This paper explores the efficient algorithms for solving overdefined systems of multivariate polynomial equations, which are crucial for the security of many cryptosystems. The authors analyze the relinearization technique introduced by Kipnis and Shamir, which aims to solve systems with more equations than variables. They find that many equations generated by relinearization are linearly dependent, reducing its efficiency. To address this, they propose the XL (eXtended Linearization) algorithm, which combines bounded-degree Gröbner bases and linearization. The XL algorithm is simpler and more powerful than relinearization, and it is expected to run in polynomial time for systems with $m \geq \epsilon n^2$ equations, where $\epsilon > 1/2 - 1/\sqrt{6}$. The paper also discusses the complexity of XL and provides experimental results showing its effectiveness in solving randomly generated systems. Additionally, the authors compare XL with Gröbner bases algorithms and discuss the cryptanalysis of the HFE cryptosystem using XL and relinearization attacks. The paper concludes by highlighting the practical implications of these algorithms for cryptographic security.This paper explores the efficient algorithms for solving overdefined systems of multivariate polynomial equations, which are crucial for the security of many cryptosystems. The authors analyze the relinearization technique introduced by Kipnis and Shamir, which aims to solve systems with more equations than variables. They find that many equations generated by relinearization are linearly dependent, reducing its efficiency. To address this, they propose the XL (eXtended Linearization) algorithm, which combines bounded-degree Gröbner bases and linearization. The XL algorithm is simpler and more powerful than relinearization, and it is expected to run in polynomial time for systems with $m \geq \epsilon n^2$ equations, where $\epsilon > 1/2 - 1/\sqrt{6}$. The paper also discusses the complexity of XL and provides experimental results showing its effectiveness in solving randomly generated systems. Additionally, the authors compare XL with Gröbner bases algorithms and discuss the cryptanalysis of the HFE cryptosystem using XL and relinearization attacks. The paper concludes by highlighting the practical implications of these algorithms for cryptographic security.