+24 1. The House of Lilliput is using RSA encryption to receive secret messages from all the realms. They have published their public encoding exponent e=37 and their public modulus M = pq = 527.
Break the code: Find their secret decoding exponent d.
2. Find an integer x such that \(0 \leq x < 527\) and \(x^{37} \equiv 3 \pmod{527}\).
3. Let s be the set of numbers of the form n(n + 1)(n + 2)(n + 3)(n + 4), where n is any positive integer. The first few terms of s are \(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120, 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720, 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 = 2520,\)and so on. What is the GCD of the elements of s?
+9519 1)\(n^{37}\equiv c\pmod {527}\)
We first find the factors of 527
527 = 17 x 31 according to this calc.
We get an equation: \(37d = 1 \pmod{480}\)
The smallest d that satisfies this equation is 37 x 13 = 1 (mod 480)
Note: 37 x 13 = 481
d = 13 ;D
2) x = 148 Same answer as Guest #1
3) 1 x 2 x 3 x 4 x 5 = 2^3 x 3 x 5
2 x 3 x 4 x 5 x 6 = 2^4 x 3^2 x 5
3 x 4 x 5 x 6 x 7 = 2^3 x 3^2 x 5 x 7
They both have a common factor of 2^3 x 3 x 5 = 120.
1) Read the article here about RSA system and how to encrypt and decrypt messages:
https://en.wikipedia.org/wiki/RSA_(cryptosystem)
2) x=148
Also, watch this video by Dr. James Grime explaining how to encrypt and decrypt messages using RSA system: https://www.youtube.com/watch?v=M7kEpw1tn50
+9519 1)\(n^{37}\equiv c\pmod {527}\)
We first find the factors of 527
527 = 17 x 31 according to this calc.
We get an equation: \(37d = 1 \pmod{480}\)
The smallest d that satisfies this equation is 37 x 13 = 1 (mod 480)
Note: 37 x 13 = 481
d = 13 ;D
2) x = 148 Same answer as Guest #1
3) 1 x 2 x 3 x 4 x 5 = 2^3 x 3 x 5
2 x 3 x 4 x 5 x 6 = 2^4 x 3^2 x 5
3 x 4 x 5 x 6 x 7 = 2^3 x 3^2 x 5 x 7
They both have a common factor of 2^3 x 3 x 5 = 120.
