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USING ORTHOGONAL PIECEWISE CONSTANT BASIS FUNCTIONS IN SHAMIR THRESHOLD SCHEME

Kh. Maleknezhad, M. Shahrezaee & M. Falah. Aliabadi

Iran University of Science and Technology, maleknejad@iust.ac.ir

Abstract: In Shamir threshold scheme one that is called dealer, chooses the key and then shares some partial information about it, called among the participants, secretly. In this paper, we use some numerical methods with piecewise constant basis functions in Shamir threshold scheme. We first introduce operational matrix of this functions and then show how dealer multiplies this matrix by vector of shares to obtain a new vector and distributes it.

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(shamir)

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>,
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⎩1i = j
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i=1
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1268732040128

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Downloaded from ijiepm.iust.ac.ir at 16:22 IRST on Saturday November 4th 2017

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Q R%- >, HK >,
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R D H1≤ i ≤ w .6 R$ ` (1≤ i ≤ w)xi ‘ $ R R xi ) R $ pi + xi 8
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t_4$#g = d ( k∈zp) k %- 67 8 D (5
.’- $ ST 9T z p a1,…,at−1g’>
:- ‘- $ 1 2 (1≤ i ≤ w), yi = p( )xi 8 D (N
P( )x = K +∑t−1ai x j (mod p)
j=1
.R $ Pi yi 6 D,1≤ I ≤ w (O

‘* + (D )! ” H – $ DRE o ‘ =d F t_4& c-W – P( )x ‘ %& R ” F %- R FM %& – $ $#g
.3 $ m% %& ‘* (xi , yi )p” + Pi t Q >, + <* 6 6RP $ ‘-
Pi1 ,….,Pit : ‘- b# . k %- ‘$
– ‘ $ ”- J k %- ‘RT zp[ ]x %& ‘* + P( )

, yi j = p(xi j )
* ( $ D G8# ” F $9T %& ‘* ) .F
=d $ P( )x KJ F t_4& c-W P( )x : F

P( )x = a0 +a1x+a2x2 +…+at−1xt−1 (4_4)
#$ (shamir) !”
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D9 [O”N] K@J_^Y $%> K RX c
:$ q =d K .F D

EB

m∑−1Δi ⎤⎥=

1 (I +Δ)(I −Δ)−1 m⎣2 i=1 ⎦ 2m
: –
Δm×m = ⎢⎡0I(m−1)(m−1)⎤⎥
⎣00⎦

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yi1 =1/1yi2 =2/2− yi3 =3/9

> $R D 67 H k %- #
:6E=[1/1,2/2−,3/9]T $’ H6′- $ Z1

B=[12,−24,42]T

$ D9 3×3 K@J _^Y $%> K +’ .’ F $I” 6′-

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:F =d PCBF
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c = f ± g

c = [c1,c2,…, cm ]T ci =< c( )t ,θi ( )t > f = [f1, f2,…, fm ]
g = [g1, g2,…, g m ]

🙁

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D< c( )t = f ( ) ( )t g t 3
c = f ⊗ g
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g( )t c = f ÷ g

$88W @Q + + 3 : . :6 D S0
kf ( )t ≈∑m (kfi ) ( )θi t = k∑m fiθi ( )t
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: # kf ( )t = kFTθ( )t

:< L “” X< : t # $” %& ‘*+” .!

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. $ `;# H @” F17 <; 67 F’ $ ! ” H U3″ L2@ ‘uR HDA” $%> K I P & K R F%I *) R ” K + ‘ ZI + ” K – F R PCBF $%>
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.F

Shamir, A., How to share a secret, comm. Of the ACM 22 1979, 612-613.

CRYPTOGRAPHY Theory and Practice, Douglas R.
Stinson, University of Nebraska, 1995, Lincoln.

Ganti, Trasada Rao, Piecewise constant orthogonal functions and their application to system and control, 1983, Stringer-Verlag.

Razzaghi, M., Nazarzadeh, J., Walsh functions, Wiley encyclopedia of electrical and electronics in engineering, 23, 1990, 429-440.

⎡1⎤

11 ⎥
482404130358

1093916130358

31 ⎢⎢⎢02211 ⎥⎥⎥⎡⎢⎢12−24⎤⎥⎥ = ⎡⎢⎢108 ⎤⎥⎥
⎢1⎥⎢⎣42 ⎥⎦⎢⎣7 ⎥⎦
⎢00


1268732040128Downloaded from ijiepm.iust.ac.ir at 16:22 IRST on Saturday November 4th 2017

Downloaded from ijiepm.iust.ac.ir at 16:22 IRST on Saturday November 4th 2017

⎢⎣2⎥⎦

:6 ‘
yi*1 =8y

:6 xi3 =5 , xi2 =3 , xi1 =1 ” 5_4! ‘- p

! ”
647702-170095 pn( )( )

:6x=0 I K= − + − − =8 1 9 1( )( 3) 1 mod( 11)

‘- b# _5Xc
‘- b# ‘uR . (1≤i ≤10)xi =i2 ,w=8 , t =5, p =11
6R ” ER 6 I Q={P1,P2,P3,P6,P7,P8,P9,P10} 6 3 .’ k X, %- ‘RT

x1=1 , x2=4 , x6=13 x7=3 , x9=12, x10=8 ,
yi2=28/12− yi3=18/18 yi5=7/15− yi6=19/11 yi8=13/21 x4=16 x8=18,
yi1=27/3 yi4=6/22− yi7=22/6−

> $R D 67 Hk %- # :F 6RP H6’- $ Z1
B=[624,−656,432,−160,−176,448,−512,320]T
3629994546913 D9 8×8 K@ J– ^Y $ %> K + ‘
A.C., Prindle, weber and Schmidt, 1978, Boston, Masschvsetts. ⎡1

1 1 1 : F $I” 6’-$

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[5] Numerical Analylis, Burden, R.L., Faires, T.D., Reynolds,

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⎢111⎥⎢⎥ ⎢ ⎥
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⎢⎢00⎥⎥⎢⎢−512⎥⎥ ⎢⎢8 ⎥⎥
000
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قیمت: تومان


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