Internet Security - Dang Thanh Binh's Page

Message
m
−1
User A
A′ private key
p A
d A
E
n A
q A
ASYMMETRIC PUBLIC-KEY CRYPTOSYSTEMS 171
p A −1
q A −1
e A d A ≡ 1 (mod j(n A ))
c ≡ m dA (mod n A )
lcm
User B
A′ public key
e A
j(n A ) = lcm(p A − 1, q A − 1)
Figure 5.3 The RSA signature scheme.
Suppose m = 55. Then the signed message is
c ≡ m dA (mod 187)
≡ 55 3 (mod 187) ≡ 132
The message will be recreated as:
m ≡ c eA (mod n)
≡ 132 27 (mod 187) ≡ 55
Thus, the message m is accepted as authentic.
D
c eA ≡ m dAeA (mod nA )
≡ m
Next, consider a case where the message is much longer. The larger m requires more computation
in signing and verification steps. Therefore, it is better to compute the message
digest using a appropriate hash function, for example, the SHA-1 algorithm. Signing
the message digest rather than the message often improves the efficiency of the process
because the message digest is usually much smaller than the message.
When the message is assumed to be m = 75 139, the message digest h of m is computed
using the SHA-1 algorithm as follows:
h ≡ H(m) (mod n)
≡ H(75 139) (mod 187)
m