Witness Complexity of Short Descriptions: A Cryptographic Perspective

In cryptographic practice, where protocols impose strict time bounds, implementations demand predictable resource usage, and real-world systems require immediate verification for security and usability, a short key or certificate is useful only if it can be expanded or verified within a bounded time; otherwise a compact representation that requires superpolynomial work to expand offers no operational guarantee within a bounded-time protocol. This paper formalises that gap by introducing \emph{witness complexity} $\gam(x)$, the minimum running time over near-shortest descriptions of a string on a universal Turing machine. $\gam$ differs from Shannon entropy and Kolmogorov complexity $\KC$: low $\KC$ can coexist with high $\gam$. We prove invariance up to polynomial factors; a conditional separation (assuming $\PneqNP$). An unconditional lower bound from incomputability of $\KC$; a biconditional characterisation of $\PeqNP$ via the class-relative variant $\gP$; and polynomial-time tractability for structured $\classNP$ families. Part II develops companion measures and shows an unconditional gap between grammar size and derivation cost, positioning $\gam$ as a metric for the usability of keys and certificates.