Page 45 - DCAP516_COMPUTER_SECURITY

Unit 4: Cryptography

The only thing is that we need to find algorithms that indeed satisfy all the three requirements. Notes
Due to the advantage of public key cryptography, researchers are working hard and some
algorithms are available.
One good method was discovered by a group of MIT (Revert, et al. 1978) based on number
theory. It goes like this:

1. Choose two large primes p and q, each greater than 10100.
2. Compute n = p x q and z = (p-1) x (q-1).
3. Choose a number relatively prime to z and call it d.
4. Find e such that e x d = 1 mod z.

With these parameters computed in advance divide the plaintext regarded as a bit string, into
blocks, so that each plaintext message, p, falls in the interval o £ p < n. To encrypt a message, p,
compute C = pe (mod n). To decrypt C, compute p = cd(mod n). It is proved that for all p in the
specified range the encryption and decryption functions are inverse. For encryption one needs
e and n and the public key is (e, n) and for decryption one needs d and n so that the secret key is
(d, n) or just d.

Figure 4.2: An Example of the MIT Algorithm
Plaintext (P) Ciphertext (C) After description
7
3
Symbolic Numeric P 3 P (mod 33) C 7 C (mod 33) Symbolic
S 19 6859 28 13492928512 19 S
U 21 9261 21 1801088541 21 U
Z 26 17576 20 128000000 26 Z
A 01 1 1 1 1 A
N 14 2744 5 78125 14 N
N 14 2744 5 78125 14 N
E 05 125 26 8031810176 5 E
Sender’s computation Receiver’s computation

For example we choose p=3, q=11, then n=33 and z=20. A suitable value for d=7 since and 20 have
no common factor. With these e can be solved by equation 7e=1 (mod 20) which yields e=3. The
cipher text, c, for a plaintext message, p is given by c=p3 (mod 33). The ciphertext c is decrypted
by p=c7 (mod 33). The plaintext here is ‘SUZANNE’. Because the primes are so small that the p
must be less than 33, each plaintext block can contain only a single character.

4.4 The AES Encryption Algorithm

Advanced Encryption Standard (AES) is a specification for the encryption of electronic data.
It has been adopted by the U.S. government and is now used worldwide. It supersedes DES. AES
is a symmetric-key algorithm, meaning the same key is used for both encrypting and decrypting
the data.

Did u know? Originally called Rijndael, the cipher was developed by two Belgian
cryptographers, Joan Daemen and Vincent Rijmen, and submitted by them to the AES
selection process. The name Rijndael is a play on the names of the two inventors.

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