The knapsack cryptography is the public-key cryptography whose security depends mainly on the hardness of the subset sum problem. Many of knapsack schemes were broken by low-density attacks, which are attack methods to use the situation that a shortest vector or a closest vector in a lattice corresponds to a solution of the subset sum problem. For the case when the Hamming weight of a solution for a random instance of the subset sum problem is arbitrary, if the density is less than 0.9408, then the instance can be solvable almost surely by a single call of lattice oracle. This fact was theoretically shown by Coster et al. In Crypto 2000, Okamoto, Tanaka and Uchiyama introduced the concept of quantum public key cryptosystems and proposed a knapsack cryptosystem, so-called OTU scheme. However, no known algorithm breaks the OTU scheme. In this paper, we introduce some combinatorial statements to describe necessary condition for the failure of low density attacks. More precisely, we give better heuristics than Gaussian heuristics for minimum norms of orthogonal lattices. Consequently, we show that the OTU scheme can be broken under these heuristics.

We introduce two frameworks in order to deal with fractal and multifractal analysis for subset sum problems where some embedding into the 1-dimensional Euclidean space plays an important role. As one of these frameworks, the notion of the combinatorial q-fractal dimension for a subset sum function is introduced. Thereby, "non-classical" generalized dimensions for a family of subset sum functions can be defined. These generalized dimensions include the box-counting dimension, the information dimension and the correlation dimension as well as the classical case. The combinatorial q-fractal dimension includes the density of the subset sum problem. As the other framework, we construct a self-similar set for a particular subset sum function in a family of subset sum functions by using a graph theoretical technique.In this paper, we give a lower bound for a combinatorial q-fractal dimension and we show the relations between the three parameters: the number of connected components in a graph, the Hausdorff dimension and a combinatorial q-fractal dimension.