Chapter 14, Fault Tolerance in Synchronous Systems Video Solutions, Introduction to Distributed Algorithms

03:26

Problem 1

How many messages are exchanged in the $\operatorname{Broadcast}(N, t)$ protocol (Algorithm 14.2 and 14.3)?

James Chok

Numerade Educator

01:12

What is the highest number of messages sent by correct processes in Algorithm 14.4 in executions that decide on 0 ? Answer both for the case where the general is correct and the case where the general is faulty.

Hast Aggarwal

Numerade Educator

05:47

Problem 3

How many messages are exchanged by the broadcast algorithm of Lamport, Shostak, and Pease, described in Subsection 14.2.1?

Chris Trentman

Numerade Educator

Problem 4

Give an execution of the protocol of Dolev and Strong, described in Subsection 14.2.1, where correct processes $p$ and $q$ end with $W_p \neq W_q$.

Check back soon!

Problem 5

Show that order-preserving renaming in the range 1 through $N$ can be solved when interactive consistency is achieved.

Check back soon!

Problem 6

Assume that $p$ signs two messages $M_1$ and $M_2$ with the ElGamal signature scheme (Subsection 14.2.3), using the same value of a. Show how to find $P$ 's secret key from the two signed messages.

Check back soon!

Problem 7

Let $n$ be the product of two large prime numbers, and assume that a black box is given that computes square roots. That is, given a quadratic residue $y$, the box outputs an $x$ with $x^2=y$ (equation is modulo $n$ ). Show how the box can be used to factor $n$.

Check back soon!

05:31

Problem 8

Show that if clocks $C_p$ and $C_q$ have $\rho$-bounded drift and are $\delta$-synchronized at real time $t$, then they are $\delta(1+\rho)$-synchronized at clock time $T=C_{\mathrm{p}}(t)$.

Khoobchandra Agrawal

Numerade Educator

Problem 9

The probabilistic clock-synchronization protocol is efficient if many messages have a delay close to $\delta_{\min }$ (that is, $F\left(\delta_{\min }+\epsilon\right)$ moves away from 0 quickly even for small $\epsilon$ ).
Give a "dual" protocol that works efficiently if many messages have a delay close to $\delta_{\max }$ (that is, if $F\left(\delta_{\max }-\epsilon\right)$ stays sufficiently far from 1 even for small $\epsilon$ ). Show that the expected message complexity is $\left[1-F\left(\delta_{\max }-\epsilon\right)\right]^{-2}$ and give the expected running time.

Check back soon!

01:09