The Hardness of Solving Subset Sum with Preprocessing
IEEE Transactions on Information Theory, Vol. IT-36, pp. 943-946, July 1990.
Antoine LOBSTEIN
Centre National de la Recherche Scientifique
Ecole Nationale Supérieure des Télécommunications
46 rue Barrault, 75634 Paris cédex 13, France
Abstract. The two problems subset sum and linear decoding (which have given birth to the knapsack and the McEliece public-key cryptosystems) are known to be NP-complete. For linear decoding, it has been proved that, even if one knows the linear code in advance and can preprocess it, the existence of a polynomial-time decoding algorithm would imply that the polynomial-time hierarchy collapses at an early stage. It is proven that the same holds for subset sum: Even if the knapsack is known in advance and can be preprocessed, there is no polynomial-time algorithm solving it, unless the polynomial hierarchy collapses. We also give the sketch of a new, straightforward proof of this result for linear decoding.