Lines of code
<https://github.com/code-423n4/2023-03-mute/blob/4d8b13add2907b17ac14627cfa04e0c3cc9a2bed/contracts/amplifier/MuteAmplifier.sol#L366-L388&gt;
<https://github.com/code-423n4/2023-03-mute/blob/4d8b13add2907b17ac14627cfa04e0c3cc9a2bed/contracts/amplifier/MuteAmplifier.sol#L417-L460&gt;

Vulnerability details

Impact

This report deals with how the calculation of the multiplier in the MuteAmplifier contract is not only different from how it is displayed in the documentation on the website but it is also different in a very important way.

The calculation on the website shows a linear relationship between the dMUTE / poolSize ratio and the APY. The dMUTE / poolSize ratio is also called the tokenRatio.
By “linear” I mean that when a user increases his tokenRatio from 0 to 0.1 this has the same effect as when increasing it from 0.9 to 1.

The implementation in the MuteAmplifier.calculateMultiplier function does not have this linear relationship between tokenRatio and APY.
An increase in the tokenRatio from 0 to 0.1 is worth much less than an increase from 0.9 to 1.

As we will see this means that all stakers that do not have a tokenRatio of exactly equal 0 or exactly equal 1 lose out on rewards that they should receive according to the documentation.

I estimate this to be of “High” severity because the issue affects nearly all stakers and results in a partial loss of rewards.

Proof of Concept

Let’s first look at the multiplier calculation from the documentation:

The example calculation shows that the amount that is added to $APY_{base}$ (5%) is scaled linearly by the $\dfrac{user_{dmute}}{pool_{rewards}}$ ratio which I called tokenRatio above.

This means that when a user increases his tokenRatio from say 0 to 0.1 he gets the same additional reward as when he increases the tokenRatio from say 0.9 to 1.

Let’s now look at how the reward and thereby the multiplier is calculated in the code.

The first step is to calculate the multiplier which happens in the MuteAmplifier.calculateMultiplier function:

Link

function calculateMultiplier(address account, bool enforce) public view returns (uint256) {
    require(account != address(0), "MuteAmplifier::calculateMultiplier: missing account");


    uint256 accountDTokenValue;


    // zkSync block.number = L1 batch number. This at times is the same for a few minutes. To avoid breaking the call to the dMute contract
    // we take the previous block into account
    uint256 staked_block =  _userStakedBlock[account] == block.number ? _userStakedBlock[account] - 1 : _userStakedBlock[account];


    if(staked_block != 0 && enforce)
        accountDTokenValue = IDMute(dToken).getPriorVotes(account, staked_block);
    else
        accountDTokenValue = IDMute(dToken).getPriorVotes(account, block.number - 1);


    if(accountDTokenValue == 0){
        return _stakeDivisor;
    }


    uint256 stakeDifference = _stakeDivisor.sub(10 ** 18);


    // ratio of dMute holdings to pool
    uint256 tokenRatio = accountDTokenValue.mul(10**18).div(totalRewards);


    stakeDifference = stakeDifference.mul(clamp_value(tokenRatio, 10**18)).div(10**18);


    return _stakeDivisor.sub(stakeDifference);
}

The multiplier that is returned is then used to calculate the reward:

Link

reward = lpTokenOut.mul(_totalWeight.sub(_userWeighted[account])).div(calculateMultiplier(account, true));

Let’s write the formula in a more readable form:

$\dfrac{lpTokenOut * weightDifference}{stakeDivisor - tokenRatio * (stakeDivisor - 1)}$

$stakeDivisor$ can be any number $&gt;=1$ and has the purpose of determining the percentage of rewards a user with $tokenRatio=0$ gets.

For the sake of this argument we can assume that all numbers except $tokenRatio$ are constant.

Let’s just say $stakeDivisor=2$ which means that a user with $tokenRatio=0§ would receive $\dfrac{1}{2}=50%$ of the maximum rewards.

Further let’s say that $lpTokenOut * weightDifference = 1$, so 100% of the possible rewards would be $1$.

We can then write the formula like this:

$\dfrac{1}{2 - tokenRatio}$

So let’s compare the calculation from the documentation with the calculation from the code by looking at a plot:

x-axis: tokenRatio y-axis: percentage of maximum rewards

We can see that the green curve is non-linear and below the blue curve.

So the rewards as calculated in the code are too low.

Tools Used

VSCode

Recommended Mitigation Steps

I recommend to change the reward calculation to this:

$(lpTokenOut * weightDifference) * (percentage_{min} + clamp(\dfrac{user_{dmute}}{pool_{rewards}},1) * (1 - percentage_{min}))$

Instead of setting the stakeDivisor upon initialization, the percentageMin should be set which can be in the interval [0,1e18].

Fix:

diff --git a/contracts/amplifier/MuteAmplifier.sol b/contracts/amplifier/MuteAmplifier.sol
index 9c6fcb5..1c86f5c 100644
--- a/contracts/amplifier/MuteAmplifier.sol
+++ b/contracts/amplifier/MuteAmplifier.sol
@@ -48,7 +48,7 @@ contract MuteAmplifier is Ownable{
 
     uint256 private _mostRecentValueCalcTime; // latest update modifier timestamp
 
-    uint256 public _stakeDivisor; // divisor set in place for modification of reward boost
+    uint256 public _percentageMin; // minimum percentage set in place for modification of reward boost
 
     uint256 public management_fee; // lp withdrawal fee
     address public treasury; // address that receives the lp withdrawal fee
@@ -131,8 +131,8 @@ contract MuteAmplifier is Ownable{
      *  @param _mgmt_fee uint
      *  @param _treasury address
      */
-    constructor (address _lpToken, address _muteToken, address _dToken, uint256 divisor, uint256 _mgmt_fee, address _treasury) {
-        require(divisor >= 10 ** 18, "MuteAmplifier: invalid _stakeDivisor");
+    constructor (address _lpToken, address _muteToken, address _dToken, uint256 percentageMin, uint256 _mgmt_fee, address _treasury) {
+        require(_percentageMin <= 10 ** 18, "MuteAmplifier: invalid _percentageMin");
         require(_lpToken != address(0), "MuteAmplifier: invalid lpToken");
         require(_muteToken != address(0), "MuteAmplifier: invalid muteToken");
         require(_dToken != address(0), "MuteAmplifier: invalid dToken");
@@ -142,7 +142,7 @@ contract MuteAmplifier is Ownable{
         lpToken = _lpToken;
         muteToken = _muteToken;
         dToken = _dToken;
-        _stakeDivisor = divisor;
+        _percentageMin = percentageMin;
         management_fee = _mgmt_fee; //bps 10k
         treasury = _treasury;
 
@@ -368,7 +368,7 @@ contract MuteAmplifier is Ownable{
         require(lpTokenOut > 0, "MuteAmplifier::_applyReward: no coins staked");
 
         // current rewards based on multiplier
-        reward = lpTokenOut.mul(_totalWeight.sub(_userWeighted[account])).div(calculateMultiplier(account, true));
+        reward = lpTokenOut.mul(_totalWeight.sub(_userWeighted[account])).div(10 **18).mul(calculateMultiplier(account, true)).div(10** 18);
         // max possible rewards
         remainder = lpTokenOut.mul(_totalWeight.sub(_userWeighted[account])).div(10**18);
         // calculate left over rewards
@@ -442,7 +442,7 @@ contract MuteAmplifier is Ownable{
             uint256 _totalWeightFee1Local = _totalWeightFee1.add(fee1.mul(10**18).div(totalCurrentStake));
 
             // current rewards based on multiplier
-            info.currentReward = totalUserStake(user).mul(totWeightLocal.sub(_userWeighted[user])).div(info.multiplier_last);
+            info.currentReward = totalUserStake(user).mul(totWeightLocal.sub(_userWeighted[user])).div(10 **18).mul(info.multiplier_last).div(10** 18);
             // add back any accumulated rewards
             info.currentReward = info.currentReward.add(_userAccumulated[user]);
 
@@ -452,7 +452,7 @@ contract MuteAmplifier is Ownable{
 
         } else {
           // current rewards based on multiplier
-          info.currentReward = totalUserStake(user).mul(_totalWeight.sub(_userWeighted[user])).div(info.multiplier_last);
+          info.currentReward = totalUserStake(user).mul(_totalWeight.sub(_userWeighted[user])).div(10 **18).mul(info.multiplier_last).div(10** 18);
           // add back any accumulated rewards
           info.currentReward = info.currentReward.add(_userAccumulated[user]);
         }
@@ -485,17 +485,17 @@ contract MuteAmplifier is Ownable{
           accountDTokenValue = IDMute(dToken).getPriorVotes(account, block.number - 1);
 
         if(accountDTokenValue == 0){
-          return _stakeDivisor;
+          return _percentageMin;
         }
 
-        uint256 stakeDifference = _stakeDivisor.sub(10 ** 18);
+        uint256 percentageDifference = (uint256(10 ** 18)).sub(_percentageMin);
 
         // ratio of dMute holdings to pool
         uint256 tokenRatio = accountDTokenValue.mul(10**18).div(totalRewards);
 
-        stakeDifference = stakeDifference.mul(clamp_value(tokenRatio, 10**18)).div(10**18);
+        uint256 additionalPercentage = percentageDifference.mul(clamp_value(tokenRatio, 10**18)).div(10**18);
 
-        return _stakeDivisor.sub(stakeDifference);
+        return _percentageMin.add(additionalPercentage);
     }  

---

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