Sums of divisors, perfect numbers and factoring
SIAM Journal on Computing
Algorithms for modular elliptic curves
Algorithms for modular elliptic curves
On Barvinok's algorithm for counting lattice points in fixed dimension
Mathematics of Operations Research
The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Some Recent Progress on the Complexity of Lattice Problems
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
A Note on the Shortest Lattice Vector Problem
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Proceedings of the forty-second ACM symposium on Theory of computing
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We consider the problem of counting the number of lattice vectors of a given length. We show that problem is @?P-complete resolving an open problem. Furthermore, we show that the problem is at least as hard as integer factorization even for lattices of bounded rank or lattices generated by vectors of bounded norm. Next, we discuss a deterministic algorithm for counting the number of lattice vectors of length d in time 2^O^(^r^s^+^l^o^g^d^), where r is the rank of the lattice, s is the number of bits that encode the basis of the lattice. The algorithm is based on the theory of modular forms.