Enkript

Implement the attacks on RSA (Rivest, Shamir & Adleman)

1. best known & widely used public-key scheme
2. based on exponentiation in a finite (Galois) field over integers modulo a prime

• exponentiation takes O((log n)3) operations (easy)

3. uses large integers (eg. 1024 bits)
4. security due to cost of factoring large numbers
5. factorization takes O(e ^ log n log log n) operations (hard)

Take two distinct, large primes p and q. Ideally these have a similar byte-length. Multiply p and q and store the result in n. Find the totient for n using the formula: 𝜑(𝑛) = (𝑝−1)⋅(𝑞−1). Take an e coprime that is greater, than 1 and less than n. Find d using the formula 𝑑⋅𝑒 ≡ 1 mod 𝜑(𝑛). At this point, the pair (e, n) is the public key and the private key (d, n) is the private key.

Potential Attacks

1. Factorization - The obvious way to do this attacks is to factor the public modulus, n, into its two prime factors, p and q. From p, q and e, the attacker can easily get d. The hard part is factoring n: 1.1 Security on RSA depends on factoring being difficult. 1.2 In fact, the task of recovering the private key is equivalent to the task of factoring the modulus. 1.3. It should be noted that the hardware improvements alone will not weaken the RSA, as long as appropriate key length are used.
2. Chosen Cipher Text
3. Encryption Exponent
4. Decryption Exponent
5. Plaintext
6. Modulus
7. Implementation

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