# Handle WatchPug # Vulnerability details ### Current Implementation #### When baseToken rebase up Per the document: and related code: Gamma is the ratio of shares received by the new liquidity provider when addLiquidity() (ΔRo) to the new totalSupply (total shares = Ro' = Ro + ΔRo). ΔRo = (Ro/(1 - γ)) * γ Ro * Gamma = -------------- 1 - Gamma ⟺ ΔRo * ( 1 - Gamma ) = Gamma * Ro ΔRo - Gamma * ΔRo = Gamma * Ro ΔRo = Gamma * Ro + Gamma * ΔRo ΔRo Gamma = --------- Ro + ΔRo In the current implementation: γ = ΔY / Y' / 2 * ( ΔX / α^ ) ΔY is the quoteToken added by the new liquidity provider. See: * Y' is the new Y after addLiquidity(), Y' = Y + ΔY. See: * * ΔX is ΔY * Omega. See: * * α^ is Alpha - X. See: * * For instance: Given: * Original State: X = Alpha = 1, Y = Beta = 1, Omega = X/Y = 1 * When baseToken rebase up: Alpha becomes 10 * Current State: Alpha = 10, X = 1, Y = Beta = 1, Omega = 1 When: new liquidity provider addLiquidity() with 4 quoteToken: 4 4 * Omega 16 Gamma = ------------ * ------------ = ---- (1+4) * 2 10 - 1 90 After addLiquidity(): * baseToken belongs to the newLP: 10 * 16 / 90 = 160 / 90 = 1.7777777777777777 * quoteToken belongs to the newLP: (1+4) * 16 / 90 = 80 / 90 = 0.8888888888888888 * In the terms of quoteToken, the total value is: 160 / 90 / Omega + 80 / 90 = 240 / 90 = 2.6666666666666665 As a result, the new liquidity provider suffers a fund loss of 4 - 240 / 90 = 1.3333333333333333 in the terms of quoteToken The case above can be reproduced by changing the numbers in [this test unit](). #### When baseToken rebase down Per the document: and related code: Gamma is the ratio of shares received by the new liquidity provider when addLiquidity() (ΔRo) to the new totalSupply (total shares = Ro' = Ro + ΔRo). ΔRo = (Ro/(1 - γ)) * γ Ro * Gamma = -------------- 1 - Gamma ⟺ ΔRo * ( 1 - Gamma ) = Gamma * Ro ΔRo - Gamma * ΔRo = Gamma * Ro ΔRo = Gamma * Ro + Gamma * ΔRo ΔRo Gamma = --------- Ro + ΔRo In the current implementation: γ = ΔX / X / 2 * ( ΔXByQuoteTokenAmount / β^ ) ΔX is the amount of baseToken added by the new liquidity provider. See: * X is the balanceOf baseToken. See: * ΔXByQuoteTokenAmount is ΔX / Omega, the value of ΔX in the terms of quoteToken. See: * * * β^ is maxΔX / Omega, the value of maxΔX in the terms of quoteToken. maxΔX = X - Alpha. See: * * * * For instance: Given: * Original State: X = Alpha = 10, Y = Beta = 10, Omega = X/Y = 1 * When baseToken rebase down, Alpha becomes 1 * Current State: Alpha = 1, X = 10, Y = Beta = 10, Omega = 1 When: new liquidity provider addLiquidity() with 4 baseToken 4 4 / Omega 8 Gamma = -------- * ---------------- = ---- 10 * 2 (10-1) / Omega 90 After addLiquidity(): * baseToken belongs to the newLP: (1 + 4) * 8 / 90 = 40 / 90 = 0.4444444444444444 * quoteToken belongs to the newLP: 10 * 8 / 90 = 80 / 90 = 0.8888888888888888 * In the terms of quoteToken, the total value is: 40 / 90 + 80 / 90 * Omega = 120 / 90 = 1.3333333333333333 < 4 As a result, the new liquidity provider suffers a fund loss of 4 - 120 / 90 = 2.6666666666666665 in the terms of quoteToken The case above can be reproduced by changing the numbers in [this test unit](). ### The correct formula for ΔRo #### When baseToken rebase up When: new liquidity provider addLiquidity with ΔY quoteToken (ΔY <= maxΔY or ΔY <= α^ / ω) After addLiquidity(): - baseToken belongs to the newLP: ΔXOfNewLP - quoteToken belongs to the newLP: ΔYOfNewLP ΔY can be divided into 2 parts: - ΔYToX: the part used for swap ΔXOfNewLP. ΔXOfNewLP = ΔYToX * Omega (a1) - ΔY - ΔYToX: the rest as ΔYOfNewLP The ratio of newly minted LP tokens for new liquidity provider to the new totalSupply (Ro'): Gamma ΔRo ΔY - ΔYToX ΔXOfNewLP = --------- = ----------- = ----------- (a2) Ro + ΔRo Y + ΔY Alpha ΔY - ΔYToX ΔYToX * Omega = ----------- = --------------- // substituting (a1) (a_exp1) Y + ΔY Alpha ⟺ (ΔY - ΔYToX) * Alpha = ΔYToX * Omega * (Y + ΔY) ΔY * Alpha - ΔYToX * Alpha = ΔYToX * Omega * (Y + ΔY) ΔY * Alpha = ΔYToX * Alpha + ΔYToX * Omega * (Y + ΔY) = ΔYToX * ( Alpha + Omega * (Y + ΔY)) ΔY * Alpha ΔY * Alpha ΔYToX = --------------------------- = -------------------- (a_r1) Alpha + Omega * (Y + ΔY) Alpha + Omega * Y' Continue from (a_exp1): ΔYToX * Omega Gamma = --------------- Alpha ΔY * Omega = ----------------------- // substituting (a_r1) (a_r2(1)) Alpha + Omega * Y' ΔY = --------------------- (a_r2(2)) Alpha/Omega + Y' Gamma is the ratio of ΔY to the total amounts of baseToken and quoteToken after addLiquidity: - (a_r2(1)) is the formula in the terms of baseToken - (a_r2(2)) is the formula in the terms of quoteToken Based on (a2): ΔRo Gamma = --------- Ro + ΔRo ⟺ ΔRo = Gamma * Ro + Gamma * ΔRo ΔRo - Gamma * ΔRo = Gamma * Ro ΔRo * ( 1 - Gamma ) = Gamma * Ro Ro * Gamma ΔRo = -------------- 1 - Gamma #### When baseToken rebase down When: new liquidity provider addLiquidity with ΔX baseToken (ΔX <= maxΔX or ΔY <= α^) After addLiquidity() - baseToken belongs to the newLP: ΔXOfNewLP - quoteToken belongs to the newLP: ΔYOfNewLP ΔX can be divided into 2 parts: - ΔXToY: the part used for swap ΔYOfNewLP. ΔYOfNewLP = ΔXToY / Omega (b1) - ΔX - ΔXToY: the rest as ΔXOfNewLP The ratio of newly minted LP tokens for new liquidity provider to the new totalSupply (Ro'): Gamma ΔRo ΔX - ΔXToY ΔYOfNewLP = --------- = ------------ = ------------ (b2) Ro + ΔRo Alpha + ΔX Y ΔXToY / Omega = ------------- // substituting (b1) Y ΔXToY = ------------- (b_exp1) Y * Omega ⟺ (ΔX - ΔXToY) * Y = (Alpha + ΔX) * ΔXToY / Omega ΔX * Y - ΔXToY * Y = (Alpha + ΔX) * ΔXToY / Omega ΔX * Y = ΔXToY * Y + (Alpha + ΔX) * ΔXToY / Omega = ΔXToY * ( Y + (Alpha + ΔX) / Omega ) ΔX * Y ΔXToY = -------------------------- (b_r1) Y + (Alpha + ΔX) / Omega Continue from (b_exp1) ΔXToY Gamma = ------------- Y * Omega ΔX * Y = ---------------------------------------- // substituting (b_r1) (Y + (Alpha + ΔX) / Omega) * Y * Omega ΔX / Omega = --------------------------- (b_r2(2)) Y + (Alpha + ΔX) / Omega ΔX = --------------------------- (b_r2(1)) Y * Omega + (Alpha + ΔX) ΔX = ------------------- // substituting (Omega = X/Y) X + (Alpha + ΔX) Gamma is the ratio of ΔX to the total amounts of baseToken and quoteToken after addLiquidity: - (b_r2(1)) is the formula in the terms of baseToken - (b_r2(2)) is the formula in the terms of quoteToken Based on (b2): ΔRo Gamma = --------- Ro + ΔRo ⟺ Ro * Gamma ΔRo = -------------- 1 - Gamma ### Recommendation Update code and document using the correct formula for ΔRo. --- The text was updated successfully, but these errors were encountered: All reactions