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4.6. SUMMARY 67
To put it more bluntly, guessing one letter correctly allows us to break Caesar
ciphers. Guessing two letters correctly allows us to break Decimation and Linear
ciphers.
4.6 Summary
A Linear Cipher is a combination of a Decimation and a Shift Cipher, and
so has both a multiplicative and an additive key. The additive key can be
chosen at random, like in a Shift Cipher, but, like in a Decimation Cipher, the
multiplicative key must be chosen so that it is relatively prime to the modulus.
A relatively quick way of computing greatest common divisors is with the
Euclidean Algorithm. To compute the gcd of two numbers, the algorithm uses a
process of repeated remainder operations in which the larger of the two numbers
is continually replaced by the remainder of the larger divided by the smaller.
By putting this into a table and making use of the quotients involved, the multiplicative
inverse can also quickly be found. Since Koblitz’s Kid-RSA depends
for its security on the inability for an adversary to compute multiplicative inverses,
we see it not secure at all, but is intended as a toy system, intended to
demonstrate the concept of public keys.
Decimation and Linear Ciphers do a better job of mixing the ciphertext
alphabet than a Caesar Cipher does. Letters that are adjacent in the plaintext
alphabet are not adjacent once enciphered, they are k letters apart, where k
is the multiplicative key of the cipher. While this prevents us from using the
hills and valley patterns of the normal frequency to break the cipher, it provides
enough order so that if two ciphertext letters can be matched with their plaintext
counterparts either of these ciphers can be broken.
4.7 Topics and Techniques
1. How many keys does a Linear Cipher need
2. How do we encipher with a Linear Cipher
3. What is the greatest common divisor of two numbers
4. What is the Euclidean Algorithm
5. What does relatively prime mean When are two numbers relatively
prime
6. How do we use the Euclidean Algorithm to find a gcd
7. What is a multiplicative inverse
8. When does a number have a multiplicative inverse modulo n