Serial Key Dust Settle «Premium Quality»

Future work: Extend model to quantum brute-force attacks and side-channel induced non-uniform priors. [1] T. Warez, "On the entropy of software keys," J. Cryptography , vol. 12, 2019. [2] L. Censor, "Partial information disclosure in product activation," IEEE S&P , 2022. [3] A. Attacker, "Dust settling in reduced keyspaces," Black Hat Briefings , 2023. If instead you meant something entirely different by "serial key dust settle" (e.g., a literal physical process of dust settling on a hardware serial key, or a term from a specific software tool), please clarify, and I will rewrite the paper accordingly.

| Attempts (log2) | KL Divergence (bits) | |----------------|----------------------| | 0 | 8.000 | | 10 | 7.998 | | 20 | 7.125 | | 30 | 3.210 | | 34 | 0.008 (< ε) |

where ( P_t ) is the attacker’s belief after ( t ) failed attempts. The ( T_s ) is the smallest ( t ) such that ( D(t) < \epsilon ) (e.g., ( \epsilon = 10^-6 ) bits). 3. Main Theorem: Exponential Dust Decay Theorem 1 (Exponential Settling). For a serial key with ( m ) unknown symbols and no validation bias (uniformly valid completions), the dust settles according to: serial key dust settle

[ D(t) = D(0) \cdot e^-t / \tau ]

Settling time ( T_s \approx 2^34 ) attempts, matching Theorem 1. We have formalized the concept of serial key dust settling — the decay of predictive entropy after partial key disclosure. The settling follows an exponential law with time constant proportional to the remaining valid keyspace. For robust licensing, designers must either (a) ensure the remaining keyspace is astronomically large even after partial leaks, or (b) introduce dynamic, server-side validation that resets the dust before it settles. Future work: Extend model to quantum brute-force attacks

No prior work has quantified how long (in terms of computational steps or guesses) it takes for this dust to settle. This paper fills that gap. 2. Formal Model 2.1 Key Representation Let a serial key be a string ( K = k_1 k_2 \ldots k_n ) where each ( k_i \in \Sigma ), ( |\Sigma| = 32 ) (alphanumeric excluding ambiguous chars). Total keyspace size ( N = 32^n ). 2.2 Partial Disclosure Event An attacker learns a set of positions ( P \subset 1,\ldots,n ) and their values. Let ( U = 1,\ldots,n \setminus P ) be the unknown positions. Before any attack, entropy ( H(K) = n \log_2 32 ). After disclosure, conditional entropy:

After each partial disclosure, the remaining unknown "dust" of the key—the unresolved characters—experiences a transient period where the probability distribution over possible completions is non-uniform. We define the "dust settling" as the moment when this distribution becomes statistically indistinguishable from uniform (maximum entropy) given the known constraints. Cryptography , vol

[ H(K | K_P) = |U| \log_2 32 ]