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# x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

## Basic facts about

Basic facts about “cryptographically strong” (1) If no security against computationally unrestricted attacker: 1) using of keys of constant length l : – attacker algorithm can always try out all 2 l keys (breaks asym. encryption systems and sym. systems in known-plaintext attack). – requires an exponential number of operations (too much effort for l > 100). � the best that the designer of encryption systems can hope for. 2) complexity theory: – mainly delivers asymptotic results – mainly deals with “worst-case”-complexity � useless for security; same for “average-case”-complexity. goal: problem is supposed to be difficult almost everywhere, i.e. except for an infinitesimal fraction of cases. – security parameter l (more general than key length; practically useful) – if l � �, then probability of breaking � 0. – hope: slow fast 141

Basic facts about “cryptographically strong” (2) 3) 2 classes of complexity: en-/decryption: easy = polynomial in l breaking: hard = not polynomial in l � exponential in l Why? a) harder than exponential is impossible, see 1). b) self-contained: substituting polynomials in polynomials gives polynomials. c) reasonable models of calculation (Turing-, RAM-machine) are polynomially equivalent. For practice polynomial of high degree would suffice for runtime of attacker algorithm on RAM-machine. 4) Why assumptions on computational restrictions, e.g., factoring is difficult? Complexity theory cannot prove any useful lower limits so far. Compact, long studied assumptions! 5) What if assumption turns out to be wrong? a) Make other assumptions. b) More precise analysis, e.g., fix model of calculation exactly and then examine if polynomial is of high enough degree. 6) Goal of proof: If attacker algorithm can break encryption system, then it can also solve the problem which was assumed to be difficult. 142

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