Start 2x speed Bob selects two primes: p = 31 q = 59 Compute: N = p · q = 31 · 59 = 1829 $ = (p - 1) · (q - 1) = 30 · 58 = 1740 Find integer e such that gcd(e, d) = 1 Guess e = 859 and check: gcd(859, 1740) = 1 If the first guess is not relatively prime to p, try another. Using Euclid's algorithm, find A and B such that A · 859+ B. 1740= 1 79. 859 + (-39)1740= 1 79 . 859 = 1 mod 1740 d = 79 is the inverse of 859 mod 1740 Public key: N= 1829 e = 859 Private key: d = 79 Captions A 1. Bob selects two primes p = 31 and q = 59. Bob computes N = p • q = 31 · 59 = 1829 and o = (p – 1) · (q – 1) = 30 · 58 = 1740. 2. Bob guesses a value for e and checks whether gcd(e, $) = 1. Bob repeats the process until he finds an e such that gcd(e, ) = 1. 3. Suppose e = 859. Bob uses Euclid's algorithm to find A and B such that A· 859 + B · 1740 = 1. 4. Since 79 · 859 + (–39) · 1740 = 1, then 79 · 859 = 1 mod 1740. Therefore, d = 79 is %3D the inverse of 859 mod 1740. 5. The public keys are N = 1829 and e = 859. The private key is d = 79.

Source:https://www.mog.dog/files/SP2019/Sipser_Introduction.to.the.Theory.of.Computation.3E.pdf

You are given the following definitions to help you: O(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ f(n) ≤ cg(n) for all n ≥ no} . N(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ cg (n) ≤ f(n) for all n ≥ no} . ©(g(n)) = {f(n): there exist positive constants C₁, C2, and no such that 0 ≤ c₁g(n) ≤ f(n) ≤ c₂g (n) for all n ≥ no} . Using above definitions, prove that: 5. T(n) = 6n + 4n+ 3 € 0 (n) 6. T(n) = 100n + 10000 € O(n²) 7. T(n) = 5n² - 2n + 16 € O(n³) 8. T(n)=5n² - 2n + 16 is not € O(n) 9. T(n) = n³ + 20n € №(n² ) 10. T(n) = 2n³ - 7n + 1 € ☺(n³)

Find all solutions for the following pair of simultaneous congruences. 5x = 9 (mod 11) 3x = 8 (mod 13)

2. Prove that 8"-3" is evenly divisible by 5 for all natural numbers n.

Provide a counter example for: The number n is an even integer if and only if 3n + 2 is an even integer.

This is a practice question from my Data Structures course: h(n) = {1 if n = 1; 3 × h(n - 1) - 1 otherwise} Rewrite h(n) using an explicit formula that is not recursive (i.e., succinctly express h(n) as a polynomial/exponential function of n). Also, prove the correctness of your expression using mathematical induction. Please explain how to do this (as opposed to simply providing the answer). Thank you for your assistance.

Consider Fib(3). What do you notice about every third Fibonacci number, i.e. Fib(6), Fib(9),Fib(12), Fib(16)... What seems to be the pattern behind these sequence s generated from the Fibonacci numbers?

suppose a,b, and c are odd integers. Prove that a+b+c is odd

Please answer completely will give rating surely Both questions answers needed