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|
# Copyright (c) 2014, NORDUnet A/S.
# See LICENSE for licensing information.
import subprocess
import json
import base64
import urllib2
import ssl
import urlparse
import struct
import sys
import hashlib
import ecdsa
import cStringIO
import zipfile
import shutil
import requests
import warnings
from datetime import datetime
from certkeys import publickeys
def get_cert_info(s):
p = subprocess.Popen(
["openssl", "x509", "-noout", "-subject", "-issuer", "-inform", "der"],
stdin=subprocess.PIPE, stdout=subprocess.PIPE,
stderr=subprocess.PIPE)
parsed = p.communicate(s)
if parsed[1]:
print "ERROR:", parsed[1]
sys.exit(1)
result = {}
for line in parsed[0].split("\n"):
(key, sep, value) = line.partition("=")
if sep == "=":
result[key] = value
return result
def get_pemlike(filename, marker):
return get_pemlike_from_file(open(filename), marker)
def get_pemlike_from_file(f, marker):
entries = []
entry = ""
inentry = False
for line in f:
line = line.strip()
if line == "-----BEGIN " + marker + "-----":
entry = ""
inentry = True
elif line == "-----END " + marker + "-----":
entries.append(base64.decodestring(entry))
inentry = False
elif inentry:
entry += line
return entries
def get_certs_from_file(certfile):
return get_pemlike(certfile, "CERTIFICATE")
def get_certs_from_string(s):
f = cStringIO.StringIO(s)
return get_pemlike_from_file(f, "CERTIFICATE")
def get_precerts_from_string(s):
f = cStringIO.StringIO(s)
return get_pemlike_from_file(f, "PRECERTIFICATE")
def get_eckey_from_file(keyfile):
keys = get_pemlike(keyfile, "EC PRIVATE KEY")
assert len(keys) == 1
return keys[0]
def get_public_key_from_file(keyfile):
keys = get_pemlike(keyfile, "PUBLIC KEY")
assert len(keys) == 1
return keys[0]
def get_root_cert(issuer):
accepted_certs = \
json.loads(open("googlelog-accepted-certs.txt").read())["certificates"]
root_cert = None
for accepted_cert in accepted_certs:
subject = get_cert_info(base64.decodestring(accepted_cert))["subject"]
if subject == issuer:
root_cert = base64.decodestring(accepted_cert)
return root_cert
class sslparameters:
cafile = None
def create_ssl_context(cafile=None):
try:
sslparameters.cafile = cafile
except AttributeError:
sslparameters.cafile = None
def urlget(url, params=None):
with warnings.catch_warnings():
try:
warnings.filterwarnings("ignore", category=requests.packages.urllib3.exceptions.SubjectAltNameWarning)
except AttributeError:
pass
return requests.get(url, verify=sslparameters.cafile, params=params)
def urlpost(url, data, session=None):
with warnings.catch_warnings():
try:
warnings.filterwarnings("ignore", category=requests.packages.urllib3.exceptions.SubjectAltNameWarning)
except AttributeError:
pass
if session:
return session.post(url, data=data, verify=sslparameters.cafile)
else:
return requests.post(url, data=data, verify=sslparameters.cafile)
def get_sth(baseurl):
result = urlget(baseurl + "ct/v1/get-sth")
result.raise_for_status()
return result.json()
def get_proof_by_hash(baseurl, hash, tree_size):
params = {"hash":base64.b64encode(hash),
"tree_size":tree_size}
result = \
urlget(baseurl + "ct/v1/get-proof-by-hash", params=params)
if result.status_code == requests.codes.ok:
return result.json()
else:
print "ERROR:", result.status_code, result.text
sys.exit(1)
def get_consistency_proof(baseurl, tree_size1, tree_size2):
params = {"first":tree_size1,
"second":tree_size2}
result = urlget(baseurl + "ct/v1/get-sth-consistency", params=params)
if result.status_code == requests.codes.ok:
return result.json()["consistency"]
else:
print "ERROR:", result.status_code, result.text
sys.exit(1)
def tls_array(data, length_len):
length_bytes = struct.pack(">Q", len(data))[-length_len:]
return length_bytes + data
def unpack_tls_array(packed_data, length_len):
padded_length = ["\x00"] * 8
padded_length[-length_len:] = packed_data[:length_len]
(length,) = struct.unpack(">Q", "".join(padded_length))
unpacked_data = packed_data[length_len:length_len+length]
assert len(unpacked_data) == length, \
"data is only %d bytes long, but length is %d bytes" % \
(len(unpacked_data), length)
rest_data = packed_data[length_len+length:]
return (unpacked_data, rest_data)
def add_chain(baseurl, submission, session=None):
try:
result = urlpost(baseurl + "ct/v1/add-chain", json.dumps(submission), session=session)
if result.status_code == requests.codes.ok:
return result.json()
else:
print "ERROR:", result.status_code, result.text
if result.status_code == 400:
return None
sys.exit(1)
except ValueError, e:
print "==== FAILED REQUEST ===="
print submission
print "======= RESPONSE ======="
print result
print "========================"
raise e
def add_prechain(baseurl, submission, session=None):
try:
result = urlpost(baseurl + "ct/v1/add-pre-chain",
json.dumps(submission), session=session)
if result.status_code == requests.codes.ok:
return result.json()
else:
print "ERROR:", result.status_code, result.text
if result.status_code == 400:
return None
sys.exit(1)
except ValueError, e:
print "==== FAILED REQUEST ===="
print submission
print "======= RESPONSE ======="
print result
print "========================"
raise e
def get_entries(baseurl, start, end):
params = {"start":start, "end":end}
result = urlget(baseurl + "ct/v1/get-entries", params=params)
if result.status_code == requests.codes.ok:
return result.json()
else:
print "ERROR:", result.status_code, result.text
sys.exit(1)
def extract_precertificate(precert_chain_entry):
(precert, certchain) = unpack_tls_array(precert_chain_entry, 3)
return (precert, certchain)
def decode_certificate_chain(packed_certchain):
(unpacked_certchain, rest) = unpack_tls_array(packed_certchain, 3)
assert len(rest) == 0
certs = []
while len(unpacked_certchain):
(cert, rest) = unpack_tls_array(unpacked_certchain, 3)
certs.append(cert)
unpacked_certchain = rest
return certs
def decode_signature(signature):
(hash_alg, signature_alg) = struct.unpack(">bb", signature[0:2])
(unpacked_signature, rest) = unpack_tls_array(signature[2:], 2)
assert rest == ""
return (hash_alg, signature_alg, unpacked_signature)
def encode_signature(hash_alg, signature_alg, unpacked_signature):
signature = struct.pack(">bb", hash_alg, signature_alg)
signature += tls_array(unpacked_signature, 2)
return signature
def check_signature(baseurl, signature, data, publickey=None):
if publickey == None:
if baseurl in publickeys:
publickey = base64.decodestring(publickeys[baseurl])
else:
print >>sys.stderr, "Public key for", baseurl, \
"not found, specify key file with --publickey"
sys.exit(1)
(hash_alg, signature_alg, unpacked_signature) = decode_signature(signature)
assert hash_alg == 4, \
"hash_alg is %d, expected 4" % (hash_alg,) # sha256
assert signature_alg == 3, \
"signature_alg is %d, expected 3" % (signature_alg,) # ecdsa
vk = ecdsa.VerifyingKey.from_der(publickey)
vk.verify(unpacked_signature, data, hashfunc=hashlib.sha256,
sigdecode=ecdsa.util.sigdecode_der)
def parse_auth_header(authheader):
splittedheader = authheader.split(";")
(signature, rawoptions) = (splittedheader[0], splittedheader[1:])
options = dict([(e.partition("=")[0], e.partition("=")[2]) for e in rawoptions])
return (base64.b64decode(signature), options)
def check_auth_header(authheader, expected_key, publickeydir, data, path):
if expected_key == None:
return True
(signature, options) = parse_auth_header(authheader)
keyname = options.get("key")
if keyname != expected_key:
raise Exception("Response claimed to come from %s, expected %s" % (keyname, expected_key))
publickey = get_public_key_from_file(publickeydir + "/" + keyname + ".pem")
vk = ecdsa.VerifyingKey.from_der(publickey)
vk.verify(signature, "%s\0%s\0%s" % ("REPLY", path, data), hashfunc=hashlib.sha256,
sigdecode=ecdsa.util.sigdecode_der)
return True
def http_request(url, data=None, key=None, verifynode=None, publickeydir=".", params=None, session=None):
if session:
return http_request_session(url, data=data, key=key, verifynode=verifynode, publickeydir=publickeydir, params=params, session=session)
else:
with requests.sessions.Session() as session:
return http_request_session(url, data=data, key=key, verifynode=verifynode, publickeydir=publickeydir, params=params, session=session)
def chunk_generator(data, maxsize):
while len(data):
yield data[:maxsize]
data = data[maxsize:]
def http_request_session(url, data=None, key=None, verifynode=None, publickeydir=".", params=None, session=None):
(keyname, keyfile) = key
privatekey = get_eckey_from_file(keyfile)
sk = ecdsa.SigningKey.from_der(privatekey)
if data == None:
method = "GET"
chunked_data = None
else:
method = "POST"
chunked_data = chunk_generator(data, 900000)
assert(params == None)
req = requests.Request(method, url, params=params, data=chunked_data)
prepared_req = session.prepare_request(req)
parsed_url = urlparse.urlparse(prepared_req.url)
if data == None:
data_to_sign = parsed_url.query
else:
data_to_sign = data
url_to_sign = parsed_url.path
signature = sk.sign("%s\0%s\0%s" % (method, url_to_sign, data_to_sign), hashfunc=hashlib.sha256,
sigencode=ecdsa.util.sigencode_der)
prepared_req.headers['X-Catlfish-Auth'] = base64.b64encode(signature) + ";key=" + keyname
with warnings.catch_warnings():
try:
warnings.filterwarnings("ignore", category=requests.packages.urllib3.exceptions.SubjectAltNameWarning)
except AttributeError:
pass
result = session.send(prepared_req, verify=sslparameters.cafile)
result.raise_for_status()
authheader = result.headers.get('X-Catlfish-Auth')
data = result.text
check_auth_header(authheader, verifynode, publickeydir, data, url_to_sign)
return data
def get_signature(baseurl, data, key=None):
try:
params = json.dumps({"plop_version":1, "data": base64.b64encode(data)})
result = http_request(baseurl + "plop/v1/signing/sth", params, key=key)
parsed_result = json.loads(result)
return base64.b64decode(parsed_result.get(u"result"))
except requests.exceptions.HTTPError, e:
print "ERROR: get_signature", e.response
raise e
def create_signature(baseurl, data, key=None):
unpacked_signature = get_signature(baseurl, data, key)
return encode_signature(4, 3, unpacked_signature)
def check_sth_signature(baseurl, sth, publickey=None):
signature = base64.decodestring(sth["tree_head_signature"])
version = struct.pack(">b", 0)
signature_type = struct.pack(">b", 1)
timestamp = struct.pack(">Q", sth["timestamp"])
tree_size = struct.pack(">Q", sth["tree_size"])
ehash = base64.decodestring(sth["sha256_root_hash"])
tree_head = version + signature_type + timestamp + tree_size + ehash
check_signature(baseurl, signature, tree_head, publickey=publickey)
def create_sth_signature(tree_size, timestamp, root_hash, baseurl, key=None):
version = struct.pack(">b", 0)
signature_type = struct.pack(">b", 1)
timestamp_packed = struct.pack(">Q", timestamp)
tree_size_packed = struct.pack(">Q", tree_size)
tree_head = version + signature_type + timestamp_packed + tree_size_packed + root_hash
return create_signature(baseurl, tree_head, key=key)
def check_sct_signature(baseurl, signed_entry, sct, precert=False, publickey=None):
if publickey == None:
publickey = base64.decodestring(publickeys[baseurl])
calculated_logid = hashlib.sha256(publickey).digest()
received_logid = base64.decodestring(sct["id"])
assert calculated_logid == received_logid, \
"log id is incorrect:\n should be %s\n got %s" % \
(calculated_logid.encode("hex_codec"),
received_logid.encode("hex_codec"))
signature = base64.decodestring(sct["signature"])
version = struct.pack(">b", sct["sct_version"])
signature_type = struct.pack(">b", 0)
timestamp = struct.pack(">Q", sct["timestamp"])
if precert:
entry_type = struct.pack(">H", 1)
else:
entry_type = struct.pack(">H", 0)
signed_struct = version + signature_type + timestamp + \
entry_type + signed_entry + \
tls_array(base64.decodestring(sct["extensions"]), 2)
check_signature(baseurl, signature, signed_struct, publickey=publickey)
def pack_mtl(timestamp, leafcert):
entry_type = struct.pack(">H", 0)
extensions = ""
timestamped_entry = struct.pack(">Q", timestamp) + entry_type + \
tls_array(leafcert, 3) + tls_array(extensions, 2)
version = struct.pack(">b", 0)
leaf_type = struct.pack(">b", 0)
merkle_tree_leaf = version + leaf_type + timestamped_entry
return merkle_tree_leaf
def pack_mtl_precert(timestamp, cleanedcert, issuer_key_hash):
entry_type = struct.pack(">H", 1)
extensions = ""
timestamped_entry = struct.pack(">Q", timestamp) + entry_type + \
pack_precert(cleanedcert, issuer_key_hash) + tls_array(extensions, 2)
version = struct.pack(">b", 0)
leaf_type = struct.pack(">b", 0)
merkle_tree_leaf = version + leaf_type + timestamped_entry
return merkle_tree_leaf
def pack_precert(cleanedcert, issuer_key_hash):
assert len(issuer_key_hash) == 32
return issuer_key_hash + tls_array(cleanedcert, 3)
def pack_cert(cert):
return tls_array(cert, 3)
def unpack_mtl(merkle_tree_leaf):
version = merkle_tree_leaf[0:1]
leaf_type = merkle_tree_leaf[1:2]
timestamped_entry = merkle_tree_leaf[2:]
(timestamp, entry_type) = struct.unpack(">QH", timestamped_entry[0:10])
if entry_type == 0:
issuer_key_hash = None
(leafcert, rest_entry) = unpack_tls_array(timestamped_entry[10:], 3)
elif entry_type == 1:
issuer_key_hash = timestamped_entry[10:42]
(leafcert, rest_entry) = unpack_tls_array(timestamped_entry[42:], 3)
return (leafcert, timestamp, issuer_key_hash)
def get_leaf_hash(merkle_tree_leaf):
leaf_hash = hashlib.sha256()
leaf_hash.update(struct.pack(">b", 0))
leaf_hash.update(merkle_tree_leaf)
return leaf_hash.digest()
def timing_point(timer_dict=None, name=None):
t = datetime.now()
if timer_dict:
starttime = timer_dict["lasttime"]
stoptime = t
deltatime = stoptime - starttime
timer_dict["deltatimes"].append((name, deltatime.seconds * 1000000 + deltatime.microseconds))
timer_dict["lasttime"] = t
return None
else:
timer_dict = {"deltatimes":[], "lasttime":t}
return timer_dict
def internal_hash(pair):
if len(pair) == 1:
return pair[0]
else:
hash = hashlib.sha256()
hash.update(struct.pack(">b", 1))
hash.update(pair[0])
hash.update(pair[1])
return hash.digest()
def chunks(l, n):
return [l[i:i+n] for i in range(0, len(l), n)]
def next_merkle_layer(layer):
return [internal_hash(pair) for pair in chunks(layer, 2)]
def build_merkle_tree(layer0):
if len(layer0) == 0:
return [[hashlib.sha256().digest()]]
layers = []
current_layer = layer0
layers.append(current_layer)
while len(current_layer) > 1:
current_layer = next_merkle_layer(current_layer)
layers.append(current_layer)
return layers
def print_inclusion_proof(proof):
audit_path = proof[u'audit_path']
n = proof[u'leaf_index']
level = 0
for s in audit_path:
entry = base64.b16encode(base64.b64decode(s))
n ^= 1
print level, n, entry
n >>= 1
level += 1
def get_one_cert(store, i):
filename = i / 10000
zf = zipfile.ZipFile("%s/%04d.zip" % (store, i / 10000))
cert = zf.read("%08d" % i)
zf.close()
return cert
def get_hash_from_certfile(cert):
for line in cert.split("\n"):
if line.startswith("-----"):
return None
if line.startswith("Leafhash: "):
return base64.b16decode(line[len("Leafhash: "):])
return None
def get_timestamp_from_certfile(cert):
for line in cert.split("\n"):
if line.startswith("-----"):
return None
if line.startswith("Timestamp: "):
return int(line[len("Timestamp: "):])
return None
def get_proof(store, tree_size, n):
hash = get_hash_from_certfile(get_one_cert(store, n))
return get_proof_by_hash(args.baseurl, hash, tree_size)
def get_certs_from_zipfiles(zipfiles, firstleaf, lastleaf):
for i in range(firstleaf, lastleaf + 1):
try:
yield zipfiles[i / 10000].read("%08d" % i)
except KeyError:
return
def get_merkle_hash_64k(store, blocknumber, write_to_cache=False, treesize=None):
firstleaf = blocknumber * 65536
lastleaf = firstleaf + 65535
if treesize != None:
assert firstleaf < treesize
usecache = lastleaf < treesize
lastleaf = min(lastleaf, treesize - 1)
else:
usecache = True
hashfilename = "%s/%04x.64khash" % (store, blocknumber)
if usecache:
try:
hash = base64.b16decode(open(hashfilename).read())
assert len(hash) == 32
return ("hash", hash)
except IOError:
pass
firstfile = firstleaf / 10000
lastfile = lastleaf / 10000
zipfiles = {}
for i in range(firstfile, lastfile + 1):
try:
zipfiles[i] = zipfile.ZipFile("%s/%04d.zip" % (store, i))
except IOError:
break
certs = get_certs_from_zipfiles(zipfiles, firstleaf, lastleaf)
layer0 = [get_hash_from_certfile(cert) for cert in certs]
tree = build_merkle_tree(layer0)
calculated_hash = tree[-1][0]
for zf in zipfiles.values():
zf.close()
if len(layer0) != lastleaf - firstleaf + 1:
return ("incomplete", (len(layer0), calculated_hash))
if write_to_cache:
f = open(hashfilename, "w")
f.write(base64.b16encode(calculated_hash))
f.close()
return ("hash", calculated_hash)
def get_tree_head(store, treesize):
merkle_64klayer = []
for blocknumber in range(0, (treesize / 65536) + 1):
(resulttype, result) = get_merkle_hash_64k(store, blocknumber, treesize=treesize)
if resulttype == "incomplete":
print >>sys.stderr, "Couldn't read until tree size", treesize
(incompletelength, hash) = result
print >>sys.stderr, "Stopped at", blocknumber * 65536 + incompletelength
sys.exit(1)
assert resulttype == "hash"
hash = result
merkle_64klayer.append(hash)
#print >>sys.stderr, print blocknumber * 65536,
sys.stdout.flush()
tree = build_merkle_tree(merkle_64klayer)
calculated_root_hash = tree[-1][0]
return calculated_root_hash
def get_intermediate_hash(store, treesize, level, index):
if level >= 16:
merkle_64klayer = []
levelsize = (2**(level-16))
for blocknumber in range(index * levelsize, (index + 1) * levelsize):
if blocknumber * (2 ** 16) >= treesize:
break
#print "looking at block", blocknumber
(resulttype, result) = get_merkle_hash_64k(store, blocknumber, treesize=treesize)
if resulttype == "incomplete":
print >>sys.stderr, "Couldn't read until tree size", treesize
(incompletelength, hash) = result
print >>sys.stderr, "Stopped at", blocknumber * 65536 + incompletelength
sys.exit(1)
assert resulttype == "hash"
hash = result
#print "block hash", base64.b16encode(hash)
merkle_64klayer.append(hash)
#print >>sys.stderr, print blocknumber * 65536,
sys.stdout.flush()
tree = build_merkle_tree(merkle_64klayer)
return tree[-1][0]
else:
levelsize = 2 ** level
firstleaf = index * levelsize
lastleaf = firstleaf + levelsize - 1
#print "firstleaf", firstleaf
#print "lastleaf", lastleaf
assert firstleaf < treesize
lastleaf = min(lastleaf, treesize - 1)
#print "modified lastleaf", lastleaf
firstfile = firstleaf / 10000
lastfile = lastleaf / 10000
#print "files", firstfile, lastfile
zipfiles = {}
for i in range(firstfile, lastfile + 1):
try:
zipfiles[i] = zipfile.ZipFile("%s/%04d.zip" % (store, i))
except IOError:
break
certs = get_certs_from_zipfiles(zipfiles, firstleaf, lastleaf)
layer0 = [get_hash_from_certfile(cert) for cert in certs]
#print "layer0", repr(layer0)
tree = build_merkle_tree(layer0)
calculated_hash = tree[-1][0]
for zf in zipfiles.values():
zf.close()
assert len(layer0) == lastleaf - firstleaf + 1
return calculated_hash
def bits(n):
p = 0
while n > 0:
n >>= 1
p += 1
return p
def merkle_height(n):
if n == 0:
return 1
return bits(n - 1)
def node_above((pathp, pathl), levels=1):
return (pathp >> levels, pathl + levels)
def node_even((pathp, pathl)):
return pathp & 1 == 0
def node_odd((pathp, pathl)):
return pathp & 1 == 1
def node_lower((path1p, path1l), (path2p, path2l)):
return path1l < path2l
def node_higher((path1p, path1l), (path2p, path2l)):
return path1l > path2l
def node_level((path1p, path1l)):
return path1l
def node_outside((path1p, path1l), (path2p, path2l)):
assert path1l == path2l
return path1p > path2p
def combine_two_hashes((path1, hash1), (path2, hash2), treesize):
assert not node_higher(path1, path2)
edge_node = (treesize - 1, 0)
if node_lower(path1, path2):
assert path1 == node_above(edge_node, levels=node_level(path1))
while node_even(path1):
path1 = node_above(path1)
assert node_above(path1) == node_above(path2)
assert (node_even(path1) and node_odd(path2)) or (node_odd(path1) and node_even(path2))
if node_outside(path2, node_above(edge_node, levels=node_level(path2))):
return (node_above(path1), hash1)
if node_even(path1):
newhash = internal_hash((hash1, hash2))
else:
newhash = internal_hash((hash2, hash1))
return (node_above(path1), newhash)
def path_as_string(pos, level, treesize):
height = merkle_height(treesize)
path = "{0:0{width}b}".format(pos, width=height - level)
if height == level:
return ""
return path
def nodes_for_subtree(subtreesize, treesize):
height = merkle_height(treesize)
nodes = []
level = 0
pos = subtreesize
while pos > 0 and pos & 1 == 0:
pos >>= 1
level += 1
if pos & 1:
nodes.append((pos ^ 1, level))
#print pos, level
while level < height:
pos_level0 = pos * (2 ** level)
#print pos, level
if pos_level0 < treesize:
nodes.append((pos, level))
pos >>= 1
pos ^= 1
level += 1
return nodes
def nodes_for_index(pos, treesize):
height = merkle_height(treesize)
nodes = []
level = 0
pos ^= 1
#print pos, level
while level < height:
pos_level0 = pos * (2 ** level)
#print pos, level
if pos_level0 < treesize:
nodes.append((pos, level))
pos >>= 1
pos ^= 1
level += 1
return nodes
def verify_consistency_proof(consistency_proof, first, second, oldhash_input):
if 2 ** bits(first - 1) == first:
consistency_proof = [oldhash_input] + consistency_proof
chain = zip(nodes_for_subtree(first, second), consistency_proof)
assert len(nodes_for_subtree(first, second)) == len(consistency_proof)
(_, hash) = reduce(lambda e1, e2: combine_two_hashes(e1, e2, second), chain)
(_, oldhash) = reduce(lambda e1, e2: combine_two_hashes(e1, e2, first), chain)
return (oldhash, hash)
def verify_inclusion_proof(inclusion_proof, index, treesize, leafhash):
chain = zip([(index, 0)] + nodes_for_index(index, treesize), [leafhash] + inclusion_proof)
assert len(nodes_for_index(index, treesize)) == len(inclusion_proof)
(_, hash) = reduce(lambda e1, e2: combine_two_hashes(e1, e2, treesize), chain)
return hash
def extract_original_entry(entry):
leaf_input = base64.decodestring(entry["leaf_input"])
(leaf_cert, timestamp, issuer_key_hash) = unpack_mtl(leaf_input)
extra_data = base64.decodestring(entry["extra_data"])
if issuer_key_hash != None:
(precert, extra_data) = extract_precertificate(extra_data)
leaf_cert = precert
certchain = decode_certificate_chain(extra_data)
return ([leaf_cert] + certchain, timestamp, issuer_key_hash)
def mv_file(fromfn, tofn):
shutil.move(fromfn, tofn)
def write_file(fn, jsondata):
tempname = fn + ".new"
open(tempname, 'w').write(json.dumps(jsondata))
mv_file(tempname, fn)
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