## CryptoDB

### Kannan Srinathan

#### Publications

**Year**

**Venue**

**Title**

2010

EPRINT

2008

EPRINT

Unconditionally Reliable and Secure Message Transmission in Undirected Synchronous Networks: Possibility, Feasibility and Optimality
Abstract

We study the interplay of network connectivity and the issues related to the possibility, feasibility and optimality for {\it unconditionally reliable message transmission} (URMT) and {\it unconditionally secure message transmission} (USMT) in an undirected {\it synchronous} network, under the influence of an adaptive {\it mixed} adversary ${\mathcal A}_{(t_b,t_o,t_f,t_p)}$, who has
{\it unbounded computing power} and can corrupt $t_b$, $t_o$, $t_f$
and $t_p$ nodes in the network in Byzantine, {\it omission}, {\it
fail-stop} and passive fashion respectively. In URMT problem, a sender {\bf S} and a receiver {\bf R} are part of a distributed network, where {\bf S} and {\bf R} are connected by intermediate nodes, of which at most $t_b, t_o, t_f$ and $t_p$ nodes can be under the control of ${\mathcal A}_{(t_b,t_o,t_f,t_p)}$. {\bf S} wants to send a message $m$ which is a sequence of $\ell$ field elements from a finite field ${\mathbb F}$ to {\bf R}. The challenge is to design a protocol,
such that after interacting in phases\footnote{A phase is a send from {\bf S} to {\bf R} or viceversa.} as per the protocol, {\bf R} should output $m' = m$ with probability at least $1 - \delta$, where
$0 < \delta < \frac{1}{2}$. Moreover, this should happen, irrespective of the adversary strategy of ${\mathcal A}_{(t_b,t_o,t_f,t_p)}$. The USMT problem has an additional requirement that ${\mathcal A}_{(t_b,t_o,t_f,t_p)}$ should not know anything about $m$ in {\it information theoretic} sense.
In this paper, we answer the following in context of URMT and USMT: (a) {\sc Possibility:} when is a protocol possible in a given network? (b) {\sc Feasibility:} Once the existence of a protocol is ensured then does there exists a polynomial time protocol on the given network? (c) {\sc Optimality: } Given a message of specific length, what is the minimum communication complexity (lower bound) needed by any protocol to transmit the message and how to design a protocol whose total communication complexity matches the lower bound on the communication complexity? Finally we also show that {\it allowing a negligible error probability significantly helps in the possibility, feasibility and optimality of both reliable and secure message transmission protocols.} To design our protocols, we propose several new techniques which are of independent interest.

2008

EPRINT

Authenticated Byzantine Generals Strike Again
Abstract

Pease {\em et al.}\/ introduced the problem of \emph{Authenticated Byzantine General} (ABG) where players could use digital signatures (or similar tools) to thwart the challenge posed by Byzantine faults in distributed protocols for agreement. Subsequently it is well known that ABG among $n$ players tolerating up to $t$ faults is (efficiently) possible if and only if $t < n$ (which is a huge improvement over the $n > 3t$ condition in the absence of authentication for the same functionality). However, in this paper we argue that the extant result of $n>t$ does not \emph{truly} simulate a broadcast channel as intended. Thus, we re-initiate the study of ABG. We show that in a completely connected synchronous network of $n$ players where up to any $t$ players are controlled by an adversary, ABG is possible if and only if $n > 2t-1$. The result is pleasantly surprising since it deviates from the standard template of ``$n > \beta t$" for some integer $\beta$, particularly $\beta = 1,2$ or $3$. The result uses new proof/design techniques which may be of independent interest.

2008

EPRINT

Unconditionally Reliable Message Transmission in Directed Hypergraphs
Abstract

We study the problem of unconditionally reliable message transmission (URMT), where two non-faulty players, the sender S and the receiver R are part of a synchronous network modeled as a directed hypergraph, a part of which may be under the influence of an adversary having unbounded computing power. S intends to transmit a message $m$ to R, such that R should correctly obtain S's message with probability at least $(1-\delta)$ for arbitrarily small $\delta > 0$. However, unlike most of the literature on this problem, we assume the adversary modeling the faults is threshold mixed, and can corrupt different set of nodes in Byzantine, passive and fail-stop fashion simultaneously. The main contribution of this work is the complete characterization of URMT in directed hypergraph tolerating such an adversary. Working out a direct characterization of URMT over directed hypergraphs tolerating threshold mixed adversary is highly un-intuitive. So we first propose a novel technique, which takes as input a directed hypergraph and a threshold mixed adversary on that hypergraph and outputs a corresponding digraph, along with a non-threshold mixed adversary, such that URMT over the hypergraph tolerating the threshold mixed adversary is possible iff a special type of URMT is possible over the obtained digraph, tolerating the corresponding non-threshold mixed adversary}. Thus characterization of URMT over directed hypergraph tolerating threshold mixed adversary reduces to characterizing special type of a URMT over arbitrary digraph tolerating non-threshold mixed adversary. We then characterize URMT in arbitrary digraphs tolerating non-threshold mixed adversary
and modify it to obtain the characterization for special type of URMT over digraphs tolerating non-threshold mixed adversary. This completes the characterization of URMT over the original hypergraph.
Surprisingly, our results indicate that even passive corruption, in
collusion with active faults, substantially affects the reliability of URMT protocols! This is interesting because it is a general belief that passive corruption (eavesdropping) does not affect reliable communication.

2008

EPRINT

Perfectly Reliable and Secure Communication Tolerating Static and Mobile Mixed Adversary
Abstract

In the problem of perfectly reliable message transmission} (PRMT), a sender {\bf S} and a receiver {\bf R} are connected by $n$ bidirectional synchronous channels. A mixed adversary
${\mathcal A}_{(t_b,t_f,t_p)}$ with {\it infinite computing power} controls $t_b, t_f$ and $t_p$ channels in Byzantine, fail-stop and passive fashion respectively. Inspite of the presence of ${\mathcal A}_{(t_b,t_f,t_p)}$, {\bf S} wants to reliably send a message $m$ to {\bf R}, using some protocol, without sharing any key with {\bf R} beforehand. After interacting in phases\footnote{A phase is a send from {\bf S} to {\bf R} or vice-versa} as per the protocol, {\bf R} should output $m' = m$, without any error. In the problem of {\it perfectly secure message transmission} (PSMT), there is an
additional constraint that ${\mathcal A}_{(t_b,t_f,t_p)}$ should not know {\it any} information about $m$ in {\it information theoretic} sense. The adversary can be either static\footnote{A static adversary corrupts the same set of channels in each phase of the protocol. The choice of the channel to corrupt is decided before the beginning of the protocol.} or mobile.\footnote{A mobile adversary can corrupt different set of channels in different phases of the protocol.}
The {\it connectivity requirement}, {\it phase complexity} and {\it communication complexity} are three important parameters of any interactive PRMT/PSMT protocol and are well studied in the literature when the channels are controlled by a static/mobile Byzantine adversary. However, when the channels are controlled by mixed adversary ${\mathcal A}_{(t_b,t_f,t_p)}$ , we encounter several surprising consequences. In this paper, we study the problem of PRMT and PSMT tolerating "static/mobile mixed adversary". We prove that even though the connectivity requirement for PRMT is same against both static and mobile mixed adversary, the lower bound on communication complexity for PRMT tolerating mobile mixed adversary
is more than its static mixed counterpart. This is interesting because against only "Byzantine adversary", the connectivity requirement and the lower bound on the communication complexity of PRMT protocols are same for both static and mobile case. Thus our result shows that
for PRMT, mobile mixed adversary is more powerful than its static counterpart. As our second contribution, we design a four phase communication optimal PSMT protocol tolerating "static mixed adversary". Comparing this with the existing three phase communication
optimal PSMT protocol against "static Byzantine adversary", we find that additional one phase is enough to design communication optimal protocol against static mixed adversary. Finally, we show that the connectivity requirement and lower bound on communication complexity
of any PSMT protocol is same against both static and mobile mixed adversary, thus proving that mobility of the adversary has no effect in PSMT. To show that our bound is tight, we also present a worst case nine phase communication optimal PSMT protocol tolerating mobile mixed adversary which is first of it's kind. This also shows that the mobility of the adversary does not hinder to design constant phase communication optimal PSMT protocol. In our protocols, we have used new techniques which can be effectively used against both static and mobile mixed adversary and are of independent interest.

2008

EPRINT

On Communication Complexity of Perfectly Reliable and Secure Communication in Directed Networks
Abstract

In this paper, we re-visit the problem of {\it perfectly reliable message transmission} (PRMT) and {\it perfectly secure message transmission}(PSMT) in a {\it directed network} under the presence of a threshold adaptive Byzantine adversary, having {\it unbounded computing power}. Desmedt et.al have given the necessary and sufficient condition for the existence of PSMT protocols in directed networks. In this paper, we first show that the necessary and sufficient condition (characterization) given by Desmedt et.al does not hold for two phase\footnote{A phase is a send from sender to receiver or vice-versa.} PSMT protocols. Hence we provide a different necessary and sufficient condition for two phase PSMT in directed networks. We also derive the lower bound on communication complexity of two phase PSMT and show that our lower bound is {\it asymptotically tight} by designing a two phase PSMT protocol whose
communication complexity satisfies the lower bound. Though the characterization for three or more phase PSMT is resolved by the result of Desmedt et. al, the lower bound on communication
complexity for the same has not been investigated yet. Here we derive the lower bound on the communication complexity of three or more phase PSMT in directed networks and show that our lower bound is {\it asymptotically tight} by designing {\it communication optimal} PSMT protocols. Finally, we characterize the class of directed networks over which communication optimal PRMT or PRMT with constant factor overhead is possible. By communication optimal PRMT
or PRMT with constant factor overhead, we mean that the PRMT protocol is able to send $\ell$ field elements by communicating $O(\ell)$ field elements from a finite field $\mathbb F$.
To design our protocols, we use several techniques, which are of independent interest.

2008

EPRINT

Unconditionally Secure Message Transmission in Arbitrary Directed Synchronous Networks Tolerating Generalized Mixed Adversary
Abstract

In this paper, we re-visit the problem of {\it unconditionally secure message transmission} (USMT) from a sender {\bf S} to a receiver {\bf R}, who are part of a distributed synchronous network, modeled as an {\it arbitrary} directed graph. Some of the intermediate nodes between {\bf S} and {\bf R} can be under the control of the adversary having {\it unbounded} computing power. Desmedt and Wang \cite{Desmedt} have given the characterization of USMT in directed networks. However, in their model, the underlying network is abstracted as directed node disjoint paths (also called as wires/channels) between {\bf S} and {\bf R}, where the intermediate nodes are oblivious, message passing nodes and perform no other computation. In this work, we first show that the characterization of USMT given by Desmedt et.al \cite{Desmedt} does not hold good for {\it arbitrary} directed networks, where the intermediate nodes perform some computation, beside acting as message forwarding nodes.
We then give the {\it true} characterization of USMT in arbitrary directed networks. As far our knowledge
is concerned, this is the first ever {\it true} characterization of USMT in arbitrary directed networks.

2008

EPRINT

Topology Knowledge Versus Fault Tolerance: The Case of Probabilistic Communication Or: How Far Must You See to Hear Reliably?
Abstract

We consider the problem of probabilistic reliable communication (PRC) over synchronous networks modeled as directed graphs in the presence of a Byzantine adversary when players' knowledge of the network topology is not complete. We show that possibility of PRC is extremely sensitive to the changes in players' knowledge of the topology. This is in complete contrast with earlier known results on the possibility of perfectly reliable communication over undirected graphs where the case of each player knowing only its neighbours gives the same result as the case where players have complete knowledge of the network. Specifically, in either case, $2t+1$-vertex connectivity is necessary and sufficient, where $t$ is the number of nodes that can be corrupted by the adversary [1][2].
We use a novel model for quantifying players' knowledge of network topology, denoted by {$\mathcal K$}. Given a directed graph $G$, influenced by a Byzantine adversary that can corrupt up to any $t$ players, we give a necessary and sufficient condition for possibility of PRC over $G$ for any arbitrary topology knowledge {$\mathcal K$}.
As a corollary, we answer the question: {\em Up to what neighbourhood level $\ell$ must the players know for the possibility of PRC} where, a player is said to know up to neighbourhood level: $1$ - if it knows only its neighbours; $2$ - if it knows its neighbours and its neighbours' neighbours and so on.
[1] Subramanian, L., Katz, R. H., Roth, V., Shenker, S., and Stoica, I. 2005. Reliable broadcast in unknown fixed-identity networks. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Principles of Distributed Computing (Las Vegas, NV, USA, July 17 - 20, 2005). PODC '05. ACM, New York, NY, 342-351.
[2] DOLEV, D. The byzantine generals strike again. In
Journal of Algorithms (1982), pp. 1430.

#### Coauthors

- Shashank Agrawal (1)
- AshwinKumar B.V (1)
- Piyush Bansal (2)
- Ashish Choudhary (5)
- Prasant Gopal (2)
- Anuj Gupta (2)
- Abhinav Mehta (1)
- Arpita Patra (5)
- C. Pandu Rangan (5)
- Pranav K. Vasishta (1)