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In Fall 2009, the final step of the factorization of rsa768 was carried out on several clusters of the Grid'5000 platform, leading to a new record in integer factorization. This step involves solving a huge sparse linear system defined over the binary field GF(2). This article aims at describing the algorithm used, the difficulties encountered, and the methodology which led to success. In particular, we illustrate how our use of the block Wiedemann algorithm led to a method which is suitable for use on a grid platform, with both adaptability to various clusters, and error detection and recovery procedures. While this was not obvious at first, it eventually turned out that the contribution of the Grid'5000 clusters to this computation was major.