Security and Post-Quantum Analysis of ARGUS: A Differential-Algebraic Stream Cipher
Xinyu Yang · Zenodo (CERN European Organization for Nuclear Research) · 2026
ARGUS is a symmetric stream cipher whose keystream mixing function is not a fixed public permutation but a secret algebraic generator obtained by symbolically differentiating a user-supplied closed-form expression. This non-standard design resists analysis by any of the classical proof templates for block or stream ciphers, because the key selects not a value but a computational shape: we argue this constitutes a \emph{key-as-algorithm problem} structurally analogous to the crisis functional encryption faced when secret keys began denoting functions rather than indices, and we show, via Rice's theorem, that no finite rule set can certify key validity over an unbounded key grammar. Resolving this is, we argue, the paper's central contribution, prior to and underlying every technical result that follows. Concretely, we give a layered security argument: (i) a strict cryptographic reduction of the position-dependent offset sequence to the PRF security of BLAKE3; (ii) a new explicit hardness assumption, the Randomized-Shift Generator Non-Reconstructibility Assumption, together with a systematic negative-result analysis against Lagrange interpolation, Berlekamp--Massey style linear-complexity attacks, resultant/lattice attacks on degenerate low-degree instances, and Gr\"obner-basis/XL algebraic attacks; (iii) an exact linear-algebraic lemma quantifying the information destroyed by the differentiation operator, which we use to derive a concrete design constraint relating derivative order to expression complexity; (iv) a post-quantum layer reducing the residual hardness to a structured, explicitly quadratized instance of the Multivariate Quadratic (MQ) problem, under two axioms ruling out hidden group/discrete-log structure and exact periodicity, with an explicit open problem flagging the Rainbow/UOV precedent of classically exploitable structured trapdoors; and (v) auxiliary results on nonce-reuse finite-difference attacks, related-key attacks, the undecidability (Richardson's theorem) of complete symbolic canonicalization, and the scope of active-attack (IND-CCA) security, including a genuine sparse-validity lemma showing ARGUS possesses non-adaptive tamper detection ``for free''. Throughout we are explicit about which claims are proofs, which are reductions to named assumptions, and which are honestly-labeled open problems.