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 ? (N)

So breaking this amounts to finding a multiple of ? (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.