def kasiski$\_$diffs$($Y$,$ case$=$"upper"$)$:

Y $=$ only$\_$letters$($Y$,$ case$=$case$)$

ctr $=$ count$\_$substrings$($Y$,3)$

tri$\_$reps $=[$k for k$,$v in ctr$.$items$()$ if v $>1]$

diffs $=[]$

for tri in tri$\_$reps:

starts $=[$m$.$start$()$ for m in re$.$finditer$($f$\text{'}(?=\{$tri$\})\text{'},$ Y$)]$

diffs.extend$([$abs$($x$-$y$)$ for x$,$y in combinations$($starts$,2)])$

return np$.$array$($sorted$($diffs$))$

def ind$\_$co$($X$)$:

X $=$ only$\_$letters$($X$,$ case$=$"upper"$)$

ctr $=$ count$\_$substrings$($X$,1)$

n $=$ sum$($ctr$.$values$())$

return $(1/($n$*($n$-1)))*$sum$($f$*($f$-1)$ for f in ctr$.$values$())$

$*$ In lecture on D$20240701,$ we used $($an estimate for$)$ the Index of Coincidence to distinguish random characters from shift ciphertext. One strength of the Vigen$$re cipher is that it looks very much like random text from this perspective. Compute $`$ind$\_$co$($Y$)`$ for $`$Y$`$ as above, and compare it with $`1/26`,$ which is the value expected for randomly chosen English characters. $(**$Comment$**.$ We've changed $`$ind$\_$co$`$ since Monday. It used to take a dictionary of letter frequencies as input, like $`$mut$\_$ind$\_$co$`.$ Now it takes a string as input.$)$

$*$ On the other hand, the above was encrypted using a Vigen$$re cipher with a key length of $`7`.$ Compute the following and compare the values with the expected Index of Coincidence value for true English of approximately $`0\mathrm{.}068`.($All of the values will be close to$,$ but slightly less than, this $`0\mathrm{.}068`$ value. I$\text{'}$m not sure if there is a theoretical reason for why they are less than $`0\mathrm{.}068`.$ The important thing is they're all significantly closer to the true English value than to the random characters value.$)$

$```$

Z $=$ only$\_$letters$($Y$)$ # we remove spaces from $`$Y$`$

for i in range$(7)$:

print$($ind$\_$co$($Z$[$i::$7]))$

$```$

$*$ Write a function to perform Vigen$$re encryption using the following template. $($If you want to make the code more elegant, replace $`$for i in range$($key$\_$length$)$:$`$ in our template with $`$for i$,$ k in enumerate$($key$)$:$`.)$

$>$ Input: a string $`$X$`$ and a list of shift amounts $`$key$`$

$>$ Output: a string $`$Y$`$

$>$

$>$ Example: If $`$X$`$ is $`"$The rain in Spain stays mainly in the plain"$`$ and $`$key$`$ is $`[5,11,0,12,8,13,6,14]`,$ then the output should be the ciphertext written on page $217$ of our textbook, beginning $`"$YSEDIVTWSDPM$..."`.$

$```$

def vigenere$($X$,$ key, case$=$"upper"$)$:

X $=$ only$\_$letters$($X$,$ case$=$case$)$

key$\_$length $=$ len$($key$)$

string$\_$list $=[]$

for i in range$($key$\_$length$)$:

s $=$ # Get the characters in $`$X$`$ at integer positions $???$ modulo $???$

shifted$\_$s $=$ # Shift the characters in s$.$ Use the imported $`$shift$\_$string$`$ function.

string$\_$list.append$($shifted$\_$s$)$

output $=$ # Use the imported $`$weave$`$ function

return output

$```*$ Test your function by evaluating the following and checking that the result matches what is on page $217$ of the textbook $($aside from ours being in upper$-$case letters$).$

$```$

X $=$ "The rain in Spain stays mainly in the plain"

key $=[5,11,0,12,8,13,6,14]$

vigenere$($X$,$ key$)$

$```$