OpenPGP.js

OpenPGP.js is a JavaScript implementation of the OpenPGP protocol. It implements RFC 9580 (superseding RFC 4880 and RFC 4880bis).

Table of Contents

• OpenPGP.js
  • Platform Support
  • Performance
  • Getting started
    • Node.js
    • Deno (experimental)
    • Browser (webpack)
    • Browser (plain files)
  • Examples
    • Encrypt and decrypt Uint8Array data with a password
    • Encrypt and decrypt String data with PGP keys
    • Encrypt symmetrically with compression
    • Streaming encrypt Uint8Array data with a password
    • Streaming encrypt and decrypt String data with PGP keys
    • Generate new key pair
    • Revoke a key
    • Sign and verify cleartext messages
    • Create and verify detached signatures
    • Streaming sign and verify Uint8Array data
  • Documentation
  • Security Audit
  • Security recommendations
  • Development
  • How do I get involved?
  • License

Platform Support

• The dist/openpgp.min.js (or .mjs) bundle works with recent versions of Chrome, Firefox, Edge and Safari 14+.

• The dist/node/openpgp.min.mjs (or .cjs) bundle works in Node.js v18+: it is used by default when you import ... from 'openpgp' (or require('openpgp'), respectively).

• Support for the Web Cryptography API's SubtleCrypto is required.

  • In browsers, SubtleCrypto is only available in secure contexts.
  • In supported versions of Node.js, SubtleCrypto is always available.

• Support for the Web Streams API is required.

  • In browsers: the latest versions of Chrome, Firefox, Edge and Safari support Streams, including TransformStreams. These are needed if you use the library with stream inputs. In previous versions of OpenPGP.js, Web Streams were automatically polyfilled by the library, but from v6 this task is left up to the library user, due to the more extensive browser support, and the polyfilling side-effects. If you're working with older browsers versions which do not implement e.g. TransformStreams, you can manually load the Web Streams polyfill. Please note that when you load the polyfills, the global ReadableStream property (if it exists) gets overwritten with the polyfill version. In some edge cases, you might need to use the native ReadableStream (for example when using it to create a Response object), in which case you should store a reference to it before loading the polyfills. There is also the web-streams-adapter library to convert back and forth between them.
  • In Node.js: OpenPGP.js v6 no longer supports native Node Readable streams in inputs, and instead expects (and outputs) Node's Web Streams. Node v17+ includes utilities to convert from and to Web Streams.

Performance

• Version 3.0.0 of the library introduced support for public-key cryptography using elliptic curves. We use native implementations on browsers and Node.js when available. Compared to RSA, elliptic curve cryptography provides stronger security per bits of key, which allows for much faster operations. Currently the following curves are supported:

  Curve	Encryption	Signature	NodeCrypto	WebCrypto	Constant-Time
  curve25519	ECDH	N/A	No	No	Algorithmically
  ed25519	N/A	EdDSA	No	Yes*	If native**
  nistP256	ECDH	ECDSA	Yes*	Yes*	If native**
  nistP384	ECDH	ECDSA	Yes*	Yes*	If native**
  nistP521	ECDH	ECDSA	Yes*	Yes*	If native**
  brainpoolP256r1	ECDH	ECDSA	Yes*	No	If native**
  brainpoolP384r1	ECDH	ECDSA	Yes*	No	If native**
  brainpoolP512r1	ECDH	ECDSA	Yes*	No	If native**
  secp256k1	ECDH	ECDSA	Yes*	No	If native**

  * when available
  ** these curves are only constant-time if the underlying native implementation is available and constant-time

• The platform's native Web Crypto API is used for performance. On Node.js the native crypto module is also used, in cases where it offers additional functionality.

• The library implements authenticated encryption (AEAD) as per RFC 9580 using AES-GCM, OCB, or EAX. This makes symmetric encryption faster on platforms with native implementations. However, since the specification is very recent and other OpenPGP implementations are in the process of adopting it, the feature is currently behind a flag. Note: activating this setting can break compatibility with other OpenPGP implementations which have yet to implement the feature. You can enable it by setting openpgp.config.aeadProtect = true. Note that this setting has a different effect from the one in OpenPGP.js v5, which implemented support for a provisional version of AEAD from RFC 4880bis, which was modified in RFC 9580.

  You can change the AEAD mode by setting one of the following options:

  openpgp.config.preferredAEADAlgorithm = openpgp.enums.aead.gcm; // Default, native in WebCrypto and Node.js
  openpgp.config.preferredAEADAlgorithm = openpgp.enums.aead.ocb; // Non-native, but supported across RFC 9580 implementations
  openpgp.config.preferredAEADAlgorithm = openpgp.enums.aead.eax; // Native in Node.js
  

Getting started

Node.js

Install OpenPGP.js using npm and save it in your dependencies:

1npm install --save openpgp

And import it as an ES module, from a .mjs file:

1import * as openpgp from 'openpgp';

Or as a CommonJS module:

1const openpgp = require('openpgp');

Deno (experimental)

Import as an ES6 module, using /dist/openpgp.mjs.

1import * as openpgp from './openpgpjs/dist/openpgp.mjs';

Browser (webpack)

Install OpenPGP.js using npm and save it in your devDependencies:

1npm install --save-dev openpgp

And import it as an ES6 module:

1import * as openpgp from 'openpgp';

You can also only import the functions you need, as follows:

1import { readMessage, decrypt } from 'openpgp';

Or, if you want to use the lightweight build (which is smaller, and lazily loads non-default curves on demand):

1import * as openpgp from 'openpgp/lightweight';

To test whether the lazy loading works, try to generate a key with a non-standard curve:

1import { generateKey } from 'openpgp/lightweight';
2await generateKey({ curve: 'brainpoolP512r1',  userIDs: [{ name: 'Test', email: 'test@test.com' }] });

For more examples of how to generate a key, see Generate new key pair. It is recommended to use curve25519 instead of brainpoolP512r1 by default.

Browser (plain files)

Grab openpgp.min.js from unpkg.com/openpgp/dist, and load it in a script tag:

1<script src="openpgp.min.js"></script>

Or, to load OpenPGP.js as an ES6 module, grab openpgp.min.mjs from unpkg.com/openpgp/dist, and import it as follows:

1<script type="module">
2import * as openpgp from './openpgp.min.mjs';
3</script>

To offload cryptographic operations off the main thread, you can implement a Web Worker in your application and load OpenPGP.js from there. For an example Worker implementation, see test/worker/worker_example.js.

TypeScript

Since TS is not fully integrated in the library, TS-only dependencies are currently listed as devDependencies, so to compile the project you’ll need to add @openpgp/web-stream-tools manually:

1npm install --save-dev @openpgp/web-stream-tools

If you notice missing or incorrect type definitions, feel free to open a PR.

Examples

Here are some examples of how to use OpenPGP.js v6. For more elaborate examples and working code, please check out the public API unit tests. If you're upgrading from v4 it might help to check out the changelog and documentation.

Encrypt and decrypt Uint8Array data with a password

Encryption will use the algorithm specified in config.preferredSymmetricAlgorithm (defaults to aes256), and decryption will use the algorithm used for encryption.

1(async () => {
2    const message = await openpgp.createMessage({ binary: new Uint8Array([0x01, 0x01, 0x01]) });
3    const encrypted = await openpgp.encrypt({
4        message, // input as Message object
5        passwords: ['secret stuff'], // multiple passwords possible
6        format: 'binary' // don't ASCII armor (for Uint8Array output)
7    });
8    console.log(encrypted); // Uint8Array
9
10    const encryptedMessage = await openpgp.readMessage({
11        binaryMessage: encrypted // parse encrypted bytes
12    });
13    const { data: decrypted } = await openpgp.decrypt({
14        message: encryptedMessage,
15        passwords: ['secret stuff'], // decrypt with password
16        format: 'binary' // output as Uint8Array
17    });
18    console.log(decrypted); // Uint8Array([0x01, 0x01, 0x01])
19})();

Encrypt and decrypt String data with PGP keys

Encryption will use the algorithm preferred by the public (encryption) key (defaults to aes256 for keys generated in OpenPGP.js), and decryption will use the algorithm used for encryption.

1const openpgp = require('openpgp'); // use as CommonJS, AMD, ES6 module or via window.openpgp
2
3(async () => {
4    // put keys in backtick (``) to avoid errors caused by spaces or tabs
5    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
6...
7-----END PGP PUBLIC KEY BLOCK-----`;
8    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
9...
10-----END PGP PRIVATE KEY BLOCK-----`; // encrypted private key
11    const passphrase = `yourPassphrase`; // what the private key is encrypted with
12
13    const publicKey = await openpgp.readKey({ armoredKey: publicKeyArmored });
14
15    const privateKey = await openpgp.decryptKey({
16        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
17        passphrase
18    });
19
20    const encrypted = await openpgp.encrypt({
21        message: await openpgp.createMessage({ text: 'Hello, World!' }), // input as Message object
22        encryptionKeys: publicKey,
23        signingKeys: privateKey // optional
24    });
25    console.log(encrypted); // '-----BEGIN PGP MESSAGE ... END PGP MESSAGE-----'
26
27    const message = await openpgp.readMessage({
28        armoredMessage: encrypted // parse armored message
29    });
30    const { data: decrypted, signatures } = await openpgp.decrypt({
31        message,
32        verificationKeys: publicKey, // optional
33        decryptionKeys: privateKey
34    });
35    console.log(decrypted); // 'Hello, World!'
36    // check signature validity (signed messages only)
37    try {
38        await signatures[0].verified; // throws on invalid signature
39        console.log('Signature is valid');
40    } catch (e) {
41        throw new Error('Signature could not be verified: ' + e.message);
42    }
43})();

Encrypt to multiple public keys:

1(async () => {
2    const publicKeysArmored = [
3        `-----BEGIN PGP PUBLIC KEY BLOCK-----
4...
5-----END PGP PUBLIC KEY BLOCK-----`,
6        `-----BEGIN PGP PUBLIC KEY BLOCK-----
7...
8-----END PGP PUBLIC KEY BLOCK-----`
9    ];
10    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
11...
12-----END PGP PRIVATE KEY BLOCK-----`;    // encrypted private key
13    const passphrase = `yourPassphrase`; // what the private key is encrypted with
14    const plaintext = 'Hello, World!';
15
16    const publicKeys = await Promise.all(publicKeysArmored.map(armoredKey => openpgp.readKey({ armoredKey })));
17
18    const privateKey = await openpgp.decryptKey({
19        privateKey: await openpgp.readKey({ armoredKey: privateKeyArmored }),
20        passphrase
21    });
22
23    const message = await openpgp.createMessage({ text: plaintext });
24    const encrypted = await openpgp.encrypt({
25        message, // input as Message object
26        encryptionKeys: publicKeys,
27        signingKeys: privateKey // optional
28    });
29    console.log(encrypted); // '-----BEGIN PGP MESSAGE ... END PGP MESSAGE-----'
30})();

If you expect an encrypted message to be signed with one of the public keys you have, and do not want to trust the decrypted data otherwise, you can pass the decryption option expectSigned = true, so that the decryption operation will fail if no valid signature is found:

1(async () => {
2    // put keys in backtick (``) to avoid errors caused by spaces or tabs
3    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
4...
5-----END PGP PUBLIC KEY BLOCK-----`;
6    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
7...
8-----END PGP PRIVATE KEY BLOCK-----`; // encrypted private key
9    const passphrase = `yourPassphrase`; // what the private key is encrypted with
10
11    const publicKey = await openpgp.readKey({ armoredKey: publicKeyArmored });
12
13    const privateKey = await openpgp.decryptKey({
14        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
15        passphrase
16    });
17
18    const encryptedAndSignedMessage = `-----BEGIN PGP MESSAGE-----
19...
20-----END PGP MESSAGE-----`;
21
22    const message = await openpgp.readMessage({
23        armoredMessage: encryptedAndSignedMessage // parse armored message
24    });
25    // decryption will fail if all signatures are invalid or missing
26    const { data: decrypted, signatures } = await openpgp.decrypt({
27        message,
28        decryptionKeys: privateKey,
29        expectSigned: true,
30        verificationKeys: publicKey, // mandatory with expectSigned=true
31    });
32    console.log(decrypted); // 'Hello, World!'
33})();

Encrypt symmetrically with compression

By default, encrypt will not use any compression when encrypting symmetrically only (i.e. when no encryptionKeys are given). It's possible to change that behaviour by enabling compression through the config, either for the single encryption:

1(async () => {
2    const message = await openpgp.createMessage({ binary: new Uint8Array([0x01, 0x02, 0x03]) }); // or createMessage({ text: 'string' })
3    const encrypted = await openpgp.encrypt({
4        message,
5        passwords: ['secret stuff'], // multiple passwords possible
6        config: { preferredCompressionAlgorithm: openpgp.enums.compression.zlib } // compress the data with zlib
7    });
8})();

or by changing the default global configuration:

1openpgp.config.preferredCompressionAlgorithm = openpgp.enums.compression.zlib

Where the value can be any of:

• openpgp.enums.compression.zip
• openpgp.enums.compression.zlib
• openpgp.enums.compression.uncompressed (default)

Streaming encrypt Uint8Array data with a password

1(async () => {
2    const readableStream = new ReadableStream({
3        start(controller) {
4            controller.enqueue(new Uint8Array([0x01, 0x02, 0x03]));
5            controller.close();
6        }
7    });
8
9    const message = await openpgp.createMessage({ binary: readableStream });
10    const encrypted = await openpgp.encrypt({
11        message, // input as Message object
12        passwords: ['secret stuff'], // multiple passwords possible
13        format: 'binary' // don't ASCII armor (for Uint8Array output)
14    });
15    console.log(encrypted); // raw encrypted packets as ReadableStream<Uint8Array>
16
17    // Either pipe the above stream somewhere, pass it to another function,
18    // or read it manually as follows:
19    for await (const chunk of encrypted) {
20        console.log('new chunk:', chunk); // Uint8Array
21    }
22})();

For more information on using ReadableStreams (both in browsers and Node.js), see the MDN Documentation on the Streams API .

Streaming encrypt and decrypt String data with PGP keys

1(async () => {
2    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
3...
4-----END PGP PUBLIC KEY BLOCK-----`; // Public key
5    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
6...
7-----END PGP PRIVATE KEY BLOCK-----`; // Encrypted private key
8    const passphrase = `yourPassphrase`; // Password that private key is encrypted with
9
10    const publicKey = await openpgp.readKey({ armoredKey: publicKeyArmored });
11
12    const privateKey = await openpgp.decryptKey({
13        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
14        passphrase
15    });
16
17    const readableStream = new ReadableStream({
18        start(controller) {
19            controller.enqueue('Hello, world!');
20            controller.close();
21        }
22    });
23
24    const encrypted = await openpgp.encrypt({
25        message: await openpgp.createMessage({ text: readableStream }), // input as Message object
26        encryptionKeys: publicKey,
27        signingKeys: privateKey // optional
28    });
29    console.log(encrypted); // ReadableStream containing '-----BEGIN PGP MESSAGE ... END PGP MESSAGE-----'
30
31    const message = await openpgp.readMessage({
32        armoredMessage: encrypted // parse armored message
33    });
34    const decrypted = await openpgp.decrypt({
35        message,
36        verificationKeys: publicKey, // optional
37        decryptionKeys: privateKey
38    });
39    const chunks = [];
40    for await (const chunk of decrypted.data) {
41        chunks.push(chunk);
42    }
43    const plaintext = chunks.join('');
44    console.log(plaintext); // 'Hello, World!'
45})();

Generate new key pair

ECC keys (smaller and faster to generate):

Possible values for curve are: curve25519, ed25519, nistP256, nistP384, nistP521, brainpoolP256r1, brainpoolP384r1, brainpoolP512r1, and secp256k1. Note that both the curve25519 and ed25519 options generate a primary key for signing using Ed25519 and a subkey for encryption using Curve25519.

1(async () => {
2    const { privateKey, publicKey, revocationCertificate } = await openpgp.generateKey({
3        type: 'ecc', // Type of the key, defaults to ECC
4        curve: 'curve25519', // ECC curve name, defaults to curve25519
5        userIDs: [{ name: 'Jon Smith', email: 'jon@example.com' }], // you can pass multiple user IDs
6        passphrase: 'super long and hard to guess secret', // protects the private key
7        format: 'armored' // output key format, defaults to 'armored' (other options: 'binary' or 'object')
8    });
9
10    console.log(privateKey);     // '-----BEGIN PGP PRIVATE KEY BLOCK ... '
11    console.log(publicKey);      // '-----BEGIN PGP PUBLIC KEY BLOCK ... '
12    console.log(revocationCertificate); // '-----BEGIN PGP PUBLIC KEY BLOCK ... '
13})();

RSA keys (increased compatibility):

1(async () => {
2    const { privateKey, publicKey } = await openpgp.generateKey({
3        type: 'rsa', // Type of the key
4        rsaBits: 4096, // RSA key size (defaults to 4096 bits)
5        userIDs: [{ name: 'Jon Smith', email: 'jon@example.com' }], // you can pass multiple user IDs
6        passphrase: 'super long and hard to guess secret' // protects the private key
7    });
8})();

Revoke a key

Using a revocation certificate:

1(async () => {
2    const { publicKey: revokedKeyArmored } = await openpgp.revokeKey({
3        key: await openpgp.readKey({ armoredKey: publicKeyArmored }),
4        revocationCertificate,
5        format: 'armored' // output armored keys
6    });
7    console.log(revokedKeyArmored); // '-----BEGIN PGP PUBLIC KEY BLOCK ... '
8})();

Using the private key:

1(async () => {
2    const { publicKey: revokedKeyArmored } = await openpgp.revokeKey({
3        key: await openpgp.readKey({ armoredKey: privateKeyArmored }),
4        format: 'armored' // output armored keys
5    });
6    console.log(revokedKeyArmored); // '-----BEGIN PGP PUBLIC KEY BLOCK ... '
7})();

Sign and verify cleartext messages

1(async () => {
2    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
3...
4-----END PGP PUBLIC KEY BLOCK-----`;
5    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
6...
7-----END PGP PRIVATE KEY BLOCK-----`; // encrypted private key
8    const passphrase = `yourPassphrase`; // what the private key is encrypted with
9
10    const publicKey = await openpgp.readKey({ armoredKey: publicKeyArmored });
11
12    const privateKey = await openpgp.decryptKey({
13        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
14        passphrase
15    });
16
17    const unsignedMessage = await openpgp.createCleartextMessage({ text: 'Hello, World!' });
18    const cleartextMessage = await openpgp.sign({
19        message: unsignedMessage, // CleartextMessage or Message object
20        signingKeys: privateKey
21    });
22    console.log(cleartextMessage); // '-----BEGIN PGP SIGNED MESSAGE ... END PGP SIGNATURE-----'
23
24    const signedMessage = await openpgp.readCleartextMessage({
25        cleartextMessage // parse armored message
26    });
27    const verificationResult = await openpgp.verify({
28        message: signedMessage,
29        verificationKeys: publicKey
30    });
31    const { verified, keyID } = verificationResult.signatures[0];
32    try {
33        await verified; // throws on invalid signature
34        console.log('Signed by key id ' + keyID.toHex());
35    } catch (e) {
36        throw new Error('Signature could not be verified: ' + e.message);
37    }
38})();

Create and verify detached signatures

1(async () => {
2    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
3...
4-----END PGP PUBLIC KEY BLOCK-----`;
5    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
6...
7-----END PGP PRIVATE KEY BLOCK-----`; // encrypted private key
8    const passphrase = `yourPassphrase`; // what the private key is encrypted with
9
10    const publicKey = await openpgp.readKey({ armoredKey: publicKeyArmored });
11
12    const privateKey = await openpgp.decryptKey({
13        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
14        passphrase
15    });
16
17    const message = await openpgp.createMessage({ text: 'Hello, World!' });
18    const detachedSignature = await openpgp.sign({
19        message, // Message object
20        signingKeys: privateKey,
21        detached: true
22    });
23    console.log(detachedSignature);
24
25    const signature = await openpgp.readSignature({
26        armoredSignature: detachedSignature // parse detached signature
27    });
28    const verificationResult = await openpgp.verify({
29        message, // Message object
30        signature,
31        verificationKeys: publicKey
32    });
33    const { verified, keyID } = verificationResult.signatures[0];
34    try {
35        await verified; // throws on invalid signature
36        console.log('Signed by key id ' + keyID.toHex());
37    } catch (e) {
38        throw new Error('Signature could not be verified: ' + e.message);
39    }
40})();

Streaming sign and verify Uint8Array data

1(async () => {
2    var readableStream = new ReadableStream({
3        start(controller) {
4            controller.enqueue(new Uint8Array([0x01, 0x02, 0x03]));
5            controller.close();
6        }
7    });
8
9    const publicKeyArmored = `-----BEGIN PGP PUBLIC KEY BLOCK-----
10...
11-----END PGP PUBLIC KEY BLOCK-----`;
12    const privateKeyArmored = `-----BEGIN PGP PRIVATE KEY BLOCK-----
13...
14-----END PGP PRIVATE KEY BLOCK-----`; // encrypted private key
15    const passphrase = `yourPassphrase`; // what the private key is encrypted with
16
17    const privateKey = await openpgp.decryptKey({
18        privateKey: await openpgp.readPrivateKey({ armoredKey: privateKeyArmored }),
19        passphrase
20    });
21
22    const message = await openpgp.createMessage({ binary: readableStream }); // or createMessage({ text: ReadableStream<String> })
23    const signatureArmored = await openpgp.sign({
24        message,
25        signingKeys: privateKey
26    });
27    console.log(signatureArmored); // ReadableStream containing '-----BEGIN PGP MESSAGE ... END PGP MESSAGE-----'
28
29    const verificationResult = await openpgp.verify({
30        message: await openpgp.readMessage({ armoredMessage: signatureArmored }), // parse armored signature
31        verificationKeys: await openpgp.readKey({ armoredKey: publicKeyArmored })
32    });
33
34    for await (const chunk of verificationResult.data) {}
35    // Note: you *have* to read `verificationResult.data` in some way or other,
36    // even if you don't need it, as that is what triggers the
37    // verification of the data.
38
39    try {
40        await verificationResult.signatures[0].verified; // throws on invalid signature
41        console.log('Signed by key id ' + verificationResult.signatures[0].keyID.toHex());
42     } catch (e) {
43        throw new Error('Signature could not be verified: ' + e.message);
44    }
45})();

Documentation

The full documentation is available at openpgpjs.org.

Security Audit

To date the OpenPGP.js code base has undergone two complete security audits from Cure53. The first audit's report has been published here.

Security recommendations

It should be noted that js crypto apps deployed via regular web hosting (a.k.a. host-based security) provide users with less security than installable apps with auditable static versions. Installable apps can be deployed as a Firefox or Chrome packaged app. These apps are basically signed zip files and their runtimes typically enforce a strict Content Security Policy (CSP) to protect users against XSS. This blogpost explains the trust model of the web quite well.

It is also recommended to set a strong passphrase that protects the user's private key on disk.

Development

To create your own build of the library, just run the following command after cloning the git repo. This will download all dependencies, run the tests and create a minified bundle under dist/openpgp.min.js to use in your project:

npm install && npm test

For debugging browser errors, run the following command:

npm run browsertest

How do I get involved?

You want to help, great! It's probably best to send us a message on Gitter before you start your undertaking, to make sure nobody else is working on it, and so we can discuss the best course of action. Other than that, just go ahead and fork our repo, make your changes and send us a pull request! :)

License

GNU Lesser General Public License (3.0 or any later version). Please take a look at the LICENSE file for more information.