Alwen, Joel FIST Austria; Serbinenko, Vladimir
We develop new theoretical tools for proving lower-bounds on the (amortized) complexity of certain functions in models of parallel computation. We apply the tools to construct a class of functions with high amortized memory complexity in the parallel Random Oracle Model (pROM); a variant of the standard ROM allowing for batches of simultaneous queries. In particular we obtain a new, more robust, type of Memory-Hard Functions (MHF); a security primitive which has recently been gaining acceptance in practice as an effective means of countering brute-force attacks on security relevant functions. Along the way we also demonstrate an important shortcoming of previous definitions of MHFs and give a new definition addressing the problem. The tools we develop represent an adaptation of the powerful pebbling paradigm (initially introduced by Hewitt and Paterson [HP70] and Cook [Coo73]) to a simple and intuitive parallel setting. We define a simple pebbling game Gp over graphs which aims to abstract parallel computation in an intuitive way. As a conceptual contribution we define a measure of pebbling complexity for graphs called cumulative complexity (CC) and show how it overcomes a crucial shortcoming (in the parallel setting) exhibited by more traditional complexity measures used in the past. As a main technical contribution we give an explicit construction of a constant in-degree family of graphs whose CC in Gp approaches maximality to within a polylogarithmic factor for any graph of equal size (analogous to the graphs of Tarjan et. al. [PTC76, LT82] for sequential pebbling games). Finally, for a given graph G and related function fG, we derive a lower-bound on the amortized memory complexity of fG in the pROM in terms of the CC of G in the game Gp.
Proceedings of the 47th annual ACM symposium on Theory of computing
595 - 603
STOC: Symposium on the Theory of Computing
Portland, OR, United States
2015-06-14 – 2015-06-17
Alwen JF, Serbinenko V. High parallel complexity graphs and memory-hard functions. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing. ACM; 2015:595-603. doi:10.1145/2746539.2746622
Alwen, J. F., & Serbinenko, V. (2015). High parallel complexity graphs and memory-hard functions. In Proceedings of the 47th annual ACM symposium on Theory of computing (pp. 595–603). Portland, OR, United States: ACM. https://doi.org/10.1145/2746539.2746622
Alwen, Joel F, and Vladimir Serbinenko. “High Parallel Complexity Graphs and Memory-Hard Functions.” In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, 595–603. ACM, 2015. https://doi.org/10.1145/2746539.2746622.
J. F. Alwen and V. Serbinenko, “High parallel complexity graphs and memory-hard functions,” in Proceedings of the 47th annual ACM symposium on Theory of computing, Portland, OR, United States, 2015, pp. 595–603.
Alwen JF, Serbinenko V. 2015. High parallel complexity graphs and memory-hard functions. Proceedings of the 47th annual ACM symposium on Theory of computing. STOC: Symposium on the Theory of Computing 595–603.
Alwen, Joel F., and Vladimir Serbinenko. “High Parallel Complexity Graphs and Memory-Hard Functions.” Proceedings of the 47th Annual ACM Symposium on Theory of Computing, ACM, 2015, pp. 595–603, doi:10.1145/2746539.2746622.