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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]

The CRYSTALS project developed the lattice-based designs CRYSTALS-Kyber and CRYSTALS-Dilithium. They solve different tasks: Kyber is a key encapsulation mechanism, while Dilithium provides digital signatures. Their compact representations use structured polynomial arithmetic and explicit computational hardness assumptions. Kyber's construction uses module learning with errors; Dilithium combines a Fiat-Shamir transform with response rejection. ML-KEM and ML-DSA are the corresponding standardized descendants, so a project submission and a final standard should be distinguished when choosing precise encodings or parameters.^[ref][ref]

Provable security is a mathematical guarantee that a cryptographic protocol satisfies a precisely defined security property under stated assumptions. A cryptographic security game specifies the adversary's observations, permitted queries and success condition. A security reduction transforms an adversary breaking that property into an algorithm solving another problem, such as one covered by a computational hardness assumption. The reduction accounts for both running time and success probability.^[ref]

For an original numerical example, suppose a proved reduction givesAn attack with advantage $0.01$ would yield an algorithm $B$ with advantage at leastIf the underlying assumption excludes that advantage at $B$'s running time, the scheme attack is excluded too. A larger multiplicative loss would weaken the inference; ignoring the reduction's resource cost could also make the resulting bound uninformative.

The definition matters as much as the deduction. A message authentication code theorem concerns forgery under the specified access model; it does not thereby establish encryption confidentiality. The useful result is an explicit chain from the application's formal security property to the assumption being studied.^[ref]

Shafi Goldwasser coauthored How to Construct Random Functions by Oded Goldreich, Shafi Goldwasser and Silvio Micali. The paper formalizes pseudorandom function families through an observer's ability to distinguish keyed evaluation from access to one uniformly random function.^[ref]

The observer may choose later inputs after seeing earlier answers. Security therefore concerns the whole interaction, not just whether each separate output appears balanced. This computational experiment makes an apparently random reusable function a precise cryptographic object, conditional on the underlying hardness assumptions.