(Solved): Note: Modular arithmetic is fundamental to cryptography. In this system, you can only have integers ...

Note: Modular arithmetic is fundamental to cryptography. In this system, you can only have integers. For example, in mod 14 system, the answer MUST be $0,1,2,3,\dots 9,11,12,13$. Non-integer values have no place in this arithmetic. If you have an answer which is a floating point, such as 12.5 , then you are doing something wrong. Question 1 [10 points]. Modular Arithmetic: Compute the following without a calculator. SHOW YOUR WORK. i. $150?92\mathrm{mod}14($ Hint: $a?b\mathrm{mod}c=((a\mathrm{mod}c)?(b\mathrm{mod}c))\mathrm{mod}c)$ ii. $6?(4/11)\mathrm{mod}14$ (Hint: In $\mathrm{mod}14$ system, a, a+14, a+28, a+42, a+56, etc. are all equivalent) iii. 24/17 mod 14 (Hint: First, simplify the numerator and denominator separately by applying the mod function independently, and then solve as in (ii) above). iv. $4^8?5^{12}\mathrm{mod}14$ (Hint: Try to compute the exponent in stages, each time simplifying it using the mod function. For example, to compute $4^8\mathrm{mod}14$, express $4^8\mathrm{mod}14?{\left(4^2\mathrm{mod}14\right)}^4\mathrm{mod}14$, compute the one in the parenthesis, and repeat this process. v. $5^{10}?6^8\mathrm{mod}14$ (same as iv above) Question 2 [10 points]. SHOW YOUR WORK. You may use EXCEL or a calculator. i. Show the elements of groups ${\mathbb{Z}}_{13}$ and ${\mathbb{Z}}_{13}^{?}$ (Note that 13 is a prime number) ii. Show the elements of groups ${\mathbb{Z}}_{18}$ and ${\mathbb{Z}}_{18}^{?}$ (Note that 18 is NOT a prime number) iii. Find the order of 5 in ${\mathbb{Z}}_{13}^{?}$ (Hint: Order of an element in a finite group $G$ is the smallest positive integer $k$ such that $a^k=1$ where 1 is the identity element of G.) iv. Find (if it exists) the multiplicative inverse of $5?{\mathbb{Z}}_{13}$ (integer ring) (Hint: For $a?\mathbb{Z}n$, its multiplicative inverse, if it exists, is defined as $a^{?1}$ such that $a?a^{?1}?1\mathrm{mod}n$. ) v. Is ${\mathbb{Z}}_{13}^{?}$ a cyclic group? If so, what is its order and the generator element? (Hint: group $G$ which contains some element $?$ with maximum order $\mathrm{ord}(?)=?G?$ is said to be cyclic. Elements with maximum order are called generators.)