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One-Way Functions David Lagakos Yutao Zhong April 2, 2001

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What are one-way functions? Do they exist? One-to-one one-way functions “Spiffy” One-Way functions An application to cryptography Topics

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Honesty 0 n if |x|=2 n for some n 1 otherwise

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Polynomial-time Invertibility f 3 (x) = ceiling(log(log(log(max(|x|,4)))))

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Definition of a One-way Function

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A One-way Function ‘Candidate’ (Note that primality can be verified quickly.)

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Do one-way functions exist? Theorem:

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“Sister” Theorem

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“Spiffy” one-way functions u Motivation: cryptography u Properties v 2-ary one-way v Strongly noninvertible v Total v Commutative v Associative u Claim: One-way function exists iff “spiffy” one-way function exists

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Definitions for 2-ary functions u f is honest if f is (polynomial-time) invertible if

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2-ary One-way functions 1. f is polynomial-time computable 2. f is NOT polynomial-time invertible 3. f is honest

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Strong Noninvertibility u is strongly (polynomial- time) noninvertible if v it is s-honest v given the output and even one of the inputs, the other input cannot in general be computed in polynomial time

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“S-Honesty”

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Strong Noninvertibility u is strongly (polynomial- time) noninvertible if v it is s-honest v given the output and even one of the inputs, the other input cannot in general be computed in polynomial time

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Associativity & Commutativity u Def: total function is associative if u Def: total function is commutative if

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Theorem u One-way functions exist if and only if strongly noninvertible, total, commutative, associative, 2-ary one-way functions exist.

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Proposition u The following are equivalent: 1. One-way functions exist 2. 2-ary one-way functions exist 3.

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strongly non-invertible, commutative, associative, 2-ary one-way function exists Proof: each computation path of N(x) has exactly p(|x|) bits ( p(n)>n ) W(x): the set of all witness for x

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v strongly noninvertible v one-way v commutative v associative f (u,v) = otherwise Claim: f is the function we need

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Eve An Application to Cryptography Alice Bob y, f(x,y) x,y f(y,z) f(x,f(y,z)) f(f(x,y),z) z

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k =010011011 Alice Bob Using the Secret Key m’ m =110101010 k =010011011 m ’ =100110001 k =010011011 m =110101010

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Conclusions u One-way functions are easy to compute and hard to invert. u Proving that one-way functions exist is the same as proving that P and NP are different. u Special types of one-way functions, like “Spiffy” one-way functions, can have quite useful applications in cryptography.

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