The modern way of life depends on the security of cryptosystems we use every day. The main argument for that security is the fact that they have not yet been broken (at least not publicly). That kind of argument seems weak at first, but studying the existing approaches we gain better insight into the level of sophistication needed for a succe- ssful attack even using generous assumptions of knowledge not available in practice. In this thesis, we examine two such attacks on the public cryptosystem RSA, which assume partial knowledge of the private key. The first of these attacks is an advanced factoring technique called the LFM (Lattice Factoring Method) of Boneh, Durfee and Howgrave-Graham based on lattice basis reduction techniques. The second attack, by Heninger and Shacham, is more elementary and works by backtracking with elegant pruning. Both attacks were implemented in the C++ programming language (the re- levant parts are supplied at the end), and experimental measurements have confirmed that the attacks are applicable to realistic key lengths. This implies that any leakage of even a small number of private key bits renders the RSA cryptosystem insecure.