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 chich 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 testes.
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.
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