it is hard to find the square root of a number using modulo arithmatic when the factors are unknown.
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However, imagine y= x^2 mod (11*13) - modulo something known.
We could set x=82 and get 3.
Trying to find the X that generated the 3, it can be one of 82, 126, 17 and 61.
Finding the square root results in 4 possible answers.
I was wondering if we had a number and ran the above twice (getting the modulo square, and square again), do we have more exponentially more uncertainty as to the original number, or is the "square square" root likely to have 1-4 possible answers still?
Any background to help me understand this please?
Many thanks.
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Zaphod1001
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When there are $n$ distinct odd prime factors of your modulus, there will be in general either none or $2^n$ solutions to $x^2=c$. So $x^2\equiv 1\pmod {3\cdot 5\cdot 7}$ has $8$ solutions. – Thomas Andrews Nov 20 '15 at 14:26
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Thank you - I think that's a great answer. There's no way of stating your comment is "the answer" however. You might want to write it as an official answer so I can upvote. Cheers. – Zaphod1001 Nov 23 '15 at 13:32