Public-key cryptography from different assumptions

This paper attempts to broaden the foundations of public-key cryptography. We construct new public-key encryption schemes based on new hardness-on-average assumptions for natural combinatorial NP-hard optimization problems. We consider the following assumptions: It is infeasible to solve a random set of sparse linear equations mod 2, of which a small fraction is noisy. It is infeasible to distinguish between a random unbalanced bipartite graph, and such a graph in which we "plant" at random in the large side a set S with only |S|/3 neighbors. There is a pseudorandom generator in NCz where every output depends on a random constant-size subset of the inputs. We obtain semantically secure public key encryption schemes based on several combinations of these assumptions with different parameters. In particular we obtain public key encryption from Assumption~1 on its own, yielding the first noisy-equations type public key scheme in which the noise rate is higher than one over the square root of the number of equations. We also obtain public-key encryption based on a combination of Assumptions~2 and~3. These are arguably of more "combinatorial"/"private-key" nature than any assumptions used before for public-key cryptography. Our proof involves novel "search to decision" and "search to prediction" reductions for sparse noisy linear equations. The strength of our assumptions raise new algorithmic and pseudorandomness questions (and new parameters for old ones). We give some evidence for these assumptions by studying their resistance to certain classes of natural algorithms, including semi-definite programs, ACO circuits, low-degree polynomials, and cycle counting. We also relate our assumptions to other problems such as planted clique and learning juntas.