Section 1
The Rivest-Shamir-Adleman Cipher System
Let $p=5$ and $q=7$ so that $m=35$, and let $e=11$. Find $d=$ $e^{-1} \bmod \varphi(m)$. Then let $x=22$ and compute $y=x^e \bmod 35$ and $z=$ $y^d \bmod m$.
Show that $a^2 \bmod 24=1$ for each $a \perp 24$. Conclude that $a^{-1} \bmod 24=$ $a \bmod 24$ for each $a \perp 24$.
Let $p=29$ and $q=31$ so that $m=899$, and let $e=101$. Find$$d=e^{-1} \bmod \varphi(m)$$Then let $x=555$ and compute $y=x^e \bmod m$ and $z=y^d \bmod m$.
Assume$$m=25972641171898723$$is a product of two primes $p$ and $q$ and that$$\varphi(m)=25972640809676568$$Use these two equations to find the primes $p$ and $q$.
Illustrate the RSA algorithm using $m$ from Problem 4 with$$\begin{aligned}& e=997 \\& x=99999999999999\end{aligned}$$
Assume that $y=x^c$ mod $m$ from problem 5 is transmitted over a noisy channel and received as $w=y+10^{12}$. How similar are $y$ and $w$ ? What message is decrypted? How similar are $x$ and the decrypted message?
Assume $m=21936520921056942428185744321881874204790829920$ 570235226904516467385564406736567597367535979699930859170 667289061009756151158068196185554149 is a product of two primes $p$ and $q$ and that $\varphi(m)=21936520921056942428185744321881874204$ 790829920570235226904516467385548925092800907044879409345 000125494759694716131857830775913843528947136 . Use these two equations to find the primes $p$ and $q$.
Illustrate the RSA algorithm using $m$ from problem 7 with$$\begin{aligned}& e=997 \\& x=99999999999999\end{aligned}$$
Let $m=p q$ where $p$ and $q$ are distinct primes. Show that if $x$ is chosen at random in the range $1 \leq x \leq m$, then the probability that $\operatorname{gcd}(x, m) \neq 1$ is equal to $1 / p+1 / q-1 / m$.
Verify the formula in problem 9 for $m=6$ and $m=15$. Show that if $x$ is chosen at random in the range $1 \leq x<m$, then the probability that $\operatorname{god}(x, m) \neq 1$ is less than $1 / p+1 / q$.