Lattices and Cryptography

The security of all of the public key cryptosystems that we have previously studied has been based, either directly or indirectly, on either the difficulty of factoring large numbers or the difficulty of finding discrete logarithms in a finite group. In this chapter we investigate a new type of hard problem arising in the theory of lattices that can be used as the basis for a public key cryptosystem. Lattice-based cryptosystems offer several potential advantages over earlier systems, including faster encryption/decryption and so-called quantum resistance. The latter means that at present there are no known quantum algorithms to rapidly solve hard lattice problems; see Sect. 8.11. Further, we will see that the theory of lattices has applications in cryptography beyond simply providing a new source of hard problems.

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