Alice received the following ciphertext from Bob, "08 14 08". Bob had encrypted it using the RSA cypher with Alice's public key
(pq, e) = (55, 3),
where
p = 5
and
q = 11.
Note that
(p − 1)(q − 1) = 40.
The value for d in Alice's private key,
(pq, d)
is a positive inverse for
3 modulo (p − 1)(q − 1).
It was found to be 27 in Example 8.4.8(b) and Example 8.4.10. What is Bob's message after Alice decrypts it? (Assume Bob encoded one letter at a time using the encoding
A = 01, B = 02, C = 03, , Z = 26.)
To decrypt Bob's message, Alice uses the decryption formula
M = C mod ,
where M is the code for a letter of the message, C is the encrypted version of the letter,
(pq, e) = (55, 3)
is the public key, and
(pq, d) = (55, 27)
is the private key.
(a)
To begin, Alice computes the values of
a, b, c, d and e
that are indicated below.
081≡a (mod 55) 082 ≡ b (mod 55) 084 ≡ c (mod 55)
088≡d (mod 55) 0816 ≡ e (mod 55)
She finds that
a = ,
b = ,
c = ,
d = ,
and
e = .
Because
27 = 16 + 8 + 2 + 1, 0827 = 0816 + 8 + 2 + 1 = 0816 · 088 · 082 · 081,
she uses the values of
a, b, d, and e
to compute
0827 mod 55 = (a · b · d · e) mod 55 = .
Thus, the first letter in Bob's message is .
(b)
Alice finds the second letter of Bob's message by computing
27 mod 55 = .
(c)
What is Bob's message after Alice finishes decrypting it?