cryptography

 
Hash
Question (Use Sage):
The following describes the simple hash function:
Choose p, q primes and compute N = pq.
Choose g relatively prime to N and less than N.
Then a number n is hashed as follows:
H = gn mod N
If there is an m that hashes to the same value as n, then
gm ≡ gn mod N
so
gm-n ≡ 1 mod N
which implies that
m –n ≡ 0 mod φ (N)
So breaking this amounts to finding a multiple of φ (N), which is the hard problem in RSA.

Write a function that takes a bit length n and generates a modulus N of bitlength n and g less than N and relatively prime to it.
Show the output of your function from part (a) for a few outputs.
Using N, g, n as arguments write a function to perform the hashing.
For the following parts (a)-(d) compute the simple hash:
N = 600107, g = 154835, n = 239715
N = 548155966307, g = 189830397891, n = 44344313866
N = 604766153, g = 12075635, n = 443096843
Write a function that creates a collision given p and q.  Show that your function works for a couple of examples.

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