Cryptography/Python: Derive the secret using Quadratic Residue knowledge

This is straightforward assignment to derive secret using Quadratic Residue knowledge.

Formula: c = (s ^ r) mod p

- s is the secret (an integer) to be encrypted using above formula

- r is a random 500-bit number

- p is a random 500-bit prime number

- c is the ciphertext computed using this formula

Above design is vulnerable, because attacker can calculate the value of s if he/she got enough pairs of <c, p> values.

Part 1:

Later, I will provide a text file of 30 groups of <c, p> values. We only know s is an integer in the interval of [2410, 2459], but do not know which value is it. The mission is to write a Python program to derive the value of s, using those 30 groups of <c, p> values as inputs.

Remarks:

1. Hint: you MUST use the knowledge of Quadratic residue. I can provide some reference to explain Quadratic residue if you need.

2. I can provide a short and simple Python code that how c, r, and p are generated.

3. This is not a brute force mission. You cannot simply compute the values from 2410 to 2459 and compare the outputs.

4. After last, write a concise and clear summary of algorithm at comment or in a separate file.

Part 2:

In part 1, the 30 random r’s were chosen so that s can be identified. Actually, if the 30 r’s are chosen uniformly and randomly, then there is a chance that we cannot uniquely identify s using 30 tuples.

The probability of successful identification increases with large number of <c, p> tuples.

Based on part 1, the mission is to give the least number of tuples required in order to achieve 99% of success, and explain it.

You can write down the answer analytically without writing a new code.

Part 3:

Same as part 2, but the interval size of s increases into 10^6.

The mission is still to write down the answer analytically.

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