This paper presents a novel approach to algebraic attacks on stream ciphers with linear feedback, focusing on the security of such ciphers against correlation attacks and higher-degree multivariate equations. The authors introduce a method to reduce the degree of algebraic equations by multiplying them with well-chosen multivariate polynomials, significantly lowering the complexity of solving these equations. This technique is applied to break the Toyocrypt stream cipher in $2^{49}$ CPU clocks, using only 20 Kbytes of keystream, and to attack the LILI-128 stream cipher within $2^{57}$ CPU clocks. The paper also demonstrates that if the Boolean function used in the cipher only uses a small subset of state bits, the cipher can be broken, regardless of the specific Boolean function. The authors propose a general algebraic attack that breaks stream ciphers satisfying all previously known design criteria, with complexity at most the square root of the complexity of the generic attack. The paper concludes with a set of design criteria for stream ciphers to resist algebraic attacks, emphasizing the importance of using a large number of state bits in the filtering function and avoiding sparse or highly nonlinear functions.This paper presents a novel approach to algebraic attacks on stream ciphers with linear feedback, focusing on the security of such ciphers against correlation attacks and higher-degree multivariate equations. The authors introduce a method to reduce the degree of algebraic equations by multiplying them with well-chosen multivariate polynomials, significantly lowering the complexity of solving these equations. This technique is applied to break the Toyocrypt stream cipher in $2^{49}$ CPU clocks, using only 20 Kbytes of keystream, and to attack the LILI-128 stream cipher within $2^{57}$ CPU clocks. The paper also demonstrates that if the Boolean function used in the cipher only uses a small subset of state bits, the cipher can be broken, regardless of the specific Boolean function. The authors propose a general algebraic attack that breaks stream ciphers satisfying all previously known design criteria, with complexity at most the square root of the complexity of the generic attack. The paper concludes with a set of design criteria for stream ciphers to resist algebraic attacks, emphasizing the importance of using a large number of state bits in the filtering function and avoiding sparse or highly nonlinear functions.