Provable State Erasure Primitive and DRBG Design Based on Bi-Group Decoupling and Algebraic Valuation Collapse

The backtracking resistance of Deterministic Random Bit Generators (DRBGs) serves as a fundamental defense in modern cryptographic systems against memory leakage attacks. Existing standards (e.g., NIST SP 800-90A) rely on the iterative obfuscation of symmetric primitives to achieve state forgetting, which suffers from theoretical limitations, including insufficient erasure granularity (merely computational-level many-to-many mappings) and the absence of an analytical lower bound for the forgetting rate. In this paper, we propose a novel cryptographic state erasure paradigm based on arithmetic dynamics. Through a "bi-group decoupling" design over finite fields ($p-1 = M \cdot R$), we pioneer the introduction of the $q$-adic valuation collapse phenomenon into the DRBG state update mechanism. We construct a state erasure primitive, $\text{StateErase}$, which leverages the algebraic property $A \equiv 0 \pmod R$ to deterministically map the "entropy pit" component of the state space to zero, achieving information-theoretic-level physical loss. Based on the spectral theory of Markov chains, we prove that even if the underlying pseudorandom exponent generator is biased, the failure probability of state erasure remains strictly upper-bounded by a pure algebraic invariant (the spectral gap parameter $\lambda_2$). This forgetting rate exhibits strong "scale invariance": the number of algebraic iterations $t$ required to achieve a 128-bit security level is completely decoupled from the bit-length of the prime $p$. Through standard cryptographic game-based proofs, we tightly reduce the security of the residual state to the Discrete Logarithm Problem (DLP) in the large subgroup. We propose a "high-frequency symmetric + low-frequency algebraic" hybrid engineering architecture, coupled with a generalized Mersenne prime acceleration strategy, rendering the primitive viable for industrial deployment in low-frequency reseeding scenarios.