Venus Prime - Maroutis's results

QA Report: Analyzing ln Function Implementation

Link : https://github.com/code-423n4/2023-09-venus/blob/b11d9ef9db8237678567e66759003138f2368d23/contracts/Tokens/Prime/libs/FixedMath0x.sol#L117-L136

Issue Summary:

• Topic: Precision Concern in Taylor's Expansion
• Impact: Potential precision loss in ln function implementation
• Function: ln(int256 x) internal pure returns (int256 r)

Detailed Analysis:

Taylor Series Expansion for Logarithm:

The Taylor Series Expansion for (\ln(1+x)) around (x=0) is given by: \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \ldots

The series converges for when x is close to 0, but as identified, may lack precision when x is close to 1.

Code Review:

The ln(int256 x) function employs Taylor Series Expansion to approximate the natural logarithm, including certain range-based corrections to enhance accuracy. However, it may lack safeguards against potential precision loss, necessitating a detailed analysis to propose improvements.

Potential Risks:

• Precision Loss: Imprecise results may arise, especially when x approaches 1 (or the residual is close to 2)
• Exploitation: Malicious actors may exploit precision loss to induce miscalculations or to gain unintended benefits.

Mitigation Strategies:

1. Use a Better Approximation for Values Near 1:
   • Before applying the taylor expansion, it would be more precise to add more condition to check for the values of x. Maby rather than dividing by 2 (e^-1, e^-0.5...), try working with smaller intervals
   • change the algorithm and apply something similar to the exponential function using the modulo function. This way the residual will always be smaller to a certain number.
2. Range Checks:
   • Implement strict range checks for x to signal or revert transactions that approach critical ranges, thereby avoiding the usage of imprecise results.

Exploitation Considerations:

• An attacker could leverage values that lead to imprecise results, influencing financial metrics to their advantage.