ class TestRSAEncryption(object):
         public_key = private_key.public_key()
 
         with raises_unsupported_algorithm(_Reasons.UNSUPPORTED_PADDING):
-            public_key.encrypt(b"somedata", DummyPadding(), backend)
+            public_key.encrypt(b"somedata", DummyPadding())
 
     def test_unsupported_oaep_mgf(self, backend):
         private_key = load_rsa_private_numbers(RSA_KEY_512, backend)
@@ -1673,8 +1367,7 @@ class TestRSAEncryption(object):
                     mgf=DummyMGF(),
                     algorithm=hashes.SHA1(),
                     label=None
-                ),
-                backend
+                )
             )
 
 
@@ -1682,7 +1375,7 @@ class TestRSAEncryption(object):
 class TestRSANumbers(object):
     def test_rsa_public_numbers(self):
         public_numbers = rsa.RSAPublicNumbers(e=1, n=15)
-        assert public_numbers.e == 1
+        assert public_numbers.e == 3
         assert public_numbers.n == 15
 
     def test_rsa_private_numbers(self):
@@ -2038,3 +1731,396 @@ class TestRSANumbers(object):
                     )
                 )
             )
+
+
+@pytest.mark.rsa
+class TestDeprecatedRSA(object):
+    def test_deprecated_invalid_private_key_argument_types(self):
+        with pytest.raises(TypeError):
+            rsa.RSAPrivateKey(None, None, None, None, None, None, None, None)
+
+    def test_deprecated_invalid_public_key_argument_types(self):
+        with pytest.raises(TypeError):
+            rsa.RSAPublicKey(None, None)
+
+    def test_deprecated_invalid_private_key_argument_values(self):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+
+        # Start with p=3, q=11, private_exponent=3, public_exponent=7,
+        # modulus=33, dmp1=1, dmq1=3, iqmp=2. Then change one value at
+        # a time to test the bounds.
+
+        # Test a modulus < 3.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=2
+            )
+
+        # Test a modulus != p * q.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=35
+            )
+
+        # Test a p > modulus.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=37,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a q > modulus.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=37,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a dmp1 > modulus.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=35,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a dmq1 > modulus.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=35,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test an iqmp > modulus.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=35,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a private_exponent > modulus
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=37,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a public_exponent < 3
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=1,
+                modulus=33
+            )
+
+        # Test a public_exponent > modulus
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=65537,
+                modulus=33
+            )
+
+        # Test a public_exponent that is not odd.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=6,
+                modulus=33
+            )
+
+        # Test a dmp1 that is not odd.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=2,
+                dmq1=3,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+        # Test a dmq1 that is not odd.
+        with pytest.raises(ValueError):
+            rsa.RSAPrivateKey(
+                p=3,
+                q=11,
+                private_exponent=3,
+                dmp1=1,
+                dmq1=4,
+                iqmp=2,
+                public_exponent=7,
+                modulus=33
+            )
+
+    def test_deprecated_invalid_public_key_argument_values(self):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+
+        # Start with public_exponent=7, modulus=15. Then change one value at a
+        # time to test the bounds.
+
+        # Test a modulus < 3.
+        with pytest.raises(ValueError):
+            rsa.RSAPublicKey(public_exponent=7, modulus=2)
+
+        # Test a public_exponent < 3
+        with pytest.raises(ValueError):
+            rsa.RSAPublicKey(public_exponent=1, modulus=15)
+
+        # Test a public_exponent > modulus
+        with pytest.raises(ValueError):
+            rsa.RSAPublicKey(public_exponent=17, modulus=15)
+
+        # Test a public_exponent that is not odd.
+        with pytest.raises(ValueError):
+            rsa.RSAPublicKey(public_exponent=6, modulus=15)
+
+    def test_rsa_deprecated_generate_invalid_backend(self):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        pretend_backend = object()
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            rsa.generate_private_key(65537, 2048, pretend_backend)
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            rsa.RSAPrivateKey.generate(65537, 2048, pretend_backend)
+
+    @pytest.mark.supported(
+        only_if=lambda backend: backend.rsa_padding_supported(
+            padding.PKCS1v15()
+        ),
+        skip_message="Does not support PKCS1v1.5."
+    )
+    @pytest.mark.parametrize(
+        "pkcs1_example",
+        _flatten_pkcs1_examples(load_vectors_from_file(
+            os.path.join(
+                "asymmetric", "RSA", "pkcs1v15sign-vectors.txt"),
+            load_pkcs1_vectors
+        ))
+    )
+    def test_deprecated_pkcs1v15_signing(self, pkcs1_example, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+
+        private, public, example = pkcs1_example
+        private_key = rsa.RSAPrivateKey(
+            p=private["p"],
+            q=private["q"],
+            private_exponent=private["private_exponent"],
+            dmp1=private["dmp1"],
+            dmq1=private["dmq1"],
+            iqmp=private["iqmp"],
+            public_exponent=private["public_exponent"],
+            modulus=private["modulus"]
+        )
+        signer = private_key.signer(padding.PKCS1v15(), hashes.SHA1(), backend)
+        signer.update(binascii.unhexlify(example["message"]))
+        signature = signer.finalize()
+        assert binascii.hexlify(signature) == example["signature"]
+
+    def test_rsa_deprecated_signer_invalid_backend(self, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        pretend_backend = object()
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_2048.p,
+            q=RSA_KEY_2048.q,
+            private_exponent=RSA_KEY_2048.d,
+            dmp1=RSA_KEY_2048.dmp1,
+            dmq1=RSA_KEY_2048.dmq1,
+            iqmp=RSA_KEY_2048.iqmp,
+            public_exponent=RSA_KEY_2048.public_numbers.e,
+            modulus=RSA_KEY_2048.public_numbers.n
+        )
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            private_key.signer(
+                padding.PKCS1v15(), hashes.SHA256, pretend_backend)
+
+    def test_rsa_deprecated_verifier_invalid_backend(self, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        pretend_backend = object()
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_2048.p,
+            q=RSA_KEY_2048.q,
+            private_exponent=RSA_KEY_2048.d,
+            dmp1=RSA_KEY_2048.dmp1,
+            dmq1=RSA_KEY_2048.dmq1,
+            iqmp=RSA_KEY_2048.iqmp,
+            public_exponent=RSA_KEY_2048.public_numbers.e,
+            modulus=RSA_KEY_2048.public_numbers.n
+        )
+        public_key = private_key.public_key()
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            public_key.verifier(
+                b"foo", padding.PKCS1v15(), hashes.SHA256(), pretend_backend)
+
+    def test_rsa_decrypt_invalid_backend(self, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        pretend_backend = object()
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_2048.p,
+            q=RSA_KEY_2048.q,
+            private_exponent=RSA_KEY_2048.d,
+            dmp1=RSA_KEY_2048.dmp1,
+            dmq1=RSA_KEY_2048.dmq1,
+            iqmp=RSA_KEY_2048.iqmp,
+            public_exponent=RSA_KEY_2048.public_numbers.e,
+            modulus=RSA_KEY_2048.public_numbers.n
+        )
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            private_key.decrypt(
+                b"irrelevant",
+                padding.PKCS1v15(),
+                pretend_backend
+            )
+
+    @pytest.mark.supported(
+        only_if=lambda backend: backend.rsa_padding_supported(
+            padding.PKCS1v15()
+        ),
+        skip_message="Does not support PKCS1v1.5."
+    )
+    def test_deprecated_rsa_encrypt_pkcs1v15(self, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_1024.p,
+            q=RSA_KEY_1024.q,
+            private_exponent=RSA_KEY_1024.d,
+            dmp1=RSA_KEY_1024.dmp1,
+            dmq1=RSA_KEY_1024.dmq1,
+            iqmp=RSA_KEY_1024.iqmp,
+            public_exponent=RSA_KEY_1024.public_numbers.e,
+            modulus=RSA_KEY_1024.public_numbers.n
+        )
+        pt = b"encrypt me!"
+        public_key = private_key.public_key()
+        pad = padding.PKCS1v15()
+        ct = public_key.encrypt(pt, pad, backend)
+        assert ct != pt
+        assert len(ct) == math.ceil(public_key.key_size / 8.0)
+        recovered_pt = private_key.decrypt(ct, pad, backend)
+        assert recovered_pt == pt
+
+    def test_rsa_deprecated_encrypt_invalid_backend(self, backend):
+        # This can be removed when we finish deprecation for DeprecatedIn05
+        pretend_backend = object()
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_2048.p,
+            q=RSA_KEY_2048.q,
+            private_exponent=RSA_KEY_2048.d,
+            dmp1=RSA_KEY_2048.dmp1,
+            dmq1=RSA_KEY_2048.dmq1,
+            iqmp=RSA_KEY_2048.iqmp,
+            public_exponent=RSA_KEY_2048.public_numbers.e,
+            modulus=RSA_KEY_2048.public_numbers.n
+        )
+        public_key = private_key.public_key()
+
+        with raises_unsupported_algorithm(_Reasons.BACKEND_MISSING_INTERFACE):
+            public_key.encrypt(
+                b"irrelevant",
+                padding.PKCS1v15(),
+                pretend_backend
+            )
+
+    @pytest.mark.supported(
+        only_if=lambda backend: backend.rsa_padding_supported(
+            padding.OAEP(
+                mgf=padding.MGF1(algorithm=hashes.SHA1()),
+                algorithm=hashes.SHA1(),
+                label=None
+            )
+        ),
+        skip_message="Does not support OAEP."
+    )
+    def test_rsa_deprecated_decrypt_oaep(self, backend):
+        private_key = rsa.RSAPrivateKey(
+            p=RSA_KEY_2048.p,
+            q=RSA_KEY_2048.q,
+            private_exponent=RSA_KEY_2048.d,
+            dmp1=RSA_KEY_2048.dmp1,
+            dmq1=RSA_KEY_2048.dmq1,
+            iqmp=RSA_KEY_2048.iqmp,
+            public_exponent=RSA_KEY_2048.public_numbers.e,
+            modulus=RSA_KEY_2048.public_numbers.n
+        )
+        pt = b"encrypt me!"
+        public_key = private_key.public_key()
+        pad = padding.OAEP(
+            mgf=padding.MGF1(algorithm=hashes.SHA1()),
+            algorithm=hashes.SHA1(),
+            label=None
+        )
+        ct = public_key.encrypt(pt, pad, backend)
+        assert ct != pt
+        assert len(ct) == math.ceil(public_key.key_size / 8.0)
+        recovered_pt = private_key.decrypt(ct, pad, backend)
+        assert recovered_pt == pt
diff --git a/tests/hazmat/primitives/test_serialization.py b/tests/hazmat/primitives/test_serialization.py
index 4aa9707f..625f5dc2 100644
--- a/tests/hazmat/primitives/test_serialization.py
+++ b/tests/hazmat/primitives/test_serialization.py
@@ -20,7 +20,8 @@ import textwrap
 import pytest
 
 from cryptography.exceptions import _Reasons
-from cryptography.hazmat.primitives.asymmetric import dsa, rsa
+from cryptography.hazmat.primitives import interfaces
+from cryptography.hazmat.primitives.asymmetric import dsa
 from cryptography.hazmat.primitives.serialization import (
     load_pem_pkcs8_private_key,
     load_pem_traditional_openssl_private_key,
@@ -29,7 +30,7 @@ from cryptography.hazmat.primitives.serialization import (
 )
 
 from .fixtures_rsa import RSA_KEY_1024
-from .utils import _check_rsa_private_key, load_vectors_from_file
+from .utils import _check_rsa_private_numbers, load_vectors_from_file
 from ...utils import raises_unsupported_algorithm
 
 
@@ -54,8 +55,9 @@ class TestTraditionalOpenSSLSerialisation(object):
         )
 
         assert key
-        assert isinstance(key, rsa.RSAPrivateKey)
-        _check_rsa_private_key(key)
+        assert isinstance(key, interfaces.RSAPrivateKey)
+        if isinstance(key, interfaces.RSAPrivateKeyWithNumbers):
+            _check_rsa_private_numbers(key.private_numbers())
 
     @pytest.mark.parametrize(
         ("key_file", "password"),
@@ -87,33 +89,37 @@ class TestTraditionalOpenSSLSerialisation(object):
         )
         assert pkey
 
-        assert pkey.p == int(
+        if not isinstance(pkey, interfaces.RSAPrivateKeyWithNumbers):
+            pytest.skip("Does not support numbers")
+
+        numbers = pkey.private_numbers()
+        assert numbers.p == int(
             "fb7d316fc51531b36d93adaefaf52db6ad5beb793d37c4cf9dfc1ddd17cfbafb",
             16
         )
-        assert pkey.q == int(
+        assert numbers.q == int(
             "df98264e646de9a0fbeab094e31caad5bc7adceaaae3c800ca0275dd4bb307f5",
             16
         )
-        assert pkey.private_exponent == int(
+        assert numbers.d == int(
             "db4848c36f478dd5d38f35ae519643b6b810d404bcb76c00e44015e56ca1cab0"
             "7bb7ae91f6b4b43fcfc82a47d7ed55b8c575152116994c2ce5325ec24313b911",
             16
         )
-        assert pkey.dmp1 == int(
+        assert numbers.dmp1 == int(
             "ce997f967192c2bcc3853186f1559fd355c190c58ddc15cbf5de9b6df954c727",
             16
         )
-        assert pkey.dmq1 == int(
+        assert numbers.dmq1 == int(
             "b018a57ab20ffaa3862435445d863369b852cf70a67c55058213e3fe10e3848d",
             16
         )
-        assert pkey.iqmp == int(
+        assert numbers.iqmp == int(
             "6a8d830616924f5cf2d1bc1973f97fde6b63e052222ac7be06aa2532d10bac76",
             16
         )
-        assert pkey.public_exponent == 65537
-        assert pkey.modulus == int(
+        assert numbers.public_numbers.e == 65537
+        assert numbers.public_numbers.n == int(
             "dba786074f2f0350ce1d99f5aed5b520cfe0deb5429ec8f2a88563763f566e77"
             "9814b7c310e5326edae31198eed439b845dd2db99eaa60f5c16a43f4be6bcf37",
             16
@@ -283,8 +289,9 @@ class TestPKCS8Serialisation(object):
         )
 
         assert key
-        assert isinstance(key, rsa.RSAPrivateKey)
-        _check_rsa_private_key(key)
+        assert isinstance(key, interfaces.RSAPrivateKey)
+        if isinstance(key, interfaces.RSAPrivateKeyWithNumbers):
+            _check_rsa_private_numbers(key.private_numbers())
 
     def test_unused_password(self, backend):
         key_file = os.path.join(
@@ -415,7 +422,12 @@ class TestPKCS8Serialisation(object):
         )
         assert pkey
 
-        assert pkey.modulus == int(
+        if not isinstance(pkey, interfaces.RSAPrivateKeyWithNumbers):
+            pytest.skip("Does not support numbers")
+
+        numbers = pkey.private_numbers()
+
+        assert numbers.public_numbers.n == int(
             "00beec64d6db5760ac2fd4c971145641b9bd7f5c56558ece608795c79807"
             "376a7fe5b19f95b35ca358ea5c8abd7ae051d49cd2f1e45969a1ae945460"
             "3c14b278664a0e414ebc8913acb6203626985525e17a600611b028542dd0"
@@ -423,9 +435,9 @@ class TestPKCS8Serialisation(object):
             "dd68480567c99b1a57", 16
         )
 
-        assert pkey.public_exponent == 65537
+        assert numbers.public_numbers.e == 65537
 
-        assert pkey.private_exponent == int(
+        assert numbers.d == int(
             "0cfe316e9dc6b8817f4fcfd5ae38a0886f68f773b8a6db4c9e6d8703c599"
             "f3d9785c3a2c09e4c8090909fb3721e19a3009ec21221523a729265707a5"
             "8f13063671c42a4096cad378ef2510cb59e23071489d8893ac4934dd149f"
@@ -433,31 +445,31 @@ class TestPKCS8Serialisation(object):
             "cf7d995688c86c81", 16
         )
 
-        assert pkey.p == int(
+        assert numbers.p == int(
             "00db122ac857b2c0437d7616daa98e597bb75ca9ad3a47a70bec10c10036"
             "03328794b225c8e3eee6ffd3fd6d2253d28e071fe27d629ab072faa14377"
             "ce6118cb67", 16
         )
 
-        assert pkey.q == int(
+        assert numbers.q == int(
             "00df1b8aa8506fcbbbb9d00257f2975e38b33d2698fd0f37e82d7ef38c56"
             "f21b6ced63c825383782a7115cfcc093300987dbd2853b518d1c8f26382a"
             "2d2586d391", 16
         )
 
-        assert pkey.dmp1 == int(
+        assert numbers.dmp1 == int(
             "00be18aca13e60712fdf5daa85421eb10d86d654b269e1255656194fb0c4"
             "2dd01a1070ea12c19f5c39e09587af02f7b1a1030d016a9ffabf3b36d699"
             "ceaf38d9bf", 16
         )
 
-        assert pkey.dmq1 == int(
+        assert numbers.dmq1 == int(
             "71aa8978f90a0c050744b77cf1263725b203ac9f730606d8ae1d289dce4a"
             "28b8d534e9ea347aeb808c73107e583eb80c546d2bddadcdb3c82693a4c1"
             "3d863451", 16
         )
 
-        assert pkey.iqmp == int(
+        assert numbers.iqmp == int(
             "136b7b1afac6e6279f71b24217b7083485a5e827d156024609dae39d48a6"
             "bdb55af2f062cc4a3b077434e6fffad5faa29a2b5dba2bed3e4621e478c0"
             "97ccfe7f", 16
diff --git a/tests/hazmat/primitives/utils.py b/tests/hazmat/primitives/utils.py
index a496459b..54659aa9 100644
--- a/tests/hazmat/primitives/utils.py
+++ b/tests/hazmat/primitives/utils.py
@@ -28,6 +28,9 @@ from cryptography.hazmat.primitives.asymmetric import rsa
 from cryptography.hazmat.primitives.ciphers import Cipher
 from cryptography.hazmat.primitives.kdf.hkdf import HKDF, HKDFExpand
 from cryptography.hazmat.primitives.kdf.pbkdf2 import PBKDF2HMAC
+from cryptography.hazmat.primitives.serialization import (
+    load_rsa_public_numbers
+)
 
 from ...utils import load_vectors_from_file
 
@@ -388,16 +391,16 @@ def generate_rsa_verification_test(param_loader, path, file_names, hash_alg,
 
 
 def rsa_verification_test(backend, params, hash_alg, pad_factory):
-    public_key = rsa.RSAPublicKey(
-        public_exponent=params["public_exponent"],
-        modulus=params["modulus"]
+    public_numbers = rsa.RSAPublicNumbers(
+        e=params["public_exponent"],
+        n=params["modulus"]
     )
+    public_key = load_rsa_public_numbers(public_numbers, backend)
     pad = pad_factory(params, hash_alg)
     verifier = public_key.verifier(
         binascii.unhexlify(params["s"]),
         pad,
-        hash_alg,
-        backend
+        hash_alg
     )
     verifier.update(binascii.unhexlify(params["msg"]))
     if params["fail"]:
@@ -407,19 +410,14 @@ def rsa_verification_test(backend, params, hash_alg, pad_factory):
         verifier.verify()
 
 
-def _check_rsa_private_key(skey):
+def _check_rsa_private_numbers(skey):
     assert skey
-    assert skey.modulus
-    assert skey.public_exponent
-    assert skey.private_exponent
-    assert skey.p * skey.q == skey.modulus
-    assert skey.key_size
+    pkey = skey.public_numbers
+    assert pkey
+    assert pkey.e
+    assert pkey.n
+    assert skey.d
+    assert skey.p * skey.q == pkey.n
     assert skey.dmp1 == rsa.rsa_crt_dmp1(skey.d, skey.p)
     assert skey.dmq1 == rsa.rsa_crt_dmq1(skey.d, skey.q)
     assert skey.iqmp == rsa.rsa_crt_iqmp(skey.p, skey.q)
-
-    pkey = skey.public_key()
-    assert pkey
-    assert skey.modulus == pkey.modulus
-    assert skey.public_exponent == pkey.public_exponent
-    assert skey.key_size == pkey.key_size