$1.$ The letter N was found to be the most frequent letter in a large ciphertext produced

using a shift cipher over the alphabet ABC$$ Z$.$ Decrypt the following portion of

the ciphertext:

CQRBRBJFNJTLRYQNA

$2.$The ciphertext string

ASISENGEHFGNONFVWTGNQWCVSE

was produced using a shift cipher over the $27-$letter alphabet

A $=0,$ B $=1,...,$ Z $=25,=26,$

where the final character indicates a space.

By conducting a frequency analysis of a larger sample of ciphertexts, it has been

found that the plaintext symbol E corresponds to the ciphertext symbol S$.$ Determine

the first word of the plaintext string corresponding to the ciphertext string given

above.

$3.$ Let $($P$,$ C$,$ K$,($ek$)$k in K$,($dk$)$k in K$)$ be a cryptosystem. Let k in K$.$ Show that ek is injective and that dk is surjective.