Consider the CFG with { S, A, B } as the non-terminal alphabet, { a, b } as the terminal alphabet, S as the start symbol and the following set of production rules:

$$\eqalign{ & S \to bA\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,S \to aB \cr & A \to a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,B \to b \cr & A \to aS\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,B \to bS \cr & S \to bAA\,\,\,\,\,\,\,\,\,\,\,B \to aBB \cr} $$

For the correct answer strings to the previous question, how many derivation trees are there?

Exponentiation is a heavily used operation in public key cryptography. Which of the following options is the tightest upper bound on the number of multiplications required to compute (bⁿ mod m),$$0 \le b,n \le m$$?

The minimum positive integer p such that (3^p modulo 17) = 1 is

Consider the following two statements:

i. A hash function (these are often used for computing digital signatures) is an injective function.
ii. encryption technique such as DES performs a permutation on the elements of its input alphabet.

Which one of the following options is valid for the above two statements?