Summary: The self-shrinking Generator (SSG) was proposed by Meier and Staffelbach at Eurocrypt’94. Two similar guess-and-determine attacks were independently proposed by Hell-Johansson and Zhang-Feng in 2006, and give the best time/data tradeoff on this cipher so far. These attacks do not depend on the Hamming weight of the feedback polynomial (defining the LFSR in SSG). In this paper we propose a new attack strategy against SSG, when the Hamming weight is at most 5. For this case we obtain a better tradeoff than all previously known attacks (including Hell-Johansson and Zhang-Feng). Our main idea consists in guessing some information about the internal bitstream of the SSG, and expressing this information by a system of polynomial equations in the still unknown key bits. From a practical point of view, we show that using a SAT solver, such as MiniSAT, is the best way of solving this polynomial system. Since Meier and Staffelbach original paper, avoiding low Hamming weight feedback polynomials has been a widely believed principle. However this rule did not materialize in previous recent attacks. With the new attacks described in this paper, we show explicitly that this principle remains true.