Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

13 2.2 RSA Cryptosystem
are implemented properly. In 2002, all three inventors of RSA received the Turing
Award for their ingenious contribution for making public-key cryptography useful
in practice. Let us systematically study the RSA cryptosystem before proceeding
any further.
2.2.1 Classical Model of RSA
In a public key cryptosystem, there exists three major components, namely the
key generation process, and the algorithms for encryption and decryption. Key
generation and decryption are performed by the recipient, whereas the encryption
occurs on the side of the sender. In case of RSA, the three phases can be described
as follows. We shall henceforth assume that Alice is the sender and Bob is the
receiver in our cryptographic scheme.
Key Generation
To create a public/private key pair for the RSA cryptosystem, Bob first chooses
randomly two ‘large’ primes p,q (recommended to use primes of same bitsize with
minimum size of 512 each). Then he calculates the product N = pq and Euler’s
totient function φ(N) = (p−1)(q −1). Bob keeps the values p,q,φ(N) secret, as
any one who knows any one of these values will be able to decrypt messages sent
to Bob.
Bob’s next step is to find two positive integers e,d such that ed ≡ 1
(mod φ(N)). Bob publishes the pair (e,N) as his public key, which can be used
to encrypt messages meant for Bob. The pair (d,N) is Bob’s secret key and these
are used to decrypt the received ciphertexts.
Encryption
To send a plaintext to Bob, Alice first transforms her message in to an element m
of Z N , and calculates c ≡ m e (mod N). The ciphertext c is sent to Bob.
Decryption
After receiving the ciphertext c, Bob decrypts c by computing c d mod N and gets
back m.