Random numbers, Encryption and Hashing

Random numbers

1. Definition: A random sequence must be generated as if they were independent draws from a well mixed urn where each number is represented once in the urn.

Mathematicians prefer abstract specifications not descriptions of processes.

2. Definition: A random sequence is one in which each number in the sequence is independent and equally probable.

3. Definition: A random sequence is one in which all finite sub-sequences of numbers are ∞-distributed.

∞-distributed is an iid process chacterized as a sequence of numbers

A pseudo random number generator is a generator that produces a series of numbers that pass a reasonable set of statistical tests.

In the 1950's Linear congruential random number generators (LCG's) were used to generate pseudo-random numbersbased on the computation;

[math] r_i = (c_1 \times r_{i-1} + c_2 ) \mbox{MOD} c_3[/math]

where [math]c_1, c_2, c_3[/math] are constants, and [math]r_{i-1}[/math] is the last pseudo-randomly generated number. All LCG's eventually repeat, and have many non random properties based on prior observations.

Encryption

Encryption is the idea of locaking away information so that the only a person with a key can unock and read it. The simplest example of a code is the alphabetic substitution cipher. This is easily crackable due to inherent structure in languages, for example spaces between words, and repeated letters.

See also;

• Enigma machine
• Vernam cypher - Uses XOR bit function
• Vigenère cipher

Encryption is used in User Authentication. The idea is rather than encrypting an entire file of passwords once, encrypt each users password via an encryption process used to generate a standard message phrase. Then only the result of the encryption is required to be stored for validation, not the actual password that generates the result. e.g.

buz:wYdB]eaWLpD[tz[b]z: Robert Uzgalis

Here, buz is the username, but the password is never stored only the result of the encryption process for validation - wYdB]eaWLpD[tz[b]z.