dissertacao.pdf

1.6 RSA Definition
We are now in conditions to present the mathematical definition of RSA. We
will consider an RSA cryptosystem to be a tuple < N,M,C,K,E,D > where:
N = pq - the public modulus, the product of two different prime numbers
p, q.
M - the set of plain text. M = ZN.
C - the set of cypher texts. C = ZN.
K - is a tuple < p, q, e, d > where (d, φ(N)) = 1 and ed ∼ = 1 (mod φ(N))
Kr =< e, N > is the public key
Kp =< p, q, d, N > is the private key
E - the encryption function: E : M → C, c = E(m|Kr) ∼ = m e (mod N)
D - the decryption function: D : C → M, m = E(c|Kp) ∼ = c d (mod N)
e and d are called public and private exponent respectively. The expo-
nents satisfy the equation ed − 1 = kφ(N), which is therefore called the key
equation. It is also possible to define the exponents modulo λ(N). Since this
is a multiple of φ(N) the rest of the procedures described in this section are the
same.
The transmission of messages is as follows: suppose Alice wishes to send a
plain text message m ∈ M to Bob. Alice encrypts m using Bob’s public key
< e, N > and obtains
c ∼ = m e
(mod N) (18)
Then, she sends c through an open channel to Bob. Now Bob gets c and decrypts
it using his private key < d, N >:
m ′ ∼ = c d ∼ = m ed ∼ = m (mod N) (19)
The last equality is a result of Euler’s Theorem:
m ed ∼ = m 1+kφ(N) ∼ = m(m φ(N) ) k ∼ = m1 k ∼ = m (mod N) (20)
The cryptosystem is easy to implement, which is one of the reasons why it
is so popular.
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