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CRYSTALS-Dilithium is a lattice-based digital signature design whose arithmetic uses vectors of polynomials modulo a fixed polynomial and integer modulus. Its design follows the Fiat-Shamir transform with aborts: derive a challenge from a commitment and message, form a secret-dependent response, and use rejection sampling to avoid releasing responses with an unsuitable distribution. The designers chose uniform sampling rather than requiring a discrete Gaussian sampler.^[ref]

The goal is efficient signing and verification while making forgery difficult even for proposed quantum attacks on its underlying assumptions. ML-DSA is the later standardized descendant. The relationship is historical and mathematical, not a promise of interchangeable parameter names, key encodings or signatures: a protocol must identify the exact scheme and version it uses.

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]

The Fiat-Shamir transform replaces a verifier's random challenge in a suitable interactive proof protocol by a cryptographic hash function of the public statement and transcript so far. For a commitment $a$ about statement $x$, the prover computes $c=H(x,a)$ and then a response $z$. A verifier recomputes $c$ and checks the response. In signature applications the message is also bound into the hashed transcript.

This removes an interaction, but hashing an arbitrary protocol is not automatically secure: the protocol structure, transcript encoding and security model matter. CRYSTALS-Dilithium uses a version with aborts, discarding unsuitable responses through rejection sampling so that released responses have the required distribution. This illustrates how the transformation and its concrete implementation need separate arguments.^[ref]