Analysis of low density codes and improved designs using irregular graphs
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Cores in random hypergraphs and Boolean formulas
Random Structures & Algorithms
Algorithmic Barriers from Phase Transitions
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Information, Physics, and Computation
Information, Physics, and Computation
Modern Coding Theory
Tight thresholds for cuckoo hashing via XORSAT
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Factorization of a 768-bit RSA modulus
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
A Better Algorithm for Random $k$-SAT
SIAM Journal on Computing
On the solution-space geometry of random constraint satisfaction problems
Random Structures & Algorithms
Efficient erasure correcting codes
IEEE Transactions on Information Theory
The freezing threshold for k-colourings of a random graph
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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The XOR-satisfiability (XORSAT) problem requires finding an assignment of n Boolean variables that satisfy m exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing n variables and m clauses of size k. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as k-satisfiability (k-SAT). For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random k-XORSAT. In particular we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is achieved through a low complexity iterative algorithm.