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Pools: Overview

A Curve pool is essentially a smart contract that implements the StableSwap invariant, housing the logic for exchanging stable tokens. While all Curve pools share this core implementation, they come in various pool flavors.

In its simplest form, a Curve pool is an implementation of the StableSwap invariant involving two or more tokens, often referred to as a 'plain pool.' Alternatively, Curve offers more complex pool variants, including pools with rebasing tokens and metapools. Metapools facilitate the exchange of one or more tokens with those from one or more underlying tokens.

New features:

Supported Assets

Stableswap-NG pools supports the following asset types:

Asset Type Description
0 Standard ERC20 token with no additional features
1 Oracle - token with rate oracle (e.g. wstETH)
2 Rebasing - token with rebase (e.g. stETH)
3 ERC4626 - token with convertToAssets method (e.g. sDAI)

Consequently, supported tokens include:

  • ERC20 support for return True/revert, return True/False, return None
  • ERC20 tokens can have arbitrary decimals (<=18)
  • ERC20 tokens that rebase (either positive or fee on transfer)
  • ERC20 tokens that have a rate oracle (e.g. wstETH, cbETH, sDAI, etc.) Oracle precision must be 10^18
  • ERC4626 tokens with arbitrary percision (<=18) of Vault token and underlying asset

Rebasing Tokens

Rebasing Tokens

Pools including rebasing tokens work a bit differently compared to others. The internal **_balance()** function - which is used to calculate the coin balances within the pool - makes sure that LP's keep all rebases.

def _balances() -> DynArray[uint256, MAX_COINS]:
    @notice Calculates the pool's balances _excluding_ the admin's balances.
    @dev If the pool contains rebasing tokens, this method ensures LPs keep all
            rebases and admin only claims swap fees. This also means that, since
            admin's balances are stored in an array and not inferred from read balances,
            the fees in the rebasing token that the admin collects is immune to
            slashing events.
    result: DynArray[uint256, MAX_COINS] = empty(DynArray[uint256, MAX_COINS])
    balances_i: uint256 = 0

    for i in range(MAX_COINS_128):

        if i == N_COINS_128:

        if 2 in asset_types:
            balances_i = ERC20(coins[i]).balanceOf(self) - self.admin_balances[i]
            balances_i = self.stored_balances[i] - self.admin_balances[i]


    return result

Dynamic Fees

Stableswap-NG introduces dynamic fees. The use of the offpeg_fee_multiplier allows the system to dynamically adjust fees based on the pool's state.

The internal _dynamic_fee() function calculates the fee based on the balances and rates of the tokens being exchanged. If the balances of the tokens being exchanged are highly imbalanced or significantly differ from its peg, the fee is adjusted using the offpeg_fee_multiplier.

Dynamic Fee Formula


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Let's define some terms and variables for clarity:

  • Let \(fee\) represent the fee, as retrieved by the method StableSwap.fee()
  • Let \(fee_m\) denote the off-peg fee multiplier, sourced from StableSwap.offpeg_fee_multiplier()
  • FEE_DENOMINATOR is a constant with a value of \(10^{10}\), representing the precision of the fee
  • The terms \(rate_{i}\) and \(balance{i}\) refer to the specific rate and balance for coin \(i\), respectively, and similarly, \(rate_j\) and \(balance_j\) for coin \(j\)
  • \(PRECISION_{i}\) and \(PRECISION_{j}\) are the precision constants for the respective coins

Given these, we define:

\(xp_{i} = \frac{{rate_{i} \times balance_{i}}}{{PRECISION_{i}}}\)

\(xp_{j} = \frac{{rate_{j} \times balance_{j}}}{{PRECISION_{j}}}\)

And we also have:

\(xps2 = (xp_{i} + xp_{j})^2\)

The dynamic fee is calculated by the following formula:

\[\text{dynamic fee} = \frac{{fee_{m} \times fee}}{\frac{(fee_{m} - 10^{10}) \times 4 \times xp_{i} \times xp_{j}}{xps2}+ 10^{10}}\]
dynamic_fee method
A_PRECISION: constant(uint256) = 100
MAX_COINS: constant(uint256) = 8
PRECISION: constant(uint256) = 10 ** 18
FEE_DENOMINATOR: constant(uint256) = 10 ** 10

def dynamic_fee(i: int128, j: int128, pool:address) -> uint256:
    @notice Return the fee for swapping between `i` and `j`
    @param i Index value for the coin to send
    @param j Index value of the coin to recieve
    @return Swap fee expressed as an integer with 1e10 precision
    N_COINS: uint256 = StableSwapNG(pool).N_COINS()
    fee: uint256 = StableSwapNG(pool).fee()
    fee_multiplier: uint256 = StableSwapNG(pool).offpeg_fee_multiplier()

    rates: DynArray[uint256, MAX_COINS] = empty(DynArray[uint256, MAX_COINS])
    balances: DynArray[uint256, MAX_COINS] = empty(DynArray[uint256, MAX_COINS])
    xp: DynArray[uint256, MAX_COINS] = empty(DynArray[uint256, MAX_COINS])
    rates, balances, xp = self._get_rates_balances_xp(pool, N_COINS)

    return self._dynamic_fee(xp[i], xp[j], fee, fee_multiplier)

def _dynamic_fee(xpi: uint256, xpj: uint256, _fee: uint256) -> uint256:

    _offpeg_fee_multiplier: uint256 = self.offpeg_fee_multiplier
    if _offpeg_fee_multiplier <= FEE_DENOMINATOR:
        return _fee

    xps2: uint256 = (xpi + xpj) ** 2
    return (
        (_offpeg_fee_multiplier * _fee) /
        ((_offpeg_fee_multiplier - FEE_DENOMINATOR) * 4 * xpi * xpj / xps2 + FEE_DENOMINATOR)

def _get_rates_balances_xp(pool: address, N_COINS: uint256) -> (
    DynArray[uint256, MAX_COINS],
    DynArray[uint256, MAX_COINS],
    DynArray[uint256, MAX_COINS],

    rates: DynArray[uint256, MAX_COINS] = StableSwapNG(pool).stored_rates()
    balances: DynArray[uint256, MAX_COINS] = StableSwapNG(pool).get_balances()
    xp: DynArray[uint256, MAX_COINS] = empty(DynArray[uint256, MAX_COINS])
    for idx in range(MAX_COINS):
        if idx == N_COINS:
        xp.append(rates[idx] * balances[idx] / PRECISION)

    return rates, balances, xp

Interactive Graph

The embedded graph has limited features, such as the inability to modify the axis. However, by clicking the "edit graph on desmos" button at the bottom right, one is redirected to the main Desmos site. There, a wider range of functionalities is available, allowing for further adjustments and detailed exploration of the graph.


The new generation (NG) of stableswap introduces oracles based on AMM State Prices and the invariant D.

  • price oracle (spot and ema price)
  • moving average D oracle

Oracles are updated when users perform a swap or when liquidity is added or removed from the pool. Most updates are carried out by the internal upkeep_oracles() function, which is called in those instances. In some cases, such as when removing liquidity in a balanced ratio, the D oracle is updated directly within the remove_liquidity() function, as there is no need to update the price oracles (removing balanced does not have a price impact).

Oracle Manipulation

The spot price cannot immediately be used for the calculation of the moving average, as this would allow for single block oracle manipulation. Consequently, _calc_moving_average uses last_prices_packed, which retains prices from previous actions.

upkeep_oracles method
def upkeep_oracles(xp: DynArray[uint256, MAX_COINS], amp: uint256, D: uint256):
    @notice Upkeeps price and D oracles.
    ma_last_time_unpacked: uint256[2] = self.unpack_2(self.ma_last_time)
    last_prices_packed_current: DynArray[uint256, MAX_COINS] = self.last_prices_packed
    last_prices_packed_new: DynArray[uint256, MAX_COINS] = last_prices_packed_current

    spot_price: DynArray[uint256, MAX_COINS] = self._get_p(xp, amp, D)

    # -------------------------- Upkeep price oracle -------------------------

    for i in range(MAX_COINS):

        if i == N_COINS - 1:

        if spot_price[i] != 0:

            # Upate packed prices -----------------
            last_prices_packed_new[i] = self.pack_2(
                    ma_last_time_unpacked[0],  # index 0 is ma_exp_time for prices

    self.last_prices_packed = last_prices_packed_new

    # ---------------------------- Upkeep D oracle ---------------------------

    last_D_packed_current: uint256 = self.last_D_packed
    self.last_D_packed = self.pack_2(
            ma_last_time_unpacked[1],  # index 1 is ma_exp_time for D

    # Housekeeping: Update ma_last_time for p and D oracles ------------------
    for i in range(2):
        if ma_last_time_unpacked[i] < block.timestamp:
            ma_last_time_unpacked[i] = block.timestamp

    self.ma_last_time = self.pack_2(ma_last_time_unpacked[0], ma_last_time_unpacked[1])

    def _calc_moving_average(
        packed_value: uint256,
        averaging_window: uint256,
        ma_last_time: uint256
    ) -> uint256:

        last_spot_value: uint256 = packed_value & (2**128 - 1)
        last_ema_value: uint256 = (packed_value >> 128)

        if ma_last_time < block.timestamp:  # calculate new_ema_value and return that.
            alpha: uint256 = self.exp(
                    (block.timestamp - ma_last_time) * 10**18 / averaging_window, int256
            return (last_spot_value * (10**18 - alpha) + last_ema_value * alpha) / 10**18

        return last_ema_value


This new function allows the exchange of tokens without actually transfering tokens in, as the exchange is based on the change of the coins balances within the pool (see code below).
Users of this method are dex aggregators, arbitrageurs, or other users who do not wish to grant approvals to the contract. They can instead send tokens directly to the contract and call exchange_received().


This function will revert if called on pools that contain rebasing tokens.

Transfer logic when using exchange_received()
def _transfer_in(
    coin_idx: int128,
    dx: uint256,
    sender: address,
    expect_optimistic_transfer: bool,
) -> uint256:
    @notice Contains all logic to handle ERC20 token transfers.
    @param coin_idx Index of the coin to transfer in.
    @param dx amount of `_coin` to transfer into the pool.
    @param dy amount of `_coin` to transfer out of the pool.
    @param sender address to transfer `_coin` from.
    @param receiver address to transfer `_coin` to.
    @param expect_optimistic_transfer True if contract expects an optimistic coin transfer
    _dx: uint256 = ERC20(coins[coin_idx]).balanceOf(self)

    # ------------------------- Handle Transfers -----------------------------

    if expect_optimistic_transfer:

        _dx = _dx - self.stored_balances[coin_idx]
        assert _dx >= dx


        assert dx > 0  # dev : do not transferFrom 0 tokens into the pool
        assert ERC20(coins[coin_idx]).transferFrom(
            sender, self, dx, default_return_value=True

        _dx = ERC20(coins[coin_idx]).balanceOf(self) - _dx

    # --------------------------- Store transferred in amount ---------------------------

    self.stored_balances[coin_idx] += _dx

    return _dx



Lets say a user wants to swap GOV-TOKEN<>USDC through an aggregator. For simplicity we assume, GOV-TOKEN<>USDT exchange is done via a uniswap pool, USDT<>USDC via a Curve pool.

graph LR
    u([USER]) --- p1[(UNISWAP)]
    p1 -->|"3. transfer out/in"| p2[(CURVE)]
    u -..-> |1. approve and transfer| a([AGGREGATOR])
    a ==> |"2. exchange"| p1
    a -.-|"4. exchange_received"| p2
    p2 --> |5. transfer dy out| u
    linkStyle 0 stroke-width:0, fill:none;
  1. User gives approval the AGGREGATOR, which then transfers tokens into the aggregator contract
  2. Aggregator exchanges GOV-TOKEN for USDT using Uniswap
  3. Transfers the USDT directly from Uniswap into the Curve pool
  4. Perform a swap on the Curve pool (USDT<>USDC) via exchange_received
  5. Transfer USDC to the user


This method saves aggregators one redundant ERC-20 transfer and eliminates the need to grant approval to a curve pool. Without this function, the aggregator would have to conduct an additional transaction, transferring USDT from the Uniswap pool to their aggregator contract after the exchange, and then sending it to the Curve pool for another exchange (USDT<>USDC). However, with this method in place, the aggregator can transfer the output tokens directly into the next pool and perform an exchange.