Analyzing Multi-key Security Degradation

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Static-Memory-Hard Functions, and Modeling the Cost of Space vs. Time

A series of recent research starting with (Alwen and Serbinenko, STOC 2015) has deepened our understanding of the notion of memory-hardness in cryptography — a useful property of hash functions for deterring large-scale password-cracking attacks — and has shown memory-hardness to have intricate connections with the theory of graph pebbling. Definitions of memory-hardness are not yet unified in the somewhat nascent field of memory-hardness, however, and the guarantees proven to date are with respect to a range of proposed definitions. In this paper, we observe two significant and practical considerations that are not analyzed by existing models of memory-hardness, and propose new models to capture them, accompanied by constructions based on new hard-to-pebble graphs. Our contribution is two-fold, as follows. First, existing measures of memory-hardness only account for dynamic memory usage (i.e., memory read/written at runtime), and do not consider static memory usage (e.g., memory on disk). Among other things, this means that memory requirements considered by prior models are inherently upper-bounded by a hash function’s runtime; in contrast, counting static memory would potentially allow quantification of much larger memory requirements, decoupled from runtime. We propose a new definition of static-memory-hard function (SHF) which takes static memory into account: we model static memory usage by oracle access to a large preprocessed string, which may be considered part of the hash function description. Static memory requirements are complementary to dynamic memory requirements: neither can replace the other, and to deter large-scale password-cracking attacks, a hash function will benefit from being both dynamic memory-hard and static-memory-hard. We give two SHF constructions based on pebbling. To prove static-memory-hardness, we define a new pebble game (“black-magic pebble game”), and new graph constructions with optimal complexity under our proposed measure. Moreover, we provide a prototype implementation of our first SHF construction (which is based on pebbling of a simple “cylinder” graph), providing an initial demonstration of practical feasibility for a limited range of parameter settings. Secondly, existing memory-hardness models implicitly assume that the cost of space and time are more or less on par: they consider only linear ratios between the costs of time and space. We propose a new model to capture nonlinear time-space trade-offs: e.g., how is the adversary impacted when space is quadratically more expensive than time? We prove that nonlinear tradeoffs can in fact cause adversaries to employ different strategies from linear tradeoffs. Finally, as an additional contribution of independent interest, we present an asymptotically tight graph construction that achieves the best possible space complexity up to log log n-factors for an existing memory-hardness measure called cumulative complexity in the sequential pebbling model

Studies on Verifiable Secret Sharing, Byzantine Agreement and Multiparty Computation

This dissertation deals with three most important as well as fundamental problems in secure distributed computing, namely Verifiable Secret Sharing (VSS), Byzantine Agreement (BA) and Multiparty Computation (MPC). VSS is a two phase protocol (Sharing and Reconstruction) carried out among $n$ parties in the presence of a centralized adversary who can corrupt up to $t$ parties. Informally, the goal of the VSS protocol is to share a secret $s$, among the $n$ parties during the sharing phase in a way that would later allow for a unique reconstruction of this secret in the reconstruction phase, while preserving the secrecy of $s$ until the reconstruction phase. VSS is used as a key tool in MPC, BA and many other secure distributed computing problems. It can take many different forms, depending on the underlying network (synchronous or asynchronous), the nature (passive or active) and computing power (bounded or unbounded) of the adversary, type of security (cryptographic or information theoretic) etc. We study VSS in information theoretic setting over both synchronous as well as asynchronous network, considering an active unbounded powerful adversary. Our main contributions for VSS are: \begin{itemize} \item In synchronous network, we carry out in-depth investigation on the round complexity of VSS by allowing a probability of error in computation and show that existing lower bounds for the round complexity of error-free VSS can be circumvented by introducing a negligible probability of error. \item We study the communication and round efficiency of VSS in synchronous network and present a robust VSS protocol that is simultaneously communication efficient and round efficient. In addition, our protocol is the best known communication and round efficient protocol in the literature. \item In asynchronous network, we study the communication complexity of VSS and propose a number of VSS protocols. Our protocols are highly communication efficient and show significant improvement over the existing protocols in terms of communication complexity. \end{itemize} The next problem that we deal with is Byzantine Agreement (BA). BA is considered as one of the most fundamental primitives for fault tolerant distributed computing and cryptographic protocols. BA among a set of $n$ parties, each having a private input value, allows them to reach agreement on a common value even if some of the malicious parties (at most $t$) try to prevent agreement among the parties. Similar to the case of VSS, several models for BA have been proposed during the last three decades, considering various aspects like the underlying network, the nature and computing power of adversary, type of security. One of these models is BA over asynchronous network which is considered to be more realistic network than synchronous in many occasions. Though important, research in BA in asynchronous network has received much less attention in comparison to the BA protocols in synchronous network. Even the existing protocols for asynchronous BA involve high communication complexity and in general are very inefficient in comparison to their synchronous counterparts. We focus on BA in information theoretic setting over asynchronous network tolerating an active adversary having unbounded computing power and mainly work towards the communication efficiency of the problem. Our contributions for BA are as follows: \begin{itemize} \item We propose communication efficient asynchronous BA protocols that show huge improvement over the existing protocols in the same setting. Our protocols for asynchronous BA use our VSS protocols in asynchronous network as their vital building blocks. \item We also construct a communication optimal asynchronous BA protocol for sufficiently long message size. Precisely, our asynchronous BA communicates O(\ell n) bits for \ell bit message, for sufficiently large \ell. \end{itemize} The studies on VSS and BA naturally lead one towards MPC problems. The MPC can model almost any known cryptographic application and uses VSS as well as BA as building blocks. MPC enables a set of $n$ mutually distrusting parties to compute some function of their private inputs, such that the privacy of the inputs of the honest parties is guaranteed (except for what can be derived from the function output) even in the presence of an adversary corrupting up to $t$ of the parties and making them misbehave arbitrarily. Much like VSS and BA, MPC can also be studied in various models. Here, we attempt to solve MPC in information theoretic setting over synchronous as well as asynchronous network, tolerating an active unbounded powerful adversary. As for MPC, our main contributions are: \begin{itemize} \item Using one of our synchronous VSS protocol, we design a synchronous MPC that minimizes the communication and round complexity simultaneously, where existing MPC protocols try to minimize one complexity measure at a time (i.e the existing protocols minimize either communication complexity or round complexity). \item We study the communication complexity of asynchronous MPC protocols and design a number of protocols for the same that show significant gain in communication complexity in comparison to the existing asynchronous MPC protocols. \item We also study a specific instance of MPC problem called Multiparty Set Intersection (MPSI) and provide protocols for the same. \end{itemize} In brief, our work in this thesis has made significant advancement in the state-of-the-art research on VSS, BA and MPC by presenting several inherent lower bounds and efficient/optimal solutions for the problems in terms of their key parameters such as communication complexity and time/round complexity. Thus our work has made a significant contribution to the field of secure distributed computing by carrying out a foundation research on the three most important problems of this field