Hiding information and signatures in trapdoor knapsacks

Compressibiity of individuai sequences by the ciam of gener-aihd finite-atate information-losales encoders ia investigated These en-codersrpnoperateinavariabie-ratemodeasweUasaflxedrateone,nndthey aiiow for any fhite-atate acheme of variabie-iength-to-variable-ien@coding. For every individuai hfiite aeqence x a quantity p (x) ia defined,calledthecompressibilityofx,whirhisshowntobetheasymptotieatlyattainable lower bound on the compression ratio tbat cao be achieved for xby any finite-state encoder. ‘flds is demonstrated by means of a amatruc-tivecodtngtbeoremanditsconversethat,apartfnnntheirafymptoticsignificance, also provide useful performance criteria for finite and practi-cai data-compression taaka. The proposed concept of compressibility ia aiaoshown to play a role analogous to that of entropy in ciaasicai informatfontheory where onedeaia with probabilistic ensembles of aequencea ratk Manuscript received June 10, 1977; revised February 20, 1978.J. Ziv is with Bell Laboratories, Murray Hill, NJ 07974, on leave fromthe Department of Electrical Engineering, Techmon-Israel Institute ofTechnology, Halfa, Israel.A. Lempel is with Sperry Research Center, Sudbury, MA 01776, onleave from the Department of Electrical Engineer@, Technion-IsraelInstitute of Technology, Haifa, Israel.tium with individuai sequences. Widie the delinition of p (x) aiiows adifferent machine for each different sequence to be compresse4 theconstructive coding theorem ieada to a universal algorithm that is aaymiktoticaiiy optfmai for au sequencea.