november 2010 volume 1 number 2 - Advances in Electronics and ...

MIECZYSŁAW JESSA: IMPROVING STATISTICAL PROPERTIES OF NUMBER SEQUENCES 53
For n < LCM(mp, L) Table T does not contain the same
elements as during the Initialization. If some element Un is
repeated for j = n, where n < LCM(mp, L), it is not
repeated for all n being a multiple of j, which results directly
fromthemethodofcomputingof Un.Consequently,theperiod
of {Un} cannot be smaller than LCM(mp, L). Changing the
size L of Table T, we can influence the period of generated
sequences. If L is relatively prime to mp, the period of {Un}
is L times greater than the period of m-sequence produced by
the MCPG, implemented in the same finite-state machine.
III. THE RESULTS OF NIST STATISTICAL TESTS
To verify the hypothesis that the statistical properties of
{pn} can be improved by the proper choice of α, K, and L,
the standard NIST statistical test suite v. 1.8 for cryptographic
applications was applied. It contains 15 tests, designed for analyzing
different statistical properties of generated sequences,
turned into binary streams [4]. The goal of the tests is to
detect non-randomness in binary sequences produced using
random number or pseudorandom number generators. The
tested sequences are composed of bits encoding successive
Un. The null hypothesis is that any sequence being tested is
random. Associated with this null hypothesis is an alternative
hypothesis, which, for the NIST tests, is that any tested
sequence is not random. The tests search for deviations from
the properties of truly random binary sequences in binary sequences
produced by a source under test. If a binary sequence
passes thetests, thereis noreasonto reject thenull hypothesis.
The empirical results can be interpreted in many ways. In
this paper two approaches proposed by NIST were used: (1)
the examination of the proportion R of sequences that pass a
statistical test and (2)the distributionof the so called P-values
computed by software. In the first case, we find the proportion
of sequences that pass a given test. The second approach,
adopted by NIST, measures the distribution of P-values in
interval [0, 1] divided into ten equal-sized subintervals. The
P-value is the probability (under the null hypothesis of randomness)thatthechosentest
statistic will assumevaluesequal
to or worse than the test statistic value observed when considering
the null hypothesis. The P-value is frequently called
the “tail probability”. When the sequences are random binary
sequences, the P-values obtained for these sequences have to
be uniformlydistributedin [0, 1] [4]. As the result of applying
a χ 2 test and an additional function, we obtain a new P-value
(PT ), corresponding to the Goodness-of-Fit Distribution Test
onthe P-valuesobtainedforanarbitrarystatistical test(i.e.the
P-value of the P-values).If PT ≥ 0.0001,then the sequences
can be considered to be uniformly distributed. The details of
computing PT can be found in [4].
The statistical tests were performed on 1000 different sequences
of length 10 6 . The sequences were successive fragments
of sequence {pn} or {Un}, produced for the smallest λ
for which {pn} was the m-sequence. Modulus q0 was a prime
number equal to 2 31 − 1 (l = 31) and p0 was equal to unity.
The size of Table T was constant during all experiments and
equal to L = 32. Because the least common multiple of mp
and L is equal to 34359738336, the period mu of {Un} is 16
TABLE I
THE RESULTSOF NIST TESTSFOR MCPG WITH λ = 7
Type of the test R(> 0.981) PT (> 0.0001) Final result
Block Frequency 0.0000 0.00000 fail
Serial* 0.9780 0.05642 fail
Approximate Entropy 0.9750 0.00711 fail
Linear Complexity 0.9900 0.7944 pass
Universal 0.9120 0.00000 fail
Overlapping
Templates
0.5490 0.00000 fail
Non-overlapping
Templates
0.9640 0.00000 fail
Cumulative Sums* 0.9670 0.00000 fail
Runs 0.9950 0.01570 pass
Longest Runs of Ones 0.9640 0.00000 fail
Rank 0.9880 0.43543 pass
Spectral DFT 0.0000 0.00000 fail
Random Excursions* 0.9836 0.07375 pass
Random Excursions
Variant**
0.9800 0.01526 pass
Frequency 0.9760 0.00000 fail
*This test consists of several subtests: the worst result is shown.
**The minimum pass rate for this test for a standard set of parameters is
approximately 0.978.
times longer than the period mp = 2 31 − 2 = 2 147 483 646
of {pn} produced by the MCPG. The results of the standard
NIST test suite performed for binary sequences, composed
of bits encoding successive pn, generated by MCPG with
λ = 7, are shown in TABLE I. The results of the same tests
for binary sequences, composed of bits encoding successive
Un, are presented in TABLE II. Parameter α was equal to 4.
Numbers from TABLE II were obtained for the smallest K
forwhichsequencesproducedbyCPRNG passedall statistical
tests.
IV. CONCLUSION
A new method for improving the quality of a multiplicative
congruential pseudorandom generator was proposed in this
paper. The method uses symbols produced by the sawtooth
map realized in a finite-state machine and numbers produced
by a multiplicative congruential generator, obtained as the
result of implementing the same map in the same machine
in the modular arithmetic. Although the proposed algorithm
improves the statistical properties of sequences produced by
a known pseudorandom generator, it can be treated as a new
generator, derived from a chaotic map. The basic weakness
of this generator is the lack of theory which could simplify
the choice of α, K and L. Simulation experiments, performed
by the author for many λ and q0 = 2 31 − 1, show that it is
always possible to choose relatively small K (of the order of
8) which yields sequences passing all tests from the standard
NIST statistical test suite v. 1.8. The speed of producing {Un}
with α = 4, L = 32 and K = 3 is only about 25% smaller
than the speed of producing {pn} on the same hardware and
software platform.
Access to a pseudorandom generator producing long period
number sequences that pass all NIST tests for many multi-