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    Wednesday, May 3, 2023

    RSA Algorithm

    The RSA algorithm is an asymmetric cryptography algorithm; this means that it uses a public key and a private key.

    The RSA algorithm is named after those who invented it in 1978: Ron Rivest, Adi Shamir, and Leonard Adleman.


    The following steps highlight how it works:

    1. Generating the keys

    1. Select two large prime numbers, x and y. The prime numbers need to be large so that they will be difficult for someone to figure out.
    2. Calculate n = x x y.
    3. Calculate the totient function: ϕ(n) = (x−1)(y−1).
    4. Select an integer e, such that e is co-prime to ϕ(n) and 1< e < ϕ(n). The pair of numbers (n,e) makes up the public key.
    1. Calculate d such that e.d=1 mod ϕ(n). The pair (n,d) makes up the private key.

    2. Encryption

    Given a plaintext P, represented as a number, the ciphertext C is calculated as:

    C=Pe mod n.

    3. Decryption

    Using the private key (n,d), the plaintext can be found using:

    P=Cd mod n.

    RSA Algorithm Example

    • Choose p = 3 and q = 11
    • Compute n = p * q = 3 * 11 = 33
    • Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20
    • Choose e such that 1 < e < φ(n) and e and φ (n) are coprime. Let e = 7
    • Compute a value for d such that (d * e) % φ(n) = 1.

    One solution is d = 3    since [(3 * 7) % 20 = 1]

    • Public key is (e, n) => (7, 33)
    • Private key is (d, n) => (3, 33)
    • The encryption of P = 2 is C = 27 % 33 = 29
    • The decryption of C = 29 is P = 293 % 33 = 2

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