Real-Life Math
Cryptography is a form of applied math. Whether you're inventing
new ways to encode information or incorporating other people's formulas
into security systems, you have to know your numbers.
You work with
a large data security firm that deals with public key encryption schemes.
With a public key encryption scheme, the encryption rule employs a public
key "e," while the decryption rule requires a different (but related) private
key "d." Knowledge of the public key "e" allows encryption of plaintext, but
doesn't allow decryption of the resulting ciphertext.
Computer
programs can do the calculations required to move from plaintext to ciphertext
and back to plaintext. These programs are based on mathematical formulas,
and in instances where programs fail, cryptographers are called in to crunch
out the numbers and decipher the text.
This is your assignment: A disgruntled
employee scrambled some important computer records the day she was fired.
If you recover her coded PIN number, you can get the computer to decipher
the records she scrambled. One of the steps you have to go through to decipher
her PIN number is changing a public key sequence of numbers into a modified
public sequence:
Public Key Numbers
14 42 43 58 35
To
find the modified public sequence, you use the private keys for this system
(4 and 55), and a mathematical formula for finding the "modulo" or "mod" for
short. This is how you find the mod:
(X x 4) mod 55
Don't
panic! This is easier than it looks. For example:
(120 x 4)
mod 55 = 480 mod 55
When you divide 480 by 55 on a calculator,
you get 8.272727. If you think back to how you first learned how to divide,
you didn't get answers with decimals right away. Your division resulted
in a whole number (called the quotient), and some leftover part called the
remainder. For example:
7/3 has a quotient of 2 and remainder
of 1, since 7 = (2 x 3) + 1. So the "mod" operation looks for the remainder
in the division, which is a whole number. In this case, the "mod" is 1.
Since
480 = (55 x 8) + 40, the quotient (whole part) is 8. The remainder is 40.
Thus 480 mod 55 = 40.
No problem. Now you find the "mod" for
the public key numbers: 14 42 43 58 35.