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Here is a list of important theorems related to the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) in the context of lattice theory.
Hardness of SVP and CVP
The following theorems establish the computational difficulty of solving SVP and CVP, forming the foundation for lattice-based cryptography.
Van Emde Boas's Theorem (1981): This theorem states that the Closest Vector Problem (CVP) is NP-hard. This was one of the earliest results demonstrating the computational difficulty of fundamental lattice problems. The proof involves a reduction from the subset sum problem.
Ajtai's Theorem (1998): This groundbreaking result established that the Shortest Vector Problem (SVP) is NP-hard under randomized reductions. This was a significant breakthrough, as proving the NP-hardness of SVP had been a long-standing open problem. Ajtai's work introduced the concept of worst-case to average-case reductions for lattice problems, which has had a profound impact on cryptography.
Arora, Babai, Stern, and Sweedyk's Theorem (1993): This theorem addresses the hardness of approximating CVP. It shows that approximating CVP within any constant factor is NP-hard. This result was later improved to show that approximating CVP is hard even for sub-polynomial factors.
Micciancio's Theorem (2001): This theorem strengthened the understanding of the hardness of approximating SVP. It proved that approximating SVP to within any constant factor less than is NP-hard.
Relationship Between SVP and CVP
These theorems describe the relationship between the computational complexities of SVP and CVP.
Goldreich-Goldwasser-Halevi (GGH) Reduction (Informal): A key result demonstrates that SVP is reducible to CVP. This means that if one has an efficient algorithm to solve CVP, one can use it to solve SVP efficiently. The reduction involves constructing a new lattice where finding the closest vector to a specific target reveals the shortest vector in the original lattice. This implies that SVP is not harder than CVP.
Kannan's Transference Theorems: These theorems relate the lengths of the shortest non-zero vectors in a lattice and its dual lattice. While not a direct comparison of SVP and CVP, they are fundamental in understanding the geometry of lattices and are used in reductions between different lattice problems.
Minkowski's Theorems
Minkowski's theorems are fundamental results in the geometry of numbers that provide bounds on the length of the shortest vector in a lattice.
Minkowski's First Theorem: This theorem states that any convex set in that is symmetric with respect to the origin and has a volume greater than
(where
is the lattice) must contain at least one non-zero lattice point. A direct consequence of this theorem is an upper bound on the length of the shortest non-zero vector
:
where is the volume of the n-dimensional unit ball. This theorem guarantees the existence of a short vector but does not provide an efficient algorithm to find it.
Minkowski's Second Theorem: This theorem provides a tighter bound on the product of the successive minima of a lattice. The successive minima, , are the lengths of the
shortest linearly independent vectors in the lattice. The theorem states:
This theorem gives more profound insight into the geometric structure of lattices.
Theorems Related to Approximation Algorithms
These theorems are related to the performance of practical algorithms for approximating solutions to SVP and CVP.
LLL Algorithm Theorem (Lenstra, Lenstra, Lovász, 1982): The LLL algorithm is a polynomial-time lattice reduction algorithm that finds a "reduced" basis for a lattice. The first vector of an LLL-reduced basis, , is an approximation of the shortest vector in the lattice. The theorem guarantees that:
where is the dimension of the lattice. This provides an exponential approximation factor, but the algorithm is efficient in practice.
Babai's Nearest Plane Algorithm Theorem: This algorithm provides an approximate solution to CVP. Given a target vector and a "good" (e.g., LLL-reduced) basis for a lattice, the algorithm can efficiently find a lattice vector
that is close to
. The approximation factor depends on the quality of the basis. For an LLL-reduced basis, the distance from the found vector to the target is bounded, providing a solution to the approximate CVP.