diff --git a/jws/__init__.py b/jws/__init__.py
index 178905a..e6cafd4 100644
--- a/jws/__init__.py
+++ b/jws/__init__.py
@@ -53,4 +53,4 @@ def verify(head, payload, encoded_signature, key=None, is_json=False):
 def _signing_input(head, payload, is_json=False):
     enc = utils.to_base64 if is_json else utils.encode
     head_input, payload_input = map(enc, [head, payload])
-    return "%s.%s" % (head_input, payload_input)
+    return b'%s.%s' % (head_input, payload_input)
diff --git a/jws/utils.py b/jws/utils.py
index 091adf4..6005fc9 100644
--- a/jws/utils.py
+++ b/jws/utils.py
@@ -21,9 +21,10 @@ def base64url_decode(input):
     input += b'=' * (4 - (len(input) % 4))
     return base64.urlsafe_b64decode(input)
 def base64url_encode(input):
-    return base64.urlsafe_b64encode(to_bytes_2and3(input)).replace(b'=', b'')
+    return base64.urlsafe_b64encode(to_bytes_2and3(input)).rstrip(b'=')
 
-def to_json(a): return json.dumps(a)
+def to_json(a):
+    return json.dumps(a, separators=(',', ':'))
 def from_json(a): return json.loads(a)
 def to_base64(a): return base64url_encode(a)
 def from_base64(a): return base64url_decode(a)
@@ -46,5 +47,5 @@ def constant_time_compare(val1, val2):
         return False
     result = 0
     for x, y in zip(val1, val2):
-        result |= ord(x) ^ ord(y)
+        result |= x ^ y
     return result == 0