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A computational hardness assumption states that a specified computational task cannot be solved efficiently in a specified model and input distribution. A useful cryptographic statement identifies the parameter controlling size, the adversary's resources and the permitted success probability. Difficulty on occasional worst-case instances alone need not protect the instances produced by a key-generation algorithm.

For example, a proposed encryption scheme may reduce message recovery to solving learning with errors instances sampled in a particular way. The reduction says that a successful attack would yield a solver for those instances; security remains conditional on the hardness assumption and the implementation meeting the model. Oded Regev's lattice reduction makes this relationship unusually concrete, connecting a noisy average-case problem to worst-case lattice problems through a quantum reduction.^[ref]

Learning with errors asks for a secret vector $s$ from noisy modular linear observations. An observation has a known random vector $a$ andwhere $e$ is sampled from a specified small-error probability distribution. The search version recovers $s$; the decision version distinguishes these samples from uniformly random pairs. Parameters and the error distribution are essential parts of the problem.

For a tiny illustration, let $q=17$, $s=(3,5)$ and $a=(2,4)$. The exact inner product is $26\equiv 9(\mathrm{mod}17)$, while error $e=-1$ gives $b=8$. Without errors, enough independent equations admit ordinary modular linear solving; adding errors obstructs that direct approach. This tiny instance is not a secure parameter choice. Oded Regev's work connects appropriately parameterized instances to worst-case lattice problems and constructs a public-key cryptography from the assumption.^[ref]

Oded Regev introduced learning with errors as a foundation for lattice-based cryptography. His paper connects appropriately parameterized noisy modular linear equations to worst-case lattice problems through a quantum reduction, and constructs a classical public-key encryption scheme from that learning assumption.^[ref]

The distinction between a quantum reduction and a classical cryptosystem matters: the encryption and decryption procedures need not run on a quantum computer, even though the reduction establishing the hardness connection uses quantum computation. The work also makes the error distribution part of the problem definition. Removing the errors gives ordinary modular linear equations and changes the computational task fundamentally. Regev's research homepage identifies this lattice-based work among his contributions.^[ref]