Hidden Powers of Miniscript Policy & Descriptors

By Sandipan Dey, Rajarshi Maitra on 1/3/2022 - Tags: Tutorial, Bdk, Bdk-cli, Miniscript, Descriptor, Bitcoin-cli

# Introduction

To send people BTC - we simply scan a QR Code (or paste an address), enter some amount and whoosh - sent! Users might think, just like traditional currency, we can only exchange money using Bitcoin. As it so happens, the underlying technology Bitcoin supports specify outputs not as addresses, but as programming scripts. This opens us to a world of possibilities using Bitcoin.

# Script

Bitcoin supports Script (opens new window), a stack-based lightweight programming language. Any script written in Script (pun intended) contains OP_* codes and raw byte arrays that Bitcoin Full Nodes understand and process. Currently, there are 117 op-codes in use. You can read more about these op-codes straight here (opens new window).

Script is intentionally left Turing incomplete (opens new window) which is why there is no halting problem (opens new window) with scripts. There are no loops and overall, it's a very constrained programming language.

A transaction is considered valid only when the Script returns true at the end of execution. Output Script (aka scriptpubkey) define the conditions under which coins associated with them can be spent. To spend a particular coin implies finding an input script (aka scriptsig) such that a script made out of concatenation of scriptsig + scriptpubkey evaluates to true.

For example, a basic legacy Pay-to-PubKey-Hash transaction would look like:

scriptPubKey: OP_DUP OP_HASH160 <pubKeyHash> OP_EQUALVERIFY OP_CHECKSIG
scriptSig: <sig> <pubKey>

# Examples of things achievable using Bitcoin Script:

  1. Pay Someone (p2pkh/p2wpkh) - A specific public key must sign to spend the coins.
  2. Escrow (2-of-3-multisig) - Two parties need to sign together to spend the coins.
  3. Vault (locked) - A specific key will not be able to spend the coins until a timeout but another master key will always be able to spend them.
  4. HTLC - The receiver needs disclose a secret before a timeout, else the coins are transferred back to the payee.

# Motivation for Policies

Unfortunately, due to its low-level and unusual stack-based nature, Script is pretty hard to reason about and use. Despite being around since Bitcoin's creation, writing and understanding Script is not trivial. This is why the scripts for the above few examples are pretty lengthy and might not make sense at the first glance. When writing a script, we would want to know that if the logic we wrote is correct, optimal and efficient in size (use lower weight (opens new window)).

The community wanted an easy alternative way of writing Script that would create the most optimized Script code. This gave rise to Miniscript.

# Miniscript

Miniscript (opens new window) tackles the above problems head-on. It is an expressive way to create policies on Bitcoin Scripts in a structured and simple fashion. Using Miniscript, it's difficult to go wrong.

Another very important goal of Miniscript is to replace any key used in a policy with another policy. This is important because people might have multiple keys and complicated timelocks in their existing setup. While signing a new policy, they would want to use their existing setup to also generate addresses for this new setup. This is accomplished using something called Output Descriptors which we will get into in next section.

Miniscript compiler compiles a spending policy down to Miniscript. It doesn't contain any signature, it's mainly a combinator language for designing spending conditions. You can try out the compiler online by using this link (opens new window).

# Fragments

Here are some fragments which can be combined to create powerful expressions.

  1. pk(key) - Specifies a given public key
  2. thresh(k, expr_1, expr_2, ..., expr_n) - Specifies k of n multisig using expressions.
  3. older(T) - Timelock for T blocks
  4. and(expr_1, expr_2) - Both expressions should evaluate to true.
  5. or(expr_1, expr_2) - Any one of the expressions should evaluate to true.
  6. aor(expr_1, expr_2) - Similar to or but expr_1 has a more probability to evaluate to true.

Bitcoin Script allows us to use another alternate stack. The combinator functions use this second stack to evaluate expressions of thresh, and, aor and or. The complete Miniscript Reference can be found here (opens new window).

# Example Policies

Here are the Miniscript Policies for the examples we looked at earlier. Note A, B, C are placeholders for keys (xpub/xprv) involved in the tx. Descriptors are high level description of scriptpubkey (p2pkh, p2sh etc). And miniscripts are semantics that describes the actual redeemscript. In general you have Descriptor(Miniscript) format.

  1. Pay A (pay-to-public-key)
pk(A)
  1. Escrow Account between A, B and third-party C.
thresh(2,pk(A),pk(B),pk(C))
  1. Vault for A time-locked for T blocks with B as the master key.
aor(and(pk(A),time(T)),pk(B))
  1. HTLC payment to B, which, if unspent for T blocks, returns to A.
aor(and(pk(A),time(T)),and(pk(B),hash(H))))

The Miniscript Policy Compiler is written in Rust and is present in this repository (opens new window). In this blog, we will later use the same using bitcoindevkit/bdk (opens new window), a lightweight descriptor-based wallet library with a cli (opens new window).

# Descriptors

The Bitcoin scriptpubkey supports various schemes like P2PKH, P2SH, P2WPKH, P2TR (Segwit v1) etc. A Descriptor is a simple "description" of what scriptpubkey to be used for a given policy. It can inclue a single pubkey within itself, or an entire miniscript policy. On the other hand, Miniscript policies are used to derive the redeemscript (the actual executable script), whereas the descriptor describes how the redeemscript will be encumbered within the scriptpubkey.

In other words, a descriptor "describes" the procedure to create an address, given a spending condition.

They make it easier to deal with Multisig or complicated key setups. Descriptors are super portable and can be easily used by any wallet to determine the list of all addresses that can be generated from the same. This feature creates a common stage for all Bitcoin apps and software.

The concept of descriptor came into existence in 2018 and since then, a lot of wallets have added support for descriptors. You can read the descriptor doc from bitcoin-core here (opens new window).

According to Bitcoin Core, Output Descriptors are "a simple language which can be used to describe collections of output scripts". They bring in themselves, derivation paths, master xpub/xprv fingerprints and paths to generate addresses from. Let's understand this with an example of an Output Descriptor:


Descriptor: pkh([d34db33f/44'/0'/0']xpub6ERaJH[...]LJRcEL/1/*)#ml40v0wf
            <1> <--------2---------><----------3---------><4> <---5--->

Sections:
1 - address type specifier (here, describing P2PK type addresses)
2 - master key fingerprint and derivation path from master
3 - xpub at m/44'/0'/0
4 - path to deriving keys/addresses at
5 - checksum for the descriptor

A descriptor have three parts:

  • address type specifier (item 1) : describes the type of address created by this descriptor.
  • policy : the spending condition that locks funds into this address.
  • checksum : for quick verification.

The address type specifiers currently supported are pk, pkh, sh, wpkh, wsh for corresponding address type and recently added tr for taproot addresses.

There is a special address specifier called combo that creates addresses of all types from spending policy policy.

After the address specifier, comes the policy that describes how the funds in the address can be spent. The descriptor above in the example has a simple spending policy of "spend by the correct private key". There can be more complicated policies, and we will touch them in later sections.

multi is a special keyword that can be used as both address type and policy. When used as an address type like multi(...), it will create an address from the raw multisig scriptpubkey. While when used as a policy like wsh(multi(...)) it will create that specific address type from the multisig script. Of course we cannot use pk(multi(...)), pkh(multi(...)) or wpkh(multi(...)), because these address types cannot hold scripts (any scripts) inside them.

For example a descriptor like wsh(multi(2, PKA, PKB, PKC)) describes a P2WSH type address created by a 2-of-3 multisig script using PKA, PKB and PKC pubkeys.

# Where it all comes together...

In this section, we are going to make a descriptor-based wallet and derive addresses from bitcoin-cli and then use bdk-cli to confirm that the addresses generated for descriptor wallets are deterministic for a given descriptor.

We will also try to create a vault miniscript policy and push funds to the vault with a lock time of 2 months. During this time, we will try to break our vault and see our transactions failing.

# Tools and Armor

# Setting Up

We require bitcoind to run in regtest mode. Use the following config file, or any other config that you are familiar with.

regtest=1
fallbackfee=0.0001
server=1

rpcuser=user
rpcpassword=password

# Start Bitcoin Core
bitcoind

# Keys and Generating Addresses

Quick installation for bdk-cli and miniscriptc:

cargo install bdk-cli --features=rpc,compiler
cargo install bdk --features="compiler" --example miniscriptc

Let us first generate an XPRV and create the wpkh wallet descriptor

XPRV=$(bdk-cli key generate | jq -r '.xprv')
EX_DESC="wpkh($XPRV/86'/1'/0'/0/*)"
EX_DESC_CS=$(elcli getdescriptorinfo $EX_DESC | jq -r '.checksum')
EX_DESC=$EX_DESC#$EX_DESC_CS

# Set this descriptor in a wallet in bitcoin-cli
bitcoin-cli -named createwallet wallet_name="mywallet" descriptors=true
bitcoin-cli -rpcwallet="mywallet" importdescriptors "[{\"desc\":\"$EX_DESC\", \"timestamp\":\"now\", \"active\": true, \"range\": [0,100]}]"

echo $EX_DESC

It should look something like this:

wpkh(tprv8ZgxMBicQKsPeuazF16EdPZw84eHj55AU8ZKgZgdhu3sXcHnFgjzskfDvZdTaAFHYNCbKqrurFo9onSaT7zGT1i3u3j7LKhVZF5sJA39WPN/86'/1'/0'/0/*)#40hv8z77

Now, we will generate 10 addresses using bitcoin-cli and thereafter bdk-cli using this above descriptor. Notice how both of them output the same set of addresses.

# Generation via bdk-cli
repeat 10 { bdk-cli -n regtest wallet --descriptor $EX_DESC -w mywallet get_new_address | jq -r ".address" }
bcrt1qc9wzxf8pthyexl00m23ug92pqrthagnzzf33wp
bcrt1qgnh7e72q92fqujwg3qxlg5kplxkm6rep0nerur
bcrt1qea6r8yvd0peupk29p94wm0xasvydgdsnyzkhez
bcrt1qm99230tpqflq0f8kpkn5d2tee02hgqcsw5sd99
bcrt1qd0afjfnl5udrsfkrj72rl34pss34yluma752qv
bcrt1qj2aymplrzxcp4m7vcxrzq93g58pmgm4fpluesy
bcrt1q4p4k63xglftez0h8yc7d4kmhsn5j5kecguu34j
bcrt1q29z2uanskweur7qrzr43gyv3l028s0pnd9ptvp
bcrt1qkzpeqz8sd73sucfythjxftez0e3ee30yhp9w67
bcrt1qptwd6ggy8ttryck2f6yjf4la68apruc3fs7elz

# Generation via bitcoin-cli
repeat 10 { bitcoin-cli -rpcwallet="mywallet" getnewaddress }
bcrt1qc9wzxf8pthyexl00m23ug92pqrthagnzzf33wp
bcrt1qgnh7e72q92fqujwg3qxlg5kplxkm6rep0nerur
bcrt1qea6r8yvd0peupk29p94wm0xasvydgdsnyzkhez
bcrt1qm99230tpqflq0f8kpkn5d2tee02hgqcsw5sd99
bcrt1qd0afjfnl5udrsfkrj72rl34pss34yluma752qv
bcrt1qj2aymplrzxcp4m7vcxrzq93g58pmgm4fpluesy
bcrt1q4p4k63xglftez0h8yc7d4kmhsn5j5kecguu34j
bcrt1q29z2uanskweur7qrzr43gyv3l028s0pnd9ptvp
bcrt1qkzpeqz8sd73sucfythjxftez0e3ee30yhp9w67
bcrt1qptwd6ggy8ttryck2f6yjf4la68apruc3fs7elz

Notes:

  • The repeat n {} syntax will only work in zsh, you can use other loops for your shell, or just manually repeat the code 10 times.
  • In case you get different outputs in either of the cases, try deleting ~/.bdk-bitcoin and retrying (thanks @Steve (opens new window) for this tip!)

Note that both bdk-cli and bitcoin-cli produced the exact same addresses. So now we have definitive proof that descriptors can make wallets portable. That single string will be able to make any wallet generate the same set of addresses and hence they can sync and broadcast transactions in the same manner!

# Making a MultiSig Descriptor for Funds

In the real-life, most of us hold two kinds of savings accounts - one to store huge funds saved throughout our lifetime (probably without internet banking functionalities) and another for regular expenses.

In the Bitcoin world, to store huge funds, most people prefer to use a Multisig descriptor with a 2-of-3 or 3-of-4 setup. They can have one key stored in their PC, one key stored in a hardware wallet, one key stored in writing in a secure vault and another key learnt by heart. In case of a mishap like a house burning on fire or permanent memory loss, they would still be able to recover their funds by using the other keys.

Here's how a secure 2-of-3 descriptor generation would look like:

# xprv generation
K1_XPRV=$(bdk-cli key generate | jq -r ".xprv")
K2_XPRV=$(bdk-cli key generate | jq -r ".xprv")
K3_XPRV=$(bdk-cli key generate | jq -r ".xprv")

# xpub generation
K1_XPUB=$(bdk-cli key derive --xprv $K1_XPRV --path "m/84'/1'/0'/0" | jq -r ".xpub")
K2_XPUB=$(bdk-cli key derive --xprv $K2_XPRV --path "m/84'/1'/0'/0" | jq -r ".xpub")
K3_XPUB=$(bdk-cli key derive --xprv $K3_XPRV --path "m/84'/1'/0'/0" | jq -r ".xpub")

# Descriptors for each key - Since we used BIP-84 generation paths for xpubs,
# we need to append the same to the xprvs so that our wallet can understand 
# which path to generate addresses and xpubs from
K1_DESC="wsh(multi(2,$K1_XPRV/84'/1'/0'/0/*,$K2_XPUB,$K3_XPUB))"
K2_DESC="wsh(multi(2,$K1_XPUB,$K2_XPRV/84'/1'/0'/0/*,$K3_XPUB))"
K3_DESC="wsh(multi(2,$K1_XPUB,$K2_XPUB,$K3_XPRV/84'/1'/0'/0/*))"

Lets create three bdk wallets aliases with above descriptors for easy future use and do initial sync to create the wallet files

alias k1wallet='bdk-cli -n regtest wallet -w K1 -d $K1_DESC'
alias k2wallet='bdk-cli -n regtest wallet -w K2 -d $K2_DESC'
alias k3wallet='bdk-cli -n regtest wallet -w K3 -d $K3_DESC'

k1wallet sync
{}
k2wallet sync
{}
k3wallet sync
{}

Now, let us send some funds to an address generated by k1wallet.

# ask regtest to generate 101 blocks, so we get 50 regtest coins to play with.
# because coinbase amounts are only spendable after 100 blocks, we generate
# 101 blocks, to use the first block's coinbase amount.
CORE_ADDR=$(bitcoin-cli getnewaddress)
bitcoin-cli generatetoaddress 101 $CORE_ADDR
bitcoin-cli getbalance
50.00000000

# And then send 10 btc to an address generated by `K1` descriptor
BDK_ADDR=$(k1wallet get_new_address | jq -r ".address")
bitcoin-cli -rpcwallet=mywallet sendtoaddress $BDK_ADDR 10

# Confirm the transaction by creating one more block
bitcoin-cli generatetoaddress 1 $CORE_ADDR

Now sync the wallets and check balances in each

k1wallet sync
{}
k1wallet get_balance
{
  "satoshi": 1000000000
}

k2wallet sync
{}
k2wallet get_balance
{
  "satoshi": 1000000000
}

k3wallet sync
{}
k3wallet get_balance
{
  "satoshi": 1000000000
}

Everyone has the same amount of balance. This happened because it was a multisig wallet. Now, let's try to spend some balance. We will give back some balance to the wallet maintained by bitcoin-cli. But remember, this is a 2-of-3 multisig wallet. That's why we will need at least two keys to sign to make a transaction.

Here's where we will require to use a PSBT (opens new window) or a partially signed bitcoin transaction. Bitcoin uses PSBTs as the standard protocol to create a transaction and add one or more signatures to it before broadcasting the same to the network which finally can become a proper valid transaction.

We will aks k2wallet to create and sign the transaction then k1wallet and k3wallet will co-sign it. Note that k2wallet even if it creates the transaction, doesn't need to sign it, because its a 2-of-3 multisig!

# create the transaction, can be started by anyone
PSBT=$(k2wallet create_tx --to "$CORE_ADDR:100000000" | jq -r ".psbt")

# Sign the transaction by K1 and look at the output
# it should say the psbt is not finalized since only one party has signed
k1wallet sign --psbt $PSBT
{
   "is_finalized": false,
   "psbt": "[...]"
}

# Saving the PSBT signed by K1
K1_SIGNED_PSBT=$(k1wallet sign --psbt $PSBT | jq -r ".psbt")

# Sign by K3 - should be finalized this time
# Notice that this time, the input psbt was the signed PSBT of K1
k3wallet sign --psbt $K1_SIGNED_PSBT
{
   "is_finalized": true,
   "psbt": "[...]"
}

# Saving the PSBT signed by K3
SIGNED_PSBT=$(k3wallet sign --psbt $K1_SIGNED_PSBT | jq -r ".psbt")

# Broadcast the transaction, again doesn't really matter who broadcasts
k2wallet broadcast --psbt $SIGNED_PSBT
{
   "txid": "49e2706fc73c49605692bf1b9ce58baf1eb0307ea39b3118628994fd56c9b642"
}

# Confirm the transaction by generating one block
bitcoin-cli generatetoaddress 1 $CORE_ADDR

# Sync and check balance - it should have gone down by 100000000 + tx fees
k1wallet sync
k1wallet get_balance
{
  "satoshi": 899999810
}
# Check similarly for `k2wallet` and `k3wallet` and they should all have same balance

So this proves we can definitely do transactions with multisig wallets with complicated descriptors. Since for Bitcoin, having keys equal having access to the accounts, we need to keep our keys safe. For legacy single key wallets, we used to keep backups of the mnemonic codes in multiple places. It was pretty insecure because in case any one of those backups leaks, our entire account would be compromised. Complicated multisig wallet descriptors are definitely a step forward - just in case a single key leak or are lost, no one would be able to take charge of the funds we hold.

Another problem with multisig was syncing between wallets to always create consistent addresses. How would one wallet know whats the next address to create without talking to other wallets? The answer is descriptors + PSBT. If all the wallet shares the correct descriptor string they will always create the exact sequence of addresses and by passing around PSBTs they would know how to sign them, without talking to each other. This solves a major problem of multisig interoperability. And BDK makes this process as seamless as possible.

# Retention Bonus - Smart Contract with Bitcoin

Let us consider that a company wants to give its employees a retention bonus for two months. If an employee stays with that company for over 2 months, the employee would get 1 BTC as a reward. This would be a smart contract between the company and an employee. The employee should be able to see that he would get his funds after two months. The company would require confidence that the employee would not be able to withdraw the reward before two months have passed.

The Miniscript policy for this contract would be as follows:

or(99@and(pk(E),older(8640)),pk(C))

where E is the employee and C is the company.

I should emphasize over here that this policy will let the company still transfer funds after the designated 2 months. It's not possible to block them after the lock time has passed, atleast not in a single policy.

Surely, after two months, the funds can be unlocked by the employee but before that, the company can revoke the funds. Let us compile this policy down to a descriptor. And this time we will ask help from the miniscript program.

# The Descriptor will be on the log, the E and C are placeholders
miniscriptc "or(99@and(pk(E),older(8640)),pk(C))" sh-wsh
[2021-08-05T12:25:40Z INFO  miniscriptc] Compiling policy: or(99@and(pk(E),older(8640)),pk(C))
[2021-08-05T12:25:40Z INFO  miniscriptc] ... Descriptor: sh(wsh(andor(pk(E),older(8640),pk(C))))#55wzucxa
Error: Descriptor(Miniscript(Unexpected("Key too short (<66 char), doesn't match any format")))

So the compiled descriptor is

sh(wsh(andor(pk(E),older(8640),pk(C))))

Let's make the keys, generate addresses using the above descriptor and fund it.

# xprvs
E_XPRV=$(bdk-cli key generate | jq -r ".xprv")
C_XPRV=$(bdk-cli key generate | jq -r ".xprv")

# xpubs
E_XPUB=$(bdk-cli key derive --xprv $E_XPRV --path "m/84'/1'/0'/0" | jq -r ".xpub")
C_XPUB=$(bdk-cli key derive --xprv $C_XPRV --path "m/84'/1'/0'/0" | jq -r ".xpub")

# descriptors using the compiled miniscript
# please note in case company or the employee was using a complicated multisig descriptor,
# it may as well have been added here like we did in the example before
E_DESC="sh(wsh(andor(pk($E_XPRV/84'/1'/0'/0/*),older(8640),pk($C_XPUB))))"
C_DESC="sh(wsh(andor(pk($E_XPUB),older(8640),pk($C_XPRV/84'/1'/0'/0/*))))"

# Create wallet aliases for easy access and sync the wallet to create initial wallet files
alias Cwallet='bdk-cli -n regtest wallet -w C -d $C_DESC'
alias Ewallet='bdk-cli -n regtest wallet -w E -d $E_DESC'

Cwallet sync
{}
Ewallet sync
{}

# get some funds in  Cwallet's address
C_ADDR=$(Cwallet get_new_address | jq -r ".address")
bitcoin-cli -rpcwallet=mywallet sendtoaddress $C_ADDR 10

# Confirm the transaction
bitcoin-cli generatetoaddress 1 $CORE_ADDR

# Sync and check balance
Cwallet sync
{}
Cwallet get_balance
{
  "satoshi": 1000000000
}

# Just as before, the employe can also see the fund in their wallet
Ewallet sync
{}
Ewallet get_balance
{
  "satoshi": 1000000000
}

According to the spending policy, for E has to wait for 8640 blocks before he can spend the coins. But let's check what happens if E tries to transact before the designated 2 months anyway.

# address to send the transaction to
E_ADDR=$(Ewallet getnewaddress | jq -r ".address")

# get external_policy id - this identifies which policy the wallet will try to sign against
POLICY_ID=$(Ewallet policies | jq -r ".external | .id")

# create the tx (external_policy id from last step in my case is j7ncy3au
PSBT=$(Ewallet create_tx --to "$E_ADDR:100000000" --external_policy "{\"$POLICY_ID\":[0]}" | jq -r ".psbt")

# sign and save the signed psbt
SIGNED_PSBT=$(Ewallet sign --psbt $PSBT | jq -r ".psbt")

# now let's try to broadcast - and see it failing
Ewallet broadcast --psbt $SIGNED_PSBT
[2021-08-05T17:48:45Z ERROR bdk_cli] Electrum(Protocol(Object({"code": Number(2), "message": String("sendrawtransaction RPC error: {\"code\":-26,\"message\":\"non-BIP68-final\"}")})))

We get an error saying the transaction we sent is Not BIP68 Final. BIP68 (opens new window) is a relative lock-time specification that ensures consensus when a signed transaction is sent which is invalid at a given time because the lock time isn't passed. So that's an expected error.

Now let's simulate two months passing and retry.

# simulate two months
# this will take around 1 mins to complete
bitcoin-cli generatetoaddress 8640 $CORE_ADDR

# create, sign and broadcast tx
PSBT=$(Ewallet create_tx --to $E_ADDR:100000000 --external_policy "{\"$POLICY_ID\":[0]}" | jq -r ".psbt")
SIGNED_PSBT=$(Ewallet sign --psbt $PSBT | jq -r ".psbt")
Ewallet broadcast --psbt $SIGNED_PSBT
{
  "txid": "2a0919bb3ce6e26018698ad1169965301a9ceab6d3da2a3dcb41343dc48e0dba"
}

# Confirm the transaction
bitcoin-cli generatetoaddress 1 $CORE_ADDR

# Sync and check balances
Cwallet sync
{}
Cwallet get_balance
{
  "satoshi": 999999810
}

Ewallet sync
{}
Ewallet get_balance
{
  "satoshi": 999999810
}

So this time it worked, because we have simulated 2 months passing by generating 8640 blocks. And both the Company and Employe wallet gets updated. Hence, we saw that we can generate some smart contracts using Bitcoin.

# Inspirations

  1. Descriptors from Bitcoin Core (opens new window)
  2. Miniscript (opens new window)
  3. Output Script Descriptors (opens new window)
  4. Descriptors in Bitcoin Dev Kit (opens new window)
  5. Role of Descriptors (opens new window)
  6. Making a Taproot Descriptor Wallet using bitcoin-cli (opens new window)
  7. Miniscripts SBC '19 - Video (opens new window)
  8. Rethinking Wallet Architecture: Native Descriptor Wallets - Video (opens new window)

Special thanks to my mentor Steve Myers (opens new window) for the constant motivation and support he gave me and for clearing so many doubts! Immense thanks to Raj (opens new window) for reviewing this blog and giving such detailed suggestions. Many of the lines added here are his. Also, thanks to the folks at the #miniscript IRC channel to help me out with the Retention Bonus policy.

This blog was written during Summer of Bitcoin 2021 (opens new window) by Sandipan Dey (opens new window).