The document discusses tests for evaluating the randomness of random number generators, specifically comparing the CHITESTS and DIEHARD test suites. It provides results from running the test suites on random numbers generated from different sources, finding that numbers from RANDOM.SYS pass the tests while the others show some non-randomness. The authors conclude that CHITESTS and DIEHARD are both effective for evaluating random number generators but DIEHARD is more rigorous.

1. . ., . .
(www.dekart.com)
) *
, .
“ t” -
$ , %& '
, . (.) .*
.
The problem of tests of the random numbers, applied in cryptography, is considered in this article.
The special attention is paid to the empirical test “the maximum of t” – the authors suggest their
own way for solving this problem that appears under discretization of the classical test, elaborated
by Professor D. Knuth. The results of the random and pseudo-random figures testing are adduced.
* & ' ( , , )
. H
' , $
, ,
' . ( & '
'
DSS. * ' DSA ( DSS) '
%
& .K % 0
160- q( % ). P
, %& ' ,
$ , (Q*RS), %&
( , -
T U .).
*
. R
,
2
[1,2]. *
,
' .
%
* :
(Ys n ps ) 2
2
= (1)
1 s k n ps
Ys - , % s;
ps - , % s;
k - ;
n– & .

2. 2
* % =k–1 p-
, . . . S n
, % ps n ps $ 5.
K , , -
, [1,3] &
“ ”( ) [1].
* “ t”. ` , %&
' , % %& :
1) t %&
;
2) .* '
% “ t”;
3) & %
“ t”
.
a “ t” [1] ) -
R . H
'
2
( k) .
b i- % t- (Xi1, Xi2, ... , Xit). )
0 k-1 % .
*
M x = max X ij , 1 j t
a , Mx = z %& . Xij –
.R
p (M x z ) = p( X ij z ) = ( z + 1) k t ,
t
0 z k 1
1 j t
p (M x = z ) = p (M x z ) p(M x z 1)
d ,
[
p(M x = z ) = ( z + 1)
t
]
z k,
t t
0 z k 1. (2)
* %* % ,
(2) & ( . . t-
) $ 5. ( z
$ 5.
– (1)
%& & , , ,
$ % , % '
.
a `.e [4] %& $
. * $ % . *
[0, k-1] ,
“ t”, $ $ .
“ t” % ,
,
2
, $ %& .( ,

3. (2), %&
:
p( z1 Mx z2) = [( z + 1) z ] k ,
2
t
t
1
t
z1 z2
C $ %
[0, k-1], &
.
U $ “ t” & %
.
1. a (2) zmin= 0. R ,
t
pmin = 1 / k n pmin= N / (t kt),
N– .* ,
, ,
N 5 kt). H
(t % ,
t k N $ $ .H , t = 8, k =
256, N 7.38 1020.
2. (
t. (
% t, %& %
t
t k N / 5. (3)
3. k
( ), N t
/ k, $ (3).
d , % ,
“ & ”,
%& t k ( (3)),
“ t”.
b CHITESTS [5],
' ,
. ) %
' & % 2
.
P ,
. CHITESTS
, ' % [1].
k p 0 – 1 % 99 – 100 %, %
.k p 99 95 % 5 1 %,
% “ ”; p, % 95 90 % 10
5 %, “ ” [1].
P Q.U RANDOM CD-ROM [6],
: ,
DIEHARD, 60 10-
. H
DIEHARD. H
& %
DIEHARD CHITESTS. T calif.bit %
, % & % ' () ,
Re`), & % G.Marsalia RANDOM CD-ROM. a random -
, , & % RANDOM.SYS,
file1.bin - smart- Payflex 1K. b $ 10 U .

4. 10000000 characters in file: calif.bits
HITESTS
. / CHI 3 4
A Frequency Test 1.000000 Test error1
B Serial Test 1.000000 Test error
C Gap Test 0.976478 Suspicious
D Poker Test 1.000000 Unsatisfactory
E Coupon Test 0.968074 Suspicious
F Permutations Test 0.999634 Unsatisfactory
G Runs Up Test 0.669232 Satisfactory
H Maximum-of-8 Test 1.000000 Test error
I Lapped M-tuple Test 1.000000 Test error
DIEHARD
. / p-value 3 4
A. BIRTHDAY SPACING .631736 Satisfactory
B. OPERM5 .541571 Satisfactory
C. BINARY RANK 1.0000 Unsatisfactory
D. SQUEEZE .997847 Unsatisfactory
E.1 CRAPS: no. of wins .847283 Satisfactory
E.2 CRAPS throws/game .441181 Satisfactory
F. MINIMUM DISTANCE .792573 Satisfactory
G. 3DSPHERES .001891 Unsatisfactory
H. OSUM .409425 Satisfactory
I.1 RUNS: Runs up .430718 Satisfactory
I.2 RUNS: Runs down .622296 Satisfactory
K. CDPARK .406615 Satisfactory
10000000 characters in file: random
HITESTS
. / CHI 3 4
A. Frequency Test 0.109633 Satisfactory
B. Serial Test 0.551388 Satisfactory
C. Gap Test 0.834301 Satisfactory
D. Poker Test 0.402890 Satisfactory
E. Coupon Test 0.433876 Satisfactory
F. Permutations Test 0.683315 Satisfactory
G. Runs Up Test 0.360567 Satisfactory
H. Max-of-8 Test 0.237868 Satisfactory
I. Lapped M-Tuple Test 0.269342 Satisfactory
DIEHARD
. / p-value 3 4
A. BIRTHDAY SPACING 0.726657 Satisfactory
B. OPERM5 0.719636 Satisfactory
C. BINARY RANK 0.854952 Satisfactory
D. SQUEEZE 0.524926 Satisfactory
E.1 CRAPS: no. of wins 0.280685 Satisfactory
E.2 CRAPS throws/game 0.159625 Satisfactory
F. MINIMUM DISTANCE 0.889075 Satisfactory
1
* % & “Test error”, CHI -
1.

5. G. 3DSPHERES 0.948734 Satisfactory
H. OSUM 0.079572 Satisfactory
I.1 RUNS: Runs up 0.668908 Satisfactory
I.2 RUNS: Runs down 0.860145 Satisfactory
K. CDPARK 0.377942 Satisfactory
11296592 characters in file: file1.bin
HITESTS
. / CHI 3 4
A. Frequency Test 0.505161 Satisfactory
B. Serial Test 0.187295 Satisfactory
C. Gap Test 0.330256 Satisfactory
D. Poker Test 0.616983 Satisfactory
E. Coupon Test 0.081551 Faintly suspicious
F. Permutations Test 0.073267 Faintly suspicious
G. Runs Up Test 0.516541 Satisfactory
H. Max-of-8 Test 0.626726 Satisfactory
I. Lapped M-Tuple Test 0.479882 Satisfactory
DIEHARD
. / p-value 3 4
A. BIRTHDAY SPACING 0.326970 Satisfactory
B. OPERM5 0.959323 Satisfactory
C. BINARY RANK 0.795310 Satisfactory
D. SQUEEZE 0.531174 Satisfactory
E.1 CRAPS: no. of wins 0.623129 Satisfactory
E.2 CRAPS: throws/game 0.862083 Satisfactory
F. MINIMUM DISTANCE 0.904371 Satisfactory
G. 3DSPHERES 0.597925 Satisfactory
H. OSUM 0.843655 Satisfactory
I.1 RUNS: Runs up 0.926601 Satisfactory
I.2 RUNS: Runs down 0.636983 Satisfactory
K. CDPARK 0.095702 Satisfactory
d -
, ' – '
random(), Borland C - -
Elite 730, 510. ) $ ', “ ” -
, & % , -
% %( ). R , -
, ' ( , , , -
% %& )
$ .* ' ) “( ” , R
®
' Dekart Digital Signature System , U -
Dekart Media Pay ®, %
smart-
Payflex 1K RANDOM.SYS. R -
% $ DIEHARD. ( RANDOM.SYS ,
% ( CHITESTS),
, & % smart- , % “ -
” CHITESTS.
R :
1. ' % & %
DIEHARD D *
. ) , & % '

6. ( . .
). RHITESTS D * ' .
2. RHITESTS 9 ,
* 2
; DIEHARD 18
, $ % p-value
[0.025, 0.975], % (“ -
” “ ”)
* ) -R .
3. DIEHARD c -
$ 80 . , RHITESTS -
% - n ps $ 5.
.
( $ & '
( , DSS)
' % ' , -
% “ ” $ .Q -
, %& “ ”
% , %&
% Q*RS &
%& . R , &
& % .S $ $
$,
' . R ' % ) “Dekart” % -
- RHITESTS, DIEHARD.
[1] ) (. P KaU/ d.2 * .- U.:
U , 1977.- 727 .
[2] a a.d., H `.T., H H.T. U $
$ .- U .: a $. e ., 1984.- 527 .
[3] Wegentkittl S. Empirical testing of pseudorandom number generator/ Master’s thesis,
University of Salzburg, Austria, 1995.
[4] Shapira A. The Discrete Runs Test and the Discrete Maximum of t Test. Technical Report
CS 96-15. ESCE Department Rensselaer Polytehnic Institute. 1996.
[5] *
(CHITESTS). b . b . ) “Dekart
S.R.L.”, 1998.
[6] DIEHARD: a battery of tests for random number generators developed by George Marsaglia.
` P : http://stat.fsu.edu/~geo/diehard.html
WeaCheslaw L. Oleinik, Dr. of C.S., Ass.Academician of International Informatization Acad-
emy, Member of the Balcanic Union for Fuzzy Systems and Artificial Intelligence.
Company “Dekart S.R.L.”, Head of Data Security Section of Smart Card Technology
Department.
Born: July 4, 1956
Author: more than 40 printed-papers.

7. Olga M. Petrova, Dr. of C.S., Senior Scientific Researcher.
Company “Dekart S.R.L.”, Leading Specialist of Data Security Section of Smart Card
Technology Department.
Born: February 16, 1966.
Author: monograph and 14 published works.