Parallel generators of pseudo random numbers with control of calculation errors

International Journal of Science and Research (IJSR), India Online ISSN: 2319-7064
Volume 1 Issue 2, November 2012
www.ijsr.net
Parallel Generators of Pseudo-random Numbers
with Control of Calculation Errors
S. Dichenko1
, O. Finko2
Kuban State University of Technology.
Moskowskaya St., 2, Krasnodar 350072, Russia
1
dichenko.sa@yandex.ru,
2
ofinko@yandex.ru
Abstract: Algorithms of parallel realization of classical "congruent" methods of generation of pseudo-random numbers are discussed.
Algebraic (equational) and matrix interpretations of ways of realization of algorithms are presented. In order to achieve high level of
performance of modern and perspective technical tools of cryptographic protection of data, as well as to provide contemporary stochastic
methods of modeling, it is necessary to realize existing "congruent" methods in parallel. In developing solutions for the problems of
high level of complexity, such as cryptographic protection of data, ensuring high levels of security of functioning in real time is of out-
most importance. In this article the means of ensuring of functional diagnosis of the tools for pseudo-random numbers generation by
means of application of classical methods of superfluous arithmetic coding (module control) are briefly discussed.
Keywords: Pseudo-random numbers, congruent method, cryptography, fault diagnosis and tolerance in cryptography.
1. Introduction
Pseudo-random number (PRN) generators have important
applied meaning for the implementation of most crypto-
graphic algorithms and key material generation systems [1] –
[6]. Stricter requirements for the encryption speed and in-
creasing amount of data which needs to be protected cause
the necessity to construct parallel algorithms of PRN genera-
tion, what should be supported by providing the required
level of reliability of their operation [7], [8].
The purpose of the article is constructing parallel algo-
rithms of PRN generation with control of calculation errors
based on the congruent method.
2. Varieties of the congruent method
Varieties of the congruent method, in particular, are ex-
pressed by formulas (1) – (4):
,1 mnn bxx  (1)
,1 mnn cbxx  (2)
,1
m
n
d
n cxbx  (3)
,1211
21
m
n
d
n
d
n cxbxbx   (4)
where m – module, b – multiple, mb 0 ; c – increment,
mc 0 ; nx – the initial value, mxn 0 ; m
 – smal-
lest non-negative residue of number  module m .
In accordance with the purpose, we pose a problem to
construct parallel algorithms of PRN generation with control
of calculation errors. Algorithms of PRN generation are
shown as algebraic and matrix formulas.
3. Development of parallel algorithms
3.1 Algebraic method
Formula (1) can take the following series of algebraic trans-
formations:


















































,
,
,
,
1
3
2
2
23
2
1
12
1
m
n
k
mknkn
m
n
mn
nmnn
m
n
mn
nmnn
mnn
xbbxx
xb
x
xbbbxx
xb
x
bxbbxx
bxx

with ,2,1k or we have:













,
,
,
22
11
mnkkn
mnn
mnn
xx
xx
xx




(5)
where
m
i
i b , ki ,,2,1  .
For (2) it is possible to get:
 
 













































,1
,1
,1
,
1
1
1
23
23
2
1
12
1
m
k
j
j
n
k
mknkn
m
nmnn
m
n
mn
nmnn
mnn
bcxbcbxx
bbcxbcbxx
bcxbc
x
cbxbcbxx
cbxx















,
,
,
222
111
mknkkn
mnn
mnn
xx
xx
xx




(6)
20

www.ijsr.net
where ;1,,
1
1
11
m
k
j
jimm
i
i ccb








 


 
 
 
































,1
,1
,1
,
1
1
1
23
23
2
12
1
m
k
j
j
n
k
m
kn
d
kn
m
dd
n
d
m
n
d
n
m
d
n
d
m
n
d
n
m
n
d
n
bcxbcxbx
bbcxbcxbx
bcxbcxbx
cxbx














,
,
,
222
111
mknkkn
mnn
mnn
xx
xx
xx




(7)
where ;1,,
1
1
11
m
k
j
jimm
di
i ccb








 


 
.,,2,1 ki 













,
,
,
1
21222
11111
mknknkkn
mnnn
mnnn
xxx
xxx
xxx




(8)
where ;1
2
12
m
  miii 2111    ,
when ,4,3i ;
mii 11   , when ,3,2i ;
c1 ;   miii c 11    , when ,3,2i .
3.2 Examples
Let’s have 4310 x , 756b , 2047m .
Find 321 ,, xxx using (5):
,363431756 204701  m
bxx
,130431756
2047
2
0
2
2 
m
xbx
24431756
2047
3
0
3
3 
m
xbx .
Let’shave 4120 x , 531b , 711c , 4093m .
Find 4321 ,,, xxxx using (6):
,2554711412531 409301  m
cbxx
 
  ,21021531711412531
1
4093
2
10
2
2


m
cxbx 
  ,35771531531711412531
1
4093
23
2
1
0
3
3









 

m
j
jcxbx 
  .9461531531531711412531
1
4093
234
3
1
0
4
4









 

m
j
jcxbx 
3.3 Matrix method
Let’s present the considered algorithms of PRN generation in
matrix form.
,2
1
m
n
k
mn
x
x





















BX
(9)
where
m
i
i b ; .,,2,1 ki 
,2
1
2
1
mk
n
k
mn
x
x






































GBX
(10)
where
.,,2,1;1,,
1
1
11 kiccb
m
k
j
jimm
i
i 







 


 
,2
1
2
1
mk
n
k
mn
x
x






































GBX
where ;1,,
1
1
11
m
k
j
jimm
di
i ccb 







 


 
ki ,,2,1  .
,
2
1
1
2
1
2
1
1
mk
n
k
n
k
mnn
xx
xx


























































GBAX
where ;1
2
12
m
  miii 2111    ,
when ,4,3i ;
mkk 11   , when ,3,2k ;
c1 ;   mkkk c 11    , when ,3,2k .
3.4 Examples
Let’s have 431nx , 756b , 2047m .
Find 321 ,, xxx using (9):
21

www.ijsr.net
.
24
130
363
431
756
756
756
2047
3
2





















 mnxBX
Let’shave 412nx , 531b , 711c , 4093m .
Find 4321 ,,, xxxx using (10):
 
 
 
.
946
3577
2102
2554
1531531531711
1531531711
1531711
711
412
531
531
531
531
23
2
4
3
2

















































m
mnx GBX
4. Control of calculation errors
Let’s use ,qmM  as a composite module in calculating
any of the obtained systems of expressions (5) – (8), where
mq  module introduced for the implementation of control
of calculation errors.
Then each of the expressions (5) – (8) can be rewritten as
a system of expressions on module M , for example (7):

















,
,
,
222
111
M
knkkn
M
nn
M
nn
xx
xx
xx




(11)
where ;1,,
1
1
11
M
k
j
jiMM
di
i ccb 







 






ki ,,1  .
Control digits are calculated by the system:


















,
,
,
222
111
q
knkkn
q
nn
q
nn
rr
rr
rr




, (12)
where ;1,,
1
1
11
q
k
j
jiqq
di
i ccb 







 






.,,1 ki 
Thus the result of (11), (12) is a redundant arithmetic
code, represented by informational 


 knn xx ,,1  and control
vectors knn rr  ,,1  . If the minimal values of the arithmetic
code distance 1min  qd control arithmetic expressions are
as follows:













,0ifis,error
,0ifno,iserror
jn
q
jn
jn
q
jn
rx
rx
where kj 1 .
The final result is:
,11
m
nn xx 
  ,22
m
nn xx 
  , .
m
knkn xx 
 
5. Conclusion
Developed algorithms of accurate parallel PRN generation
provide the required level of perspective highly productive
cryptographic means of protection of data.
References
[1] A.V.Babash, G.P.Shankin, “Cryptology”. SOLON-
PRESS. 1997. (book style)
[2] Henk C.A. van Tilborg, “Fundamentals of Cryptology”.
KLUWER ACADEMIC PUBLISHERS.2000. (book
style)
[3] B.Schneier, “Applied Cryptography”. John Wiley &
Sons, Inc. 1996. (book style)
[4] B.А.Forouzan, “Cryptography and Network Security”.
McGraw Hill. 2008. (book style)
[5] D.E.Knuth,“The Art of Computer Programming. Vo-
lume 2. Seminumerical Algorithms”. Addison-Wesley.
Redwood City. 1981. (book style)
[6] C.P.Schonorr, “On the Construction of Random Num-
ber Generators and Random Function Generators”.
EUROCRUPT. 1988. (journal style)
[7] “Cryptographic Hardware and Embedded Systems –
CHES 2004”. 6th International Workshop Cambridge,
MA, USA, August 11-13, 2004. Proceedings
[8] “Fault Diagnosis and Tolerance in Cryptography”.
Third International Workshop, FDTC 2006, Yokohama,
Japan, October 10, 2006, Proceedings.
Sergey Dichenko- Is a Graduate student. Research
interests are development of parallel algorithms for
generating pseudo-random numbers and binary se-
quences. Error control algorithms.
Oleg Finko - Professor, Doctor of Technical
Sciences. Professor of Department of computer tech-
nologies and information security of the Kuban State
University of Technology. Research interests ‐a resi-
due number system, the use of error-correcting coding
techniques in cryptography, multi-biometric encryption, digital
signature algorithms improve, secure electronic document systems,
Parallel computing logic by modular numerical polyno-
mials.URL: http://www.mathnet.ru/eng/person/40004
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