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We propose a novel variant of the gradient descent bit-flipping (GDBF) algorithm for decoding low-density parity-check (LDPC) codes over the binary symmetric channel. The new bit-flipping rule is based on the reliability information passed from neighboring nodes in the corresponding Tanner graph. The name SuspicionDistillation reflects the main feature of the algorithm—that in every iteration, we assign a level of suspicion to each variable node about its current bit value. The level of suspicion of a variable node is used to decide whether the corresponding bit will be flipped. In addition, in each iteration, we determine the number of satisfied and unsatisfied checks that connect a suspicious node with other suspicious variable nodes. In this way, in the course of iteration, we “distill” such suspicious bits and flip them. The deterministic nature of the proposed algorithm results in a low-complexity implementation, as the bit-flipping rule can be obtained by modifying the original GDBF rule by using basic logic gates, and the modification is not applied in all decoding iterations. Furthermore, we present a more general framework based on deterministic re-initialization of the decoder input. The performance of the resulting algorithm is analyzed for the codes with various code lengths, and significant performance improvements are observed compared to the state-of-the-art hard-decision-decoding algorithms.