The problem of finding a solution to multivariate polynomial systems, and more specifi- cally quadratic systems is a algorithmic problem generally denoted as the MQ problem. If this problem is well kown and well studied for uniformly random systems, many MQ-based cryptosystems rely on the problem of solving structured polynomial systems. The main goal of this thesis was to study the security of specific cryptosystems that rely on structured systems. We first studied the Biscuit signature scheme, submitted to the NIST competition for additional post-quantum signatures. The security of Biscuit relies on the hardess of finding a solution to an arbitary polynomial system that has a known structure. We proposed a modified hybrid method specific to these systems that exploit their structure. This yields more efficient attacks that what the designers had anticipated and thus to a modification of the sinature parameters to verify the security level of the NIST competition. We also proposed a MQ-based commitment scheme. The security analysis of this commitment leaded us to study the hardness of finding a solution to a bilinear system. In this thesis, we proposed notions of semi-bi-regularity, verified empirically that unimormly random bilinear systems followed this property and deduced bounds on the computation cost of finding a solution to these systems. We thus deduced secured parameters for this new MQ-based commitment scheme.