If $x$ terminates with $d$ digits after the decimal place, we can write it as $10^d x / 10^d$. For (a), $d=4$ and so we can write it as $31415 / 10000$. This is not in lowest terms, but you need not reduce it.
We call a decimal in which a pattern repeats a repeating decimal. Thus (b) and (c) are repeating decimals. The period is the number of digits in the repeating pattern. In (b) the period is two because of the pattern 30. In (c) the period is three because of the pattern 215 . If $x$ is a repeating decimal with period $k$, then $10^k x-x$ will be a
terminating decimal, which can be written as a rational number $a / b$ by the previous discussion. Thus $x=a /\left(b\left(10^k-1\right)\right)$. We now apply this.
(b) Since $x=0.303030 \ldots, k=2,10^2 x-x=30$, and so $x=30 / 99$.
(c) $x=6.3215215215 \ldots, k=3$,
10^3 x-x=6321.5215215 \ldots-6.3215215 \ldots=6315.2=63152 / 10,
and so $x=63152 / 9990$.
Question b) Do you believe that the GCHQ authors” fifth principle-that “any exceptional access solution should not fundamentally change the trust relationship between a service provider and its users” – is met by their proposed exceptional access mechanism? Why or why not? (4-6 sentences)
(b) Argument against:
Even if, as the GCHQ authors claim, “you don’t even have to touch the encryption” in their proposed system, ghost access nonetheless constitutes a significant change in the trust relationship between a service provider and its users. That is because, as discussed in lecture 3 , encryption is only half of the equation: users need to trust that their messages are not just encrypted, but also that their messages are only sent to the intended recipients. Additionally, this proposal requires the ability to suppress notifications that users have elected to receive, thus preventing the user from trusting the integrity of
the notification system.
Arguments for are difficult when using “trust” in a theoretical cybersecurity sense, but possible if viewing it through the same lens as the GCHQ cryptographers: if the risk of exploitation is minimal, and you believe that the government is not interested in your communications, then the trust relationship between user and service provider is not fundamentally changed. Arguments along these lines are valid, and are interesting in that they highlight a key point of contention in the exceptional access debate.
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