3 Some Secret Sharing Schemes and Further Studies - Dr Avishek Adhikari

8/13/2019 3 Some Secret Sharing Schemes and Further Studies - Dr Avishek Adhikari

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Some Secret Sharing Schemes and Further Studies

Avishek Adhikariwebsite: www.imbic.org/avishek.html

Research Team MembersPartha Sarathi Roy, Angsuman Das,

Ushnish Sarkar, Sabyasachi Dutta

Department of Pure MathematicsUniversity of Calcutta, Kolkata.

TWC 2011Avishek Adhikari (University of Calcutta) Secret Sharing Schemes 16.07.11 1 / 39
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Introduction to Secret Sharing What is secret sharing?

What is secret sharing?

Avishek Adhikari (University of Calcutta) Secret Sharing Schemes 16.07.11 2 / 39
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I d i S Sh i Wh i h i ?
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(t,w)threshold scheme

Lettandwbe two positive integers, such thattw.A(t,w)threshold schemeis a method of sharing a

scheme keykamong a set ofwparticipants in such away that anytparticipants can compute the value ofk, but no group of(t 1)participants can do so.


I t d ti t S t Sh i Wh t i t h i ?
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Simple Way!


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Perfectly Secure (2,2)-SSS


Introduction to Secret Sharing Shamirs (t n) threshold scheme
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Introduction to Secret Sharing Shamir s (t, n) threshold scheme

Shamirs(k,n)-Secret Sharing Scheme

It takestwo pointsto define astraight line,three pointsto fully define aquadratic,four

pointsto define acubic, and so on.One can fit a unique polynomial of degree(k 1)to any set of kpoints that lie on thepolynomial.


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Shamirs(k,n)-Secret Sharing Scheme

It takestwo pointsto define astraight line,three pointsto fully define aquadratic,four

pointsto define acubic, and so on.One can fit a unique polynomial of degree(k 1)to any set of kpoints that lie on thepolynomial.


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Shamirs (3, 4) threshold scheme

LetP={P1, P2, P3, P4}be a set of 4participants.

The key set,K=Zp, wherep=5 is a prime &p>nKet the secret be 1.

The set of all possible shares,S=Z5.

The dealer constructs a random polynomial

f(x) Z5[x]of degreet 1=3 1= 2, inwhich the constant term is the secret K =1.

f(x) =1 + 2x+ 3x2


Introduction to Secret Sharing Shamirs (t, n) threshold scheme
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Shamirs (3, 4) threshold scheme

Every participantPiobtains a point(xi, yi)on this polynomial,

whereyi=f(xi)and distinctxiZ

p.P1 gets (1,a(1)=6=1),(P2)gets (2,2),P3 gets(3, 4)andP4 gets(4,2).

Recovery of Secret

Suppose a subset B of t=3 participants wants to recollect thesecret.

Let the participantsP1, P2, P3 want to determineK =1.

They know that 1=f(1), 2=f(2)and 4=f(3).

They will assume the form of the secret polynomial asy=f(x) =a0+ a1x+ a2x

2, wherea0, a1 anda2are unknown and

belong to Z.

Thus, these participants can obtain 3 linear equations in the 3

unknowns a0, a1, a2.


Introduction to Secret Sharing Shamirs (t, n) threshold scheme


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g ( , )

Shamirs (t, n) threshold scheme

1 1 1

2

1 2 22

1 3 32

a0a1

a2

=

f(1)f(2)

f(3)

Now, the coefficient matrix Ais the so calledVandermondesmatrix.

detA=

1j


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g ( )

Shamirs (t, n) threshold scheme

1 1 1

2

1 2 22

1 3 32

a0a1

a2

=

f(1)f(2)

f(3)

Now, the coefficient matrix Ais the so calledVandermondesmatrix.

detA=

1j


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Example of(2, 2)-VCS


Visual Cryptography Shamirs Scheme
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Example of(2, 2)-VCS


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Visual Cryptography

TheVisual cryptographic scheme, introduced byNaorand Shamirin1994, for a setPofnparticipantsis a cryptographic paradigm that enablesa secret image to be split intonshadow images calledshares, where eachparticipant inPreceives oneshare. Certain qualified subsets of participants canvisually" recover the secret image with some loss ofcontrast, but other forbidden sets of participants have

no information about the secret image.


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(2,2)-VCS

For Black pixel

For White pixel

S1 =

0 1

1 0

andS0 =

0 1

0 1

.



( )
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(2,2)-VCS

For Black pixel

For White pixel

S1 =

0 1

1 0

andS0 =

0 1

0 1

.



(2 2) VCS
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(2,2)-VCS

For Black pixel

For White pixel

S1 =

0 1

1 0

andS0 =

0 1

0 1

.



R l ti t t


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Relative contrast

Let us consider a(2,n)-VCS on a setP={1,2, . . . , n}ofnparticipants with basis matricesS0 andS1 and

having pixel expansionm. Then therelative contrast

for the participants corresponding toX,X P, isdenoted byX(m)and is defined as

w(S1X) w(S0X)

m .



(2 ) VCS b N d Sh i
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(2,n)-VCS by Naor and Shamir

S0 =

1 0 0 0

1 0 0 0

1 0 0 0

1 0 0 0

and S 1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

.

Here therelative contrastfor any two participants is 14and thepixel expansionis 4.



(2 ) VCS b Sti t l
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(2,n)-VCS by Stinson at el

Letv, kandbe positive integers such that

v>k2.a(v, k, )-balanced incomplete block design

(BIBD) is a pair (X,A) such that the followingproperties are satisfied :

1 Xis a set of v elements called points,2 Ais a collection of subsets of Xcalled block,3 each block contains exactly k points,and4 every pair of distinct points is contained in exactly blocks.



(2 n) VCS by Stinson at el
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(2,n)-VCS by Stinson at el

Example of a(v=7,b=7, r=3, k=3, =1)-BIBD:

X ={1,2,3,4, 5,6,7}.

A={{1,2,4}, {2,3,5}, {3,4,6},{4,5,7}, {1,5, 6}, {2, 6,7}, {1,3,7}}.

the incidence matrix is :

1 0 0 0 1 0 11 1 0 0 0 1 00 1 1 0 0 0 1

1 0 1 1 0 0 00 1 0 1 1 0 00 0 1 0 1 1 00 0 0 1 0 1 1



Example of a (2 7) VCS using BIBD
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Example of a (2,7)-VCS using BIBD

S1 =

1 0 0 0 1 0 11 1 0 0 0 1 00 1 1 0 0 0 11 0 1 1 0 0 0

0 1 0 1 1 0 00 0 1 0 1 1 00 0 0 1 0 1 1

& S0 =

1 1 1 0 0 0 01 1 1 0 0 0 01 1 1 0 0 0 01 1 1 0 0 0 0

1 1 1 0 0 0 01 1 1 0 0 0 01 1 1 0 0 0 0

Here thepixel expansionis 7 and the relative contrast

is 2/7.


Visual Cryptography Our Proposed Scheme

Example of a (2 6) VCS
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Example of a(2,6)-VCS

S1 =

1 1 0 01 0 1 01 0 0 10 1 1 00 1 0 10 0 1 1

andS0 =

1 1 0 01 1 0 01 1 0 01 1 0 01 1 0 01 1 0 0

.

Clearly,pixel expansionis 4 (m=4) and therelative

contrastis either 1

2 or 1

4 .


Visual Cryptography Our Proposed Scheme

(2 6) VCS using PBIBD


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(2,6)-VCS using PBIBD

Secret image Share 1

Share 2 Share 6

Share 1 + Share 6 Share 1 + Share 2



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DNA Secret Sharing

Introduction


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Introduction

Secret Sharing Schemes,

DNA Computing,Mathematics.

Statistics.


DNA Secret Sharing

Introduction
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Introduction



Statistics.


DNA Secret Sharing

Introduction
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Introduction



Statistics.


DNA Secret Sharing

Introduction
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Introduction



Statistics.


DNA Secret Sharing

Aim of Our Work
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Aim of Our Work

Suppose we want to distribute asecret binary stringto a set{P1,P2, . . . ,Pn}ofnparticipants in such a

way that certain designated set of participants canreveal the secret by pulling their shares, but noforbidden set of participants hasany informationabout the secret binary string.


DNA Secret Sharing

Why DNA?
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Why DNA?

The very small size,

The huge storage capacity,Easy to carry or hide,

Made up of A, T, G, C,

Huge parallel computing,Stable as a DNA double

strand,

High longevity,Easy to get synthesizedDNA


DNA Secret Sharing

DNA encoding of binary strings
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e cod g o b a y st gs

a binary string can be represented as a set of

integers that corresponds the positions where thebits are 1 from left to right.

1011 can be represented as a set {1,3,4},

each integerican be represented in a DNAdouble strand notation as follows

dsi=(GAATT)i.if=1011, the DNA double strand representation

ofis the test tubeT[] ={ds1, ds3, ds4}.


DNA Secret Sharing

Mixing operation
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g p

Take the content of two test

tubes.Mixing can be done bydehydrating the tubecontents (if not already in

solution) and thencombining the fluidstogether into a new tube, bypouring and pumping.


DNA Secret Sharing

Bio-mathematical operations
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p

Booleanoroperation between

two binary stringsif=1011 and=1001,

the binaryorof two strings will

be 1011.T[](T[]) the test tubecorresponding to the binary string

().

pore the contents of the two testtubes to get binaryor.


DNA Secret Sharing
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Automated DNA Sequencing

To read the DNA double strands in the test tubewe need the process called DNA sequencing.


DNA Secret Sharing Proposed DNA Secret Sharing Scheme

Outlay of our scheme
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y

Consider the secret sharing scheme on

P={1,2,3,4}of 4 participants, where0={{1,2}, {2,3}, {2,4}, {1,4}}.

Qual={Y P :XYfor someX0}and

Forb=2

P

\ Qual.let the secret binary string bex=x1x2x3=011.Assume that the DNA encoding, the mixingprocess are public.



Share Distribution
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The dealer chooses two Boolean matrices

G0=

0 1 0 10 1 0 1

0 1 0 0

0 0 0 1

andG1=

1 0 1 00 1 1 0

1 0 0 0

0 0 0 1

.

Sincex1=0, the dealer considers the matrixG0and apply a random permutation to the columns of

G0 and produces a matrix M1.

Similarly, forx2 andx3 onG1 to produce matricesM2andM3.Assume that the DNA encoding, the mixingprocess are public.



Share Distribution
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LetM=M1||M2||M3.

first row ofM is1=010110101010. Similarly2=010101100110,3=010010001000 and4=000100010001.dealer converts the binary strings to DNArepresentations to get the test tubesT[1] ={ds2, ds4,ds5,ds7,ds9,ds11},T[2] ={ds2, ds4,ds6,ds7,ds10,ds11},T[3] ={ds2, ds4,ds5,ds9},

T[4] ={ds4, ds8,ds12},



Share Distribution
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T[i]is given to the participants Pi,i=1,2, . . . , n

through a secret channel.Also the valuesm=4 andk=3 are given to theparticipants even through an insecure channel.



Decryption by the Qualified participants
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LetP1,P2come together.

They use mixing procedure with test tubesT[1]andT[2]to getT[1] T[2] ={ds2,ds4, ds5,ds6,ds7,ds9,ds10,ds11}.

Execute automated DNA sequencing method toread the DNA double strands.With the knowledge of decoding the DNArepresentation to the binary string, the values of

k=3 andm=4, the participantsP1 andP2canconvert the DNA representation to the binarystringy=010111101110.



Decryption by the Qualified participants
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Since, the value ofmis known to the participants,

P1 andP2can breakyasy= (0101)(1110)(1110).

Next they will find the value ofwas 3 and thenthey will computez=011, asBW(0101)


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LetY ={P3,P4}come together.

They use mixing procedure with test tubesT[3]andT[4]to getT[1] T[2] ={ds2, ds4,ds5, ds8,ds9,ds12}.

they will convert the DNA representation to thebinary stringy= (0101)(1001)(1001).Thus looking at those it is not possible to predictwhether they correspond to 0 or 1.


Further Studies Secure Multiparty Computation

Secure Multiparty Computation: An Example
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Yaos Two Millionaires Problem

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Further Studies Verifiable Secret Sharing

Questions


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Questions???


Further Studies Verifiable Secret Sharing
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3 Some Secret Sharing Schemes and Further Studies - Dr Avishek Adhikari

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