Quasi-Random Number Sequences from a Long Period TLP Generator with Remarks on

Application to Cryptography

By Herbert S. Bright and

Richard L. Enison

Presented by Saunders Roesser

The Problem

• Generation of successful random number sequences that pass all statistical testing criteria.

• Generation in an Application domain.

Background

• Physical Generations are unsuitable for modern computers

• Linear Congruential formulas:– Xi+1 = axi + c (mod m)

• Additive Formulas– Xi = a1Xx-1 + a2xi-2+…..+apxi-p+ c (mod m)

• Don’t work unless you have large primes.

TLP Sequence

• Tausworthe-Lewis-Payne distribution

• Sequence for generation of random numbers.

• Trinomial: x521+x32+1

• Generate 64-bit numbers

• Period is 2521-1

• Better then linear congruential generators

Statistical Testing Criteria

• Equidistribution/Frequency Test– The number of time a given number falls into

a given interval

• Serial Test– The number of times a sequence appears in a

certain number of numbers

• Gap Test– The distribution of gaps in the sequence of

various lengths.

More Tests

• Runs Test– Plots the distribution of maximal ascending

runs of various lengths

• Coupon Collector’s Test– Choose a small interger, divide the number

into intervals then plot the distribution runs of various lengths required to have all intervals represented

More Tests

• Permutation Test – Order relations between the members of the

sequence in groups of k.

• Serial Correlation Test– Computer the correlation coefficient between

consecutive members of the sequence.

• Others..

Results

• At the time, all present generators failed the battery of tests.

• Hope came from recursive function theory.

• TLP Generator showed good results in string tests

• Passed equidistributivity tests, along with other tests.

Other Physical Random Number Generators

• Dice

• Ionizing radiation

• Gas discharge tubes

• Leaky capacitors

• Physical noise generators