diff --git a/jose/backends/cryptography_backend.py b/jose/backends/cryptography_backend.py
index abd24260..f392b098 100644
--- a/jose/backends/cryptography_backend.py
+++ b/jose/backends/cryptography_backend.py
@@ -1,3 +1,4 @@
+import re
 import math
 import warnings
 
@@ -22,6 +23,68 @@
 _binding = None
 
 
+# Based on https://github.com/jpadilla/pyjwt/commit/9c528670c455b8d948aff95ed50e22940d1ad3fc
+# Based on https://github.com/hynek/pem/blob/7ad94db26b0bc21d10953f5dbad3acfdfacf57aa/src/pem/_core.py#L224-L252
+_PEMS = {
+    b"CERTIFICATE",
+    b"TRUSTED CERTIFICATE",
+    b"PRIVATE KEY",
+    b"PUBLIC KEY",
+    b"ENCRYPTED PRIVATE KEY",
+    b"OPENSSH PRIVATE KEY",
+    b"DSA PRIVATE KEY",
+    b"RSA PRIVATE KEY",
+    b"RSA PUBLIC KEY",
+    b"EC PRIVATE KEY",
+    b"DH PARAMETERS",
+    b"NEW CERTIFICATE REQUEST",
+    b"CERTIFICATE REQUEST",
+    b"SSH2 PUBLIC KEY",
+    b"SSH2 ENCRYPTED PRIVATE KEY",
+    b"X509 CRL",
+}
+
+
+_PEM_RE = re.compile(
+    b"----[- ]BEGIN ("
+    + b"|".join(_PEMS)
+    + b""")[- ]----\r?
+.+?\r?
+----[- ]END \\1[- ]----\r?\n?""",
+    re.DOTALL,
+)
+
+
+def is_pem_format(key):
+    return bool(_PEM_RE.search(key))
+
+
+# Based on https://github.com/pyca/cryptography/blob/bcb70852d577b3f490f015378c75cba74986297b/src/cryptography/hazmat/primitives/serialization/ssh.py#L40-L46
+_CERT_SUFFIX = b"-cert-v01@openssh.com"
+_SSH_PUBKEY_RC = re.compile(br"\A(\S+)[ \t]+(\S+)")
+_SSH_KEY_FORMATS = [
+    b"ssh-ed25519",
+    b"ssh-rsa",
+    b"ssh-dss",
+    b"ecdsa-sha2-nistp256",
+    b"ecdsa-sha2-nistp384",
+    b"ecdsa-sha2-nistp521",
+]
+
+
+def is_ssh_key(key):
+    if any(string_value in key for string_value in _SSH_KEY_FORMATS):
+        return True
+
+    ssh_pubkey_match = _SSH_PUBKEY_RC.match(key)
+    if ssh_pubkey_match:
+        key_type = ssh_pubkey_match.group(1)
+        if _CERT_SUFFIX == key_type[-len(_CERT_SUFFIX) :]:
+            return True
+
+    return False
+
+
 def get_random_bytes(num_bytes):
     """
     Get random bytes
@@ -552,14 +615,7 @@ def __init__(self, key, algorithm):
         if isinstance(key, str):
             key = key.encode("utf-8")
 
-        invalid_strings = [
-            b"-----BEGIN PUBLIC KEY-----",
-            b"-----BEGIN RSA PUBLIC KEY-----",
-            b"-----BEGIN CERTIFICATE-----",
-            b"ssh-rsa",
-        ]
-
-        if any(string_value in key for string_value in invalid_strings):
+        if is_pem_format(key) or is_ssh_key(key):
             raise JWKError(
                 "The specified key is an asymmetric key or x509 certificate and"
                 " should not be used as an HMAC secret."
diff --git a/tests/test_jwt.py b/tests/test_jwt.py
index 8c2e262f..378504f8 100644
--- a/tests/test_jwt.py
+++ b/tests/test_jwt.py
@@ -5,7 +5,7 @@
 import pytest
 
 from jose import jws, jwt
-from jose.exceptions import JWTError
+from jose.exceptions import JWTError, JWKError
 
 
 @pytest.fixture
@@ -522,3 +522,36 @@ def test_require(self, claims, key, claim, value):
         new_claims[claim] = value
         token = jwt.encode(new_claims, key)
         jwt.decode(token, key, options=options, audience=str(value))
+
+    def test_CVE_2024_33663(self):
+        """Test based on https://github.com/mpdavis/python-jose/issues/346"""
+        from Crypto.PublicKey import ECC
+        from Crypto.Hash import HMAC, SHA256
+
+        # ----- SETUP -----
+        # generate an asymmetric ECC keypair
+        # !! signing should only be possible with the private key !!
+        KEY = ECC.generate(curve='P-256')
+
+        # PUBLIC KEY, AVAILABLE TO USER
+        # CAN BE RECOVERED THROUGH E.G. PUBKEY RECOVERY WITH TWO SIGNATURES:
+        # https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm#Public_key_recovery
+        # https://github.com/FlorianPicca/JWT-Key-Recovery
+        PUBKEY = KEY.public_key().export_key(format='OpenSSH').encode()
+
+        # ---- CLIENT SIDE -----
+        # without knowing the private key, a valid token can be constructed
+        # YIKES!!
+
+        b64 = lambda x:base64.urlsafe_b64encode(x).replace(b'=',b'')
+        payload = b64(b'{"alg":"HS256"}') + b'.' + b64(b'{"pwned":true}')
+        hasher = HMAC.new(PUBKEY, digestmod=SHA256)
+        hasher.update(payload)
+        evil_token = payload + b'.' + b64(hasher.digest())
+
+        # ---- SERVER SIDE -----
+        # verify and decode the token using the public key, as is custom
+        # algorithm field is left unspecified
+        # but the library will happily still verify without warning, trusting the user-controlled alg field of the token header
+        with pytest.raises(JWKError):
+            data = jwt.decode(evil_token, PUBKEY)