For the issues of non-additive system noise and time-varying measurement noise during the cooperative navigation model of multiple AUVs, this paper proposes a square-root decomposition-based Extended dimension Cubature Kalman Filtering (SRECKF) algorithm. Firstly, the multiplicative noise is incorporated into the state variables for estimation, leveraging the model to estimate the noise. Subsequently, the square-root decomposition replaces the original Cholesky decomposition, propagating the square root of the variance. Finally, simulation experiments demonstrate that the proposed algorithm offers higher localization accuracy and stability compared to the Embedded Cubature Kalman Filter (ECKF) and the Extended Dimension Cubature Kalman Filter based on the SVD decomposition algorithm (SVDECKF).