🤖 AI Summary
In cryptography, short keys or certificates that cannot be generated or verified within bounded time fail to meet protocols’ demands for resource predictability and immediate verifiability. This work introduces the notion of witness complexity, denoted γ(x), defined as the minimal runtime required on a universal Turing machine to produce a near-shortest description of a string x, thereby capturing the practical feasibility of succinct representations. Distinct from both Shannon entropy and Kolmogorov complexity, γ(x) is intrinsically linked to the P≠NP conjecture and polynomial-time tractability. We establish that γ(x) is invariant up to polynomial factors, provide conditional separation results, derive a biconditional characterization of P=NP, and demonstrate polynomial-time solvability for structured families of NP problems.
📝 Abstract
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.