diff --git a/jose/backends/cryptography_backend.py b/jose/backends/cryptography_backend.py
index abd24260..1117e106 100644
--- a/jose/backends/cryptography_backend.py
+++ b/jose/backends/cryptography_backend.py
@@ -15,7 +15,7 @@
 from cryptography.x509 import load_pem_x509_certificate
 
 from ..constants import ALGORITHMS
-from ..exceptions import JWEError, JWKError
+from ..exceptions import JWEError, JWKError, JWKAlgMismatchError
 from ..utils import base64_to_long, base64url_decode, base64url_encode, ensure_binary, long_to_base64
 from .base import Key
 
@@ -52,7 +52,7 @@ class CryptographyECKey(Key):
 
     def __init__(self, key, algorithm, cryptography_backend=default_backend):
         if algorithm not in ALGORITHMS.EC:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a valid EC algorithm" % algorithm)
 
         self.hash_alg = {
             ALGORITHMS.ES256: self.SHA256,
@@ -97,7 +97,7 @@ def __init__(self, key, algorithm, cryptography_backend=default_backend):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "EC":
-            raise JWKError("Incorrect key type. Expected: 'EC', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'EC', Received: %s" % jwk_dict.get("kty"))
 
         if not all(k in jwk_dict for k in ["x", "y", "crv"]):
             raise JWKError("Mandatory parameters are missing")
@@ -226,7 +226,7 @@ class CryptographyRSAKey(Key):
 
     def __init__(self, key, algorithm, cryptography_backend=default_backend):
         if algorithm not in ALGORITHMS.RSA:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a valid RSA algorithm" % algorithm)
 
         self.hash_alg = {
             ALGORITHMS.RS256: self.SHA256,
@@ -273,7 +273,7 @@ def __init__(self, key, algorithm, cryptography_backend=default_backend):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "RSA":
-            raise JWKError("Incorrect key type. Expected: 'RSA', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'RSA', Received: %s" % jwk_dict.get("kty"))
 
         e = base64_to_long(jwk_dict.get("e", 256))
         n = base64_to_long(jwk_dict.get("n"))
@@ -441,9 +441,9 @@ class CryptographyAESKey(Key):
 
     def __init__(self, key, algorithm):
         if algorithm not in ALGORITHMS.AES:
-            raise JWKError("%s is not a valid AES algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a valid AES algorithm" % algorithm)
         if algorithm not in ALGORITHMS.SUPPORTED.union(ALGORITHMS.AES_PSEUDO):
-            raise JWKError("%s is not a supported algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a supported algorithm" % algorithm)
 
         self._algorithm = algorithm
         self._mode = self.MODES.get(self._algorithm)
@@ -538,7 +538,7 @@ class CryptographyHMACKey(Key):
 
     def __init__(self, key, algorithm):
         if algorithm not in ALGORITHMS.HMAC:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("hash_alg: %s is not a valid hash algorithm" % algorithm)
         self._algorithm = algorithm
         self._hash_alg = self.ALG_MAP.get(algorithm)
 
@@ -569,7 +569,7 @@ def __init__(self, key, algorithm):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "oct":
-            raise JWKError("Incorrect key type. Expected: 'oct', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'oct', Received: %s" % jwk_dict.get("kty"))
 
         k = jwk_dict.get("k")
         k = k.encode("utf-8")
diff --git a/jose/backends/ecdsa_backend.py b/jose/backends/ecdsa_backend.py
index 756c7ea8..ecb5aac6 100644
--- a/jose/backends/ecdsa_backend.py
+++ b/jose/backends/ecdsa_backend.py
@@ -4,7 +4,7 @@
 
 from jose.backends.base import Key
 from jose.constants import ALGORITHMS
-from jose.exceptions import JWKError
+from jose.exceptions import JWKError, JWKAlgMismatchError
 from jose.utils import base64_to_long, long_to_base64
 
 
@@ -35,7 +35,7 @@ class ECDSAECKey(Key):
 
     def __init__(self, key, algorithm):
         if algorithm not in ALGORITHMS.EC:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a valid EC algorithm" % algorithm)
 
         self.hash_alg = {
             ALGORITHMS.ES256: self.SHA256,
@@ -75,7 +75,7 @@ def __init__(self, key, algorithm):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "EC":
-            raise JWKError("Incorrect key type. Expected: 'EC', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'EC', Received: %s" % jwk_dict.get("kty"))
 
         if not all(k in jwk_dict for k in ["x", "y", "crv"]):
             raise JWKError("Mandatory parameters are missing")
diff --git a/jose/backends/native.py b/jose/backends/native.py
index eb3a6ae3..7661e2c5 100644
--- a/jose/backends/native.py
+++ b/jose/backends/native.py
@@ -4,7 +4,7 @@
 
 from jose.backends.base import Key
 from jose.constants import ALGORITHMS
-from jose.exceptions import JWKError
+from jose.exceptions import JWKError, JWKAlgMismatchError
 from jose.utils import base64url_decode, base64url_encode
 
 
@@ -22,7 +22,7 @@ class HMACKey(Key):
 
     def __init__(self, key, algorithm):
         if algorithm not in ALGORITHMS.HMAC:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("hash_alg: %s is not a valid hash algorithm" % algorithm)
         self._algorithm = algorithm
         self._hash_alg = self.HASHES.get(algorithm)
 
@@ -53,7 +53,7 @@ def __init__(self, key, algorithm):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "oct":
-            raise JWKError("Incorrect key type. Expected: 'oct', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'oct', Received: %s" % jwk_dict.get("kty"))
 
         k = jwk_dict.get("k")
         k = k.encode("utf-8")
diff --git a/jose/backends/rsa_backend.py b/jose/backends/rsa_backend.py
index 4e8ccf1c..c908e4b3 100644
--- a/jose/backends/rsa_backend.py
+++ b/jose/backends/rsa_backend.py
@@ -13,7 +13,7 @@
 )
 from jose.backends.base import Key
 from jose.constants import ALGORITHMS
-from jose.exceptions import JWEError, JWKError
+from jose.exceptions import JWEError, JWKError, JWKAlgMismatchError
 from jose.utils import base64_to_long, long_to_base64
 
 ALGORITHMS.SUPPORTED.remove(ALGORITHMS.RSA_OAEP)  # RSA OAEP not supported
@@ -124,7 +124,7 @@ class RSAKey(Key):
 
     def __init__(self, key, algorithm):
         if algorithm not in ALGORITHMS.RSA:
-            raise JWKError("hash_alg: %s is not a valid hash algorithm" % algorithm)
+            raise JWKAlgMismatchError("%s is not a valid RSA algorithm" % algorithm)
 
         if algorithm in ALGORITHMS.RSA_KW and algorithm != ALGORITHMS.RSA1_5:
             raise JWKError("alg: %s is not supported by the RSA backend" % algorithm)
@@ -174,7 +174,7 @@ def __init__(self, key, algorithm):
 
     def _process_jwk(self, jwk_dict):
         if not jwk_dict.get("kty") == "RSA":
-            raise JWKError("Incorrect key type. Expected: 'RSA', Received: %s" % jwk_dict.get("kty"))
+            raise JWKAlgMismatchError("Incorrect key type. Expected: 'RSA', Received: %s" % jwk_dict.get("kty"))
 
         e = base64_to_long(jwk_dict.get("e"))
         n = base64_to_long(jwk_dict.get("n"))
diff --git a/jose/exceptions.py b/jose/exceptions.py
index e8edc3b6..2099208d 100644
--- a/jose/exceptions.py
+++ b/jose/exceptions.py
@@ -30,6 +30,11 @@ class JWKError(JOSEError):
     pass
 
 
+class JWKAlgMismatchError(JWKError):
+    '''JWK Key type doesn't support the given algorithm.'''
+    pass
+
+
 class JWEError(JOSEError):
     """Base error for all JWE errors"""
 
diff --git a/jose/jws.py b/jose/jws.py
index bfaf6bd0..944130e1 100644
--- a/jose/jws.py
+++ b/jose/jws.py
@@ -5,7 +5,7 @@
 from jose import jwk
 from jose.backends.base import Key
 from jose.constants import ALGORITHMS
-from jose.exceptions import JWSError, JWSSignatureError
+from jose.exceptions import JWSError, JWSSignatureError, JWKAlgMismatchError
 from jose.utils import base64url_decode, base64url_encode
 
 
@@ -205,7 +205,10 @@ def _load(jwt):
 def _sig_matches_keys(keys, signing_input, signature, alg):
     for key in keys:
         if not isinstance(key, Key):
-            key = jwk.construct(key, alg)
+            try:
+                key = jwk.construct(key, alg)
+            except JWKAlgMismatchError:
+                continue
         try:
             if key.verify(signing_input, signature):
                 return True
diff --git a/tests/test_jws.py b/tests/test_jws.py
index 01b5fd05..75a7398d 100644
--- a/tests/test_jws.py
+++ b/tests/test_jws.py
@@ -6,7 +6,7 @@
 from jose import jwk, jws
 from jose.backends import RSAKey
 from jose.constants import ALGORITHMS
-from jose.exceptions import JWSError
+from jose.exceptions import JWSError, JWKAlgMismatchError
 
 try:
     from jose.backends.cryptography_backend import CryptographyRSAKey
@@ -25,6 +25,16 @@ def test_unicode_token(self):
         token = "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJhIjoiYiJ9.jiMyrsmD8AoHWeQgmxZ5yq8z0lXS67_QGs52AzC8Ru8"
         jws.verify(token, "secret", ["HS256"])
 
+    def test_hetero_keys(self):
+        class BadKey(jwk.Key):
+            def __init__(self, key, algorithm):
+                if key != "xyzw":
+                    raise JWKAlgMismatchError("%s is not a valid XYZW algorithm" % algorithm)
+
+        token = "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJhIjoiYiJ9.jiMyrsmD8AoHWeQgmxZ5yq8z0lXS67_QGs52AzC8Ru8"
+        jwk.register_key("XYZW", BadKey)
+        jws.verify(token, {"keys": [{"alg": "XYZW"}, "secret"]}, ["XYZW", "HS256"])
+
     def test_multiple_keys(self):
         old_jwk_verify = jwk.HMACKey.verify
         try: