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

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

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

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

5. Definition of a One-way Function

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

7. Do one-way functions exist? Theorem:

12. “Sister” Theorem

13. “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

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

15. 2-ary One-way functions 1. f is polynomial-time computable 2. f is NOT polynomial-time invertible 3. f is honest

16. 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

17. “S-Honesty”

18. 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

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

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

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

22. 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

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

24. 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

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

26. 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.