How do I validate an access token using the at_hash claim of an id token?

The at_hash ID Token claim is defined by OpenID Connect as such:

Access Token hash value. Its value is the base64url encoding of the left-most half of the hash of the octets of the ASCII representation of the access_token value, where the hash algorithm used is the hash algorithm used in the alg Header Parameter of the ID Token's JOSE Header. For instance, if the alg is RS256, hash the access_token value with SHA-256, then take the left-most 128 bits and base64url encode them. The at_hash value is a case sensitive string.

The c_hash ID Token claim for the hybrid flow is defined similarly, the same steps can be used to verify either.

Steps to generate an at_hash or c_hash from the token:

1. Hash the ASCII representation of the token using the same alg as the ID Token itself, SHA-256 in Google's case.
2. Truncate the hash to the first half of the raw hash value (importantly: not the string hex representation of the hash).
3. Base64url encode (without padding) the truncated hash bytes.

Here's some sample code in python to create that hash, you'll need two libraries, pycrypto and the google-api-python-client (for the base64 encoding & id token comparison, you could potentially substitute with an alternative). You can install them with pip like so:

pip install pycrypto
pip install --upgrade google-api-python-client

Then, run python interactively, and try the following:

# Config: app's client id & tokens (in this case OAuth Playground's client id, and the tokens were extracted from the Token Endpoint response).
client_id = "407408718192.apps.googleusercontent.com"
id_token_string = "eyJhbGciOiJSUzI1NiIsImtpZCI6IjcwZjZjNDI2NzkyNWIzMzEzNmExZDFjZmVlNGViYzU3YjI0OWU1Y2IifQ.eyJpc3MiOiJhY2NvdW50cy5nb29nbGUuY29tIiwiYXRfaGFzaCI6Iml5VkFfTnNtY2JJMDFHcFJDQVJaOEEiLCJhdWQiOiI0MDc0MDg3MTgxOTIuYXBwcy5nb29nbGV1c2VyY29udGVudC5jb20iLCJzdWIiOiIxMTAxNjk0ODQ0NzQzODYyNzYzMzQiLCJhenAiOiI0MDc0MDg3MTgxOTIuYXBwcy5nb29nbGV1c2VyY29udGVudC5jb20iLCJpYXQiOjE0NjcyMTg1NzMsImV4cCI6MTQ2NzIyMjE3M30.e4hJJYeUaFVwJ9OC8LBnmOjwZln_E2-isEUJtb-Um7vt3GDZnBZkHdCokAPBL4OW3DXBNPk9iY0QL2P5Gpb-nX_s-PZKOIES8CE0i2DmGahCZgJY_Y3V2qwiP1fTEQjcUmHEG2e7OdCn6siSZveFQ0W7SiSbbSeJVLws9aoHROo_UXy8CVjaU5KinROG6m6igqCxFoskIWRzAynfx70xMadY4UdS8kbKK_v5id0_Rdg_gYlF1ND0lsPM9vdm3jOifQEAAkjHr-RuSDWlX4Bs4cQtEkeQkN6--MWhoqAshJITuGSazVIiDkVUNNBIXmB_dp9TO6ZjeQEEfeGCs6axKA"
access_token = "ya29.Ci8QA5eGBdBglK59FXdqXIR5KnbMJs-swx6Alk6_AV_6YPkjhxdO1e0Hqxi-8NB3Ww"

# Verifies & parses id token.
idtoken = oauth2client.client.verify_id_token(id_token_string, client_id)

# Token to hash & expected hash value (replace with code & c_hash to verify code).
token_to_hash = access_token
token_hash_expected = idtoken["at_hash"]

# Step 1. hashes the access token using SHA-256 (Google uses `RS256` as the ID Token `alg`).
hash = hashlib.sha256()
hash.update(token_to_hash)
digest = hash.digest()   # this returns the hash digest bytes (not a hex string)

# Step 2. truncates the hash digest to the first half.
digest_truncated = digest[:(len(digest)/2)]

# Step 3. base64url encodes the truncated hash digest bytes.
token_hash_computed = oauth2client.crypt._urlsafe_b64encode(digest_truncated)

# Compares computed to expected, outputs result.
str("Computed at_hash: %s" % token_hash_computed)
str(token_hash_computed == token_hash_expected)

To try this sample with a fresh ID Token from your own account, create a request using the OAuth Playground with the profile scope (or use this one), exchange the code for refresh and access tokens, and copy the response into token_response_http_body in the sample above (remove the linebreaks).

PHP solution:

$accessToken = 'xxx';
$idToken = 'yyy';
$client = new Google_Client();

$verification = $client->verifyIdToken($idToken);

$hash = hash('sha256', $accessToken);
$hash = substr($hash, 0, 32);
$hash = hex2bin($hash);
$hash = base64_encode($hash);
$hash = rtrim($hash, '=');
$hash = str_replace('/', '_', $hash);
$hash = str_replace('+', '-', $hash);

if ($hash === $verification['at_hash']) {
    // access token is valid
}

Google_Client available here: https://packagist.org/packages/google/apiclient