One-Way Functions and Cryptography Lane A. Hemaspaandra, Christopher M. Homan, and Mayur Thakur University of Rochester

1. R D Easy f Hard Aha…but researchers at the University of Rochester have proved these spiffy one-way functions exist if and only if vanilla one-way functions that we all believe probably exist do exist! Hello Prof. Way! I have developed exciting cryptographic protocols based on spiffy one-way functions. If we had these functions we could rule the world. Prof. One, I believe vanilla one-way functions probably exist, but that does not mean that I believe that your spiffy one-way functions with all these additional properties exist. Welcome to Functionstone, the one stop shop for all your one-way function needs. We exchange old, ragged, partial one-way permutations for brand-new, loaded, total one-way permutations…at absolutely no cost! One-Way Functions and Cryptography Lane A. Hemaspaandra, Christopher M. Homan, and Mayur Thakur University of Rochester Computer Science A one-way function is a function that is computationally easy to compute but hard to invert. One-way functions have been used as building blocks in the construction of cryptographic protocols like multi-party secret-key agreement and digital signatures. Researchers at the University of Rochester have been studying various aspects of one-way functions like their link to well-known complexity classes, how to construct one-way functions that have nice algebraic properties, the ambiguity of these one-way functions, one-way permutations, strong noninvertibility of one-way functions etc. One-way functions having strong properties such as associativity and commutativity are the underlying building blocks for cryptographic protocols like digital key signatures and multi-party secret key agreement. Not even a single one-way function is known to date. However, Hemaspaandra and Rothe (1999) prove that these spiffy one-way functions that have all sorts of nice algebraic properties exist if and only if vanilla one-way functions that do not have such nice properties exist. Characterizing the existence of one-way permutations. A major focus of one-way function research has been the characterization of the existence of different kinds of one-way functions with the separation of some well-known complexity classes (like NP, UP, UP Å coUP) from P. For example, it is known that total, one-to-one one-way functions exist if and only if P ¹ UP. One-way permutations (i.e., one-way functions that are total, one-to-one, and onto) had resisted such neat characterization in terms of complexity class separation. Homan and Thakur (2002) prove such a characterization for one-way permutations, and using their characterization show that (total) one-way permutations are logically as likely to exist as partial one-way permutations.

2. Because I know your trick. EVIL can be easily inverted if none of the inputs are known but cannot be inverted if one of the inputs is known. No thank you… you can keep your booby trap. Play One-Way-Function Lotto for only $5!!!!!!!!!! Invert the EVIL function and pocket a cool Million!!!!!! Early Bird Special: We will give you not only the output but also one of the two inputs. One lotto ticket please… NO early bird. Play One-Way-Function Lotto for only $5!!!!!!!!!! Invert the EVIL function and pocket a cool Million!!!!!! Early Bird Special: We will give you not only the output but also one of the two inputs. Why would you not want the early bird special? One-Way Functions and Cryptography Two-ary one-way functions have two inputs and one output. In this setting, a function is noninvertible if, given only the output, it is not feasible to compute the corresponding inputs. On the other hand, a function is strongly noninvertible if, given the output and even one of the inputs, it is not feasible to compute the corresponding remaining input. One would expect strong noninvertibility to subsume noninvertibility. But, Hemaspaandra, Pasanen, and Rothe (2000) prove the somewhat surprising and seemingly counterintuitive result. The “If I tell you anything more, you are dead” Theorem: Under standard complexity theoretic hypotheses, there exists a function that is invertible but that is strongly noninvertible. Bibliography 1. Lane A. Hemaspaandra and Joerg Rothe. Creating strong, total, commutative, associative one-way functions from any one-way function in complexity theory. Journal of Computer and System Sciences, 58(3): 648—659, 1999. 2. Lane A. Hemaspaandra, Kari Pasanen, Joerg Rothe. If P ¹ NP Then Some Strongly Noninvertible Functions are Invertible. In Proceedings of the 13th International Symposium on Foundations of Computation Theory, pages 162—171. Springer-Verlag Lecture Notes in Computer Science #2138, August 2001. 3. Christopher M. Homan and Mayur Thakur. One-Way Permutations and Self-Witnessing Languages. In Proceedings of the 2nd IFIP International Conference on Theoretical Computer Science, pages 243—254. Kluwer Academic Publishers, 2002. 4. Christopher M. Homan. Low Ambiguity in Strong, Total, Associative, One-Way Functions. Technical Report TR734, Department of Computer Science, University of Rochester, August 2000. 5. Joerg Rothe and Lane A. Hemaspaandra. Characterizing the Existence of Partial One-Way Permutations. Information Processing Letters, 82(3): 165—171, 2002. • Other Results • One-Way Functions and Degrees of Injectivity: Homan (2000) studies the levels of injectivity (or the amount of “many-to-one”-ness) of a one-way function. Informally speaking, the amount of “many-to-one”-ness of a function is the number of elements in the domain that map to an element in the range. Homan proves that associativity and totality of a function preclude constant-to-oneness. He also provides tight lower bounds on the “many-to-one”-ness of associative one-way functions. • Partial One-Way Permutations: Rothe and Hemaspaandra (2002) completely characterize the existence of partial one-way permutations in terms of a complexity class separation. • Self-Witnessing Languages and One-Way Permutations: Homan and Thakur (2002) introduce the self-witnessing language classes (nondeterministic classes such that for any language in the class the set of witnesses is the same as the the language itself) and show that the existence of one-way permutations is linked to the relation of these classes with well-known classes like P and NP.