Assignment 6
1) I want to determine p and q. I know that φ(n) = (p-1) * (q-1) = pq – (p+q) + 1. With this simple
formula I can know compute (p+q). Now I can also say that (p-q)^2 = (p+q)^2 - 4pq. And now I also
compute p-q by making the square root of (p-q)^2. By simply adding (p+q) to (p-q) I can compute
the value of p and as a consequence the value of q.
2) n = 84773093
φ(n) = 84754668
84754668 = 84773093 - (p + q) + 1
p + q = 84773093 – 84754668 + 1 = 18426
(p - q) ^ 2 = (18426)^2 – 4 * (84773093) = 425104
P - q = 652
p = (18426+652) / 2 = 9539
q = 18426 – 9539 = 8887
3) S U N → 18 × 26^2 + 20 × 26 + 13 = 12701
T A P → 19 × 26^2 + 0 × 26 + 15 = 12859
4) 127 × 149 = 18923
I also know that 1261d mod 18648 = 1. Computing the multiplicative inverse of e mod n with
Bézout identity I got d = 5797.
12423 = 18 x 26^2 + 9 x 26 + 21 → S J V = 18921
Using the above information I have
C^d ≡ M (mod n)
18921^5797 mod 18923 = 14668 and so the message was ‘’OGGI’’.