A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications

1.      The Frequency (Monobit) Test

Ones and zeroes’ proportion is actually the aim of test concerning the whole sequence. This test’s purpose surrounds the determination whether zeroes and ones in the set are almost the same for a sequence which is actually random or not. The assessment of closeness between ½ and ones’ fraction is actually carried out in the test. Meanwhile in a sequence, the number of zeroes and ones should be near approximately. This test’s passing and authenticity varies all other tests which are subsequent.

2.      Frequency Test within a Block

The ones which are present in M-bit blocks, their proportion is actually the aim of this test. Moreover, the determination of their frequency and whether it is M/2 in an assumption which is supposed under randomness is actually the purpose. M=1 for the size of block and degeneration of this test takes place to the Monobit or Frequency test, the test 1.

3.      The Runs Test

Runs’ total number in the sequence is this test’s aim while an undisturbed sequence is a run compromising of similar bits. Identical bits k are contained in a run whose length is k. Moreover, it is limited with only a single bit of value which is opposite after and even before. The determination whether the zeroes and ones in various bits with several lengths fulfill expectations for a sequence which is random or not, it is this test’s purpose. Particularly, the speed of oscillations between these ones and zeroes is determined in this test.

4.      Tests for the Longest-Run-of-Ones in a Block

In M-bit blocks, ones’ longest run is this test’s aim. This test’s purpose is the determination of consistency of longest run’s length within the sequence which is tested with the length which would be supposed or predicted in a random or abstract sequence. If there is an irregularity in the supposed or predicted length of ones’ longest run, it would imply the existence of an irregularity within the same of zeroes. Thus for only ones, the test is important.

5.      The Binary Matrix Rank Test

In this test, the focus is upon disjoint sub-matrices’ rank of the whole sequence. Meanwhile, the purpose is checking linear reliance between original or real sequence’s substrings of fixed lengths.

6.      The Discrete Fourier Transform (Spectral) Test

In this test, the focus is peak or top heights in the Transform concerning the sequence’s Discrete Fourier. Meanwhile this test’s purpose is the detection of features which are periodic such as patterns which are in a loop near other patterns in the sequence which is tested. The indication of deviation from the randomness which is assumed is assisted with it. Compared to the other 5%, detection of peaks going over the threshold of 95% and their number is intention.

7.      The Non-overlapping Template Matching Test

This test’s focus is upon the occurrences’ number of target strings which are pre-specified. In this test, the purpose revolves around the detection of generators which have a part in producing excessive occurrences of an aperiodic pattern which is given. For the test concerning section 8’s Overlapping Template Matching and this test, a window of m-bit is actually utilized for searching a certain pattern of m-bit. Position of one bit is slid by the window if founding of the pattern doesn’t takes place. The search actually continues with the identification of pattern. Moreover to the bit, the resetting of window takes place.

8.      The Overlapping Template Matching Test

The test of Overlapping Template Matching has the focus on occurrences’ number of target strings which are pre-specified. Both the Non-overlapping and Overlapping tests utilize a window of m-bit for searching a certain pattern of m-bit pattern. Just like the 7 Section, a bit position is slid if there is no pattern. The different among these sections actually the sliding of only a single bit before the search continues.

9.      Maurer's "Universal Statistical" Test

This test has the focus on bits’ number among patterns which are matching (a step or measure which is related to compressed sequence’s length). This test’s purpose is the detection whether the information is lost or not when it comes t compressing a sequence. A sequence with the capability of being highly compressed is actually recognized as non-random or specific.

10.  The Linear Complexity Test

This test’s focus is upon the LFSR or linear feedback shift register’s length. This test’s purpose is all about determining whether randomness is approved by a sequence’s complexity for it to be recognized as a random or not. Long LFSRs are the ones which characterize random sequences. Non-randomness is implied by a shorter LFSR.

11.  The Serial Test

Over the whole sequence, determining the frequency of every other m-bit pattern which is overlapping is this test’s focus. This test has the purpose of determining whether overlapping patterns of 2^mm-bit and their occurrences’ number is the same for an expected sequence which is random or not. Uniformity is actually found in random sequences which mean that every pattern of m-bit has the opportunity for approaching like how other patterns become visible. As m = 1, Section 1’s frequency test is similar to the Serial test.

12.  The Approximate Entropy Test

Like Section 2.11’s Serial test, this test’s focus is directly upon the overlapping patterns of m-bit which are possible over the whole sequence.  Meanwhile, the purpose revolves around the comparison of frequencies of blocks which are overlapping for lengths which are adjacent or consecutive (m and m+1). Such appears against the result which is expected of a sequence which is random.

13.  The Cumulative Sums (Cusums) Test

Maximal excursion is actually this test’s focus of the walk that is random from zero. Digits of (-1, +1) and their adjusted cumulative sum in the sequence define the random walk. This test’s purpose is the determination whether partial sequences’ cumulative sum in the sequence which is tested is too small or too large relative to the cumulative sum’s expected behavior of the sequences which are random. As a varying walk, this sum might be recognized. For a sequence which is random, random walk’s excursions should be close to nothing or simply zero. For some specific non-random sequence’s types, random walk’s excursions will be quite sufficient.

14.  The Random Excursions Test

This test’s focus is upon the cycles’ number having an appropriate number of visits or K in random walk’s cumulative sum. The random walk cumulative sum is gotten from sums which are partial after the sequence (0, 1) is sent to an authentic sequence of (-1, +1). A random walk’s cycle includes some steps with length at unit taken randomly at the beginning and to origin’s return. This test’s purpose is about determining whether visit numbers in a specific cycle change according to the expectations concerning a sequence which is random. In this test, eight tests are in series along with conclusions. For each state, one conclusion and one test: +1, +2, +3, +4 and -4, -3, -2, -1

15.  The Random Excursions Variant Test.

This test’s focus is upon the times and their numbers regarding visiting at a specific state in a random walk cumulative sum. This test’s purpose is about detecting changes and variations from visits’ expected number to random walk’s several states. In this test, there are eighteen conclusions and tests. For each state, there is 1 conclusion and 1 test: +1, +2, …, +9 and -9, -8, …, -1.