A Characterization of Guesswork on Swiftly Tilting Curves

Given a set of strings, its guesswork is defined as the logarithm of the position at which a string appears in the ordered list of all strings from the most likely to the least likely. Guesswork is central to several applications in information theory: Average guesswork provides a lower bound on the expected computational cost of a sequential decoder to decode successfully the intended message; the complementary cumulative distribution function of guesswork gives the error probability in list decoding; the logarithm of guesswork is the number of bits needed in optimal lossless one-to-one source coding; and guesswork is the number of trials required of an adversary to breach a password protected system in a brute-force attack. In this paper, we consider memoryless string-sources that generate i.i.d. strings from a certain categorical distribution, and characterize their corresponding guesswork. Our main tool is the tilt operation on a memoryless string-source. We show that the tilt operation on a memoryless string-source parametrizes an exponential family of memoryless string-sources, which we refer to as the tilted family of the string-source. We provide an operational meaning to the tilted families by proving that two memoryless string-sources result in the same guesswork on all strings of all lengths if and only if their respective categorical distributions belong to the same tilted family. Establishing some general properties of the tilt operation, we generalize the notions of weakly typical set and asymptotic equipartition property to tilted weakly typical sets of different orders. We use this new definition to This work was presented in part in the 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton 2015) [1]. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under awards FA 9550-14-1-043 and FA 9550-13-1-0076; the work of K. Duffy and M. Médard was also supported in part by Netapp through a Faculty Fellowship. A. Beirami is with the Department of Electrical and Computer Engineering, Duke University, USA, and also with the Research Laboratory of Electronics, Massachusetts Institute of Technology, USA (email: [email protected], [email protected]). R. Calderbank is with the Department of Electrical and Computer Engineering, Duke University, USA (email: [email protected]). M. Christiansen and K. Duffy are with the Hamilton Institute, National University of Ireland Maynooth, Ireland (email: {mark.christiansen, ken.duffy}@nuim.ie). M. Médard is with the Research Laboratory of Electronics, Massachusetts Institute of Technology, USA (email: [email protected]). December 13, 2017 DRAFT