$4.$ For RC$4$ stream cipher, what is the value of the key that will leave the initialPublic cryptography

$5.$Regarding one$-$directional hash functions, if $H(m)$ is a collision$-$resistant hash function that maps a message of arbitrary bit length size$($m$)$ into an $n-$bit hash value i$.$e$.$ size$($m$)>$n$.$ Is it true that, for all messages $x,x^{\text{'}}$ with $x{x}^{\text{'}},$ we have $H(x)H(x^{\text{'}})?$ Explain your answer.

$6.$ Consider a $32-$bit hash function defined as the concatenation of two $16-$bit functions: XOR and RXOR, Where RXOR is the rotated XOP function performed as follows: Initially set the $16-$bit hash value to zero. Process each successive $16-$bit block of data:

a$.$ Rotate the current hash value to the left by one bit.

b$.$ XOR the block into the hash value.

a$)$ Will this checksum detect all errors caused by an odd number of error bits? Explain.

b$)$ Will this checksum detect all errors caused by an even number of error bits? If not, characterize the error patterns that will cause the checksum to fail.

$7.$ Evaluate the following:

a$)9^{682}$mod$10$

b$)4^{283}$mod$10($hint: calculate $\frac{O}{10},$ knowing that $x^{\frac{O}{10}}$mod$10=1)$

c$)gcd(2^{2000},8)$

$8.$ To generate its RSA key pairs, A generates two primes $p=59$ and $q=67.$ What is the "smallest" encryption exponent $e$ that can be chosen by A$?$ What is the corresponding decryption exponent $d?($hint: use the extended Euclidean algorithm$).$

$9.$ In a public$-$key system using RSA, you intercept the ciphertext $C=10$ sent to a user whose public key is $e=5,n=35.$ What is the plaintext $M?$

$10.$ A and B use Diffie Helman key exchange with parameters $p=79,g=3.$ A chooses private key $5$ and $B$ choose its private key a $7.$ What is the common secret computed by A and $B?$

state S unchanged during initialization? That is$,$ after the initial permutation of S$,$ the entries of S will be equal to the values from $0$ through $255$ in ascending order?