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?

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

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),$$0 \le b,n \le m$$?

In a look $$-$$ ahead carry generator, the carry generate function $${G_i}$$ and the carry propagate function $${P_i}$$ for inputs, $${A_i}$$ and $${B_i}$$ are given by: $${P_i} = {A_i} \oplus {B_i}$$ and $${G_i} = {A_i}{B_i}.$$

The expressions for the sum bit $${S_i}$$ and the carry bit $${C_{i + 1}}$$ of the look ahead carry adder are given by $${S_i} = {P_i} \oplus {C_i}$$ and $${C_{i + 1}} = {G_i} + {P_i}{C_i},$$ where $${C_0}$$ is the input carry. Consider a two $$-$$ level logic implementation of the look $$-$$ ahead carry generator. Assume that all $${P_i}$$ and $${G_i}$$ are available for the carry generator circuit and that the $$AND$$ and $$OR$$ gates can have any number of inputs. The number of $$AND$$ gates and $$OR$$ gates needed to implement the look $$-$$ ahead carry generator for a $$4$$-bit adder with $${S_3},\,\,{S_2},\,\,{S_1},\,\,{S_0}$$ and $${C_4}$$ as its outputs are respectively