Ch12. Secret Sharing Schemes
Imagine that you have made billions of ￡$ from
Internet stocks, and you wish to leave your estate
to your 4 children. You like to divide it among them
in such a way that two of them have to get
together to reconstruct the real combination, i.e.,
someone who wants some of the inheritance must
somehow cooperate with one of the other children.
(t,w)=(2,4) - threshold scheme
Definition
Let t≦w be positive integers.
A (t,w)-threshold scheme is a method of
sharing a message M among a set of w
participants such that any subset
consisting of t participants can reconstruct
the message M, but no subsets of smaller
size can reconstruct M.
Shamir threshold scheme in 1979
Based on Lagrange interpolation polynomial
Also called Lagrange interpolation scheme
Choose a large prime p, the message M is
represented as a number (mod p)
s(x)≡M+s1x+s2x2+…+st-1xt-1 (mod p)
(xi,yi), i=1,2, …, w; yi≡s(xi) (mod p)
Lagrange Interpolating Polynomials
Suppose that the function y=f(x) is known at the
n+1 points (x0,y0), (x1,y1), … (xn,yn), where a≤
x0 <x1 <x2 …< xn≤b, then there is a polynomial
Pn(xi)=yi, 0≤i≤n
 n x  xi 
Pn ( x)   yk Ln,k ( x)   yk  
 mod p
k 0
k 0
i 0,i  k xk  xi 
n
n
Computing Secret Value M
 t
 xi 
M   yk  
 mod p
x

x
k 1
i

1
,
i

k
k
i

t
Simple Exercises from p.303-306
2. You set up a (2,30) Shamir threshold scheme,
working mod the prime 101. Two of the shares
are (1,13) and (3,12). Another person received
the share (2,*), what is the value of *?
3. In a (3,5) Shamir secret sharing scheme with
modulus p=17, the following were given to Alice,
Bob, Charles: (1,8), (3,10), (5,11). Calculate the
corresponding Lagrange interpolating
polynomial, and identify the secret.
(Secret) Value Sharing
• A (k, n) threshold secret sharing should
satisfy the following requirements:
(1) A secret value M is used to generate n
shadows.
(2) Any ≧k shadows can reconstruct the
secret value M.
(3) Any ＜k shadows can not get sufficient
information to reveal the secret value M.
Secret Sharing (1/4)
• A (k, n) threshold polynomial can be written by
s(x) ≡ M+s1x+s2x2+…+sk-1xk-1 (mod p)
• Select n distinct integer x1,x2,…,xn form [0,p-1]
• Deliver (xi,s(xi)) to the i-th participant
p= a (large) prime number
M: secret value
s1,…,sk-1: randomly chosen from [0, p-1]
Secret Sharing (2/4)
• To reveal the secret value M, we must collect
(at least k) ≧k shadows.
• Without loss of generality, we use
(x1,s(x1)),…, (xk,s(xk)) as k shadows.
• We can reveal the secret value M by using
Lagrange interpolation.
k

x  xi 
s ( x)    s ( x j ) 
 mod p
j 1 
i 1,i  j x j  xi 


k
• where M=s(0)
Secret Sharing (3/4)
Example:
• (k, n)=(2, 4)-threshold secret sharing
• M=9,p=17
• Given x1=1,x2=2,x3=3,x4=4
• A polynomial equation can be defined as
s(x) ≡ 9+13x mod 17
• Then s(1)=5, s(2)=1, s(3)=14, s(4)=10
• Four shadows: (1,5), (2,1), (3,14), (4,10)
Secret Sharing (4/4)
Example
• We can get the equation by taking (1,5), (4,10)
by using Lagrange interpolation.
 x 1   x  4 
sx   10
  5
 mod17
 4 1   1  4 
• s(0)=9
Secret Image Sharing
• A (k, n)-threshold secret image sharing
msut satisfy the following requirements:
(1)The secret image S is used to generate n
shadow images.
(2)Any ≧k shadow images can reconstruct
the secret image.
(3)Any ＜k shadow images can not get
sufficient information to reveal the secret
image.
Generation of Shadow Images
Pre-operation
Scramble a Secret Image
Pre-operation:
• All gray levels are in the range [0,255]
• Let p=251
• Suppress all values larger than 250 to 250.
-The values are in the range 0~250
-A Lossy method
• Select a key P to create a permutation matrix.
-To decrease the correlation between any
neighboring pixels.
Meaningless Shadow Images
Share
Shadow Images Acquisition
• The equation can be written by
g(x)≡a0+a1x+a2x2+…+ak-1xk-1 mod 251
• Select n distinct secret keys x1,x2,…,xn.
• Deliver (xi,s(xi)) to the ith participant.
Get the Permutation Image
Secret Image Reconstruction
• Without loss of generality, we have
(x1,s(x1)),(x2,s(x2)),…, (xk,s(xk)).
• Use Lagrange interpolation to reconstruct
the image.
Flowchart of Recovery
An Example Without Permutation
Experimental results
The (2,4)-threshold
on image Lenna
512x512,
Histogram of Lenna
The secret image
(1,s(1))
(2,s(2))
The permutation image
(3,s(3))
(4,s(4))
Security Analysis
• Without loss of generality, if we only have (k-1)
shadow images.
 y1 ≡ (a0+a1+…+ak-1) mod p

 y2 ≡ (a0+2a1+…+2k-1ak-1) mod p

…


 yk-1 ≡ (a0+(k-1)a1+…+(k-1)(k-1)ak-1) mod p


The probability to get the right image is
 1 


251


512 x512
k
Property and Conclusion
• P=251
• A lossy method
• The size of each shadow image is 1/r of the
secret image
•
•
•
•
Fault-tolerance
Use network hard disks for storage.
Steganography
…
References
• W. Trappe and l.C. Washington,
Introduction to Cryptography with Coding
Theory, Pearson International Edition
(2006)
• C.C. Thien and J.C. Lin, “Secret image
sharing,” Computers & Graphics, vol. 26,
no. 1, 765-770, 2002.