Blakley Secret Sharing

If you want to find out about other secret sharing methods using Shamir’s Secret Shares and with Chinese Remainder Theory, click here.
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Blakley Secret Sharing

If you want to find out about other secret sharing methods using Shamir’s Secret Shares and with Chinese Remainder Theory, click here.

With secret sharing, we distribute n secret shares and then define that t of these shares can rebuild a secret. The value of t is known as the threshold. In this article I will use the example here. Blakley wrote about the method in 1979 [1]:

In Blakley’s method, we define a secret x, and then construct a number of hyperplanes. For example, we could define each share with a unique hyperplane of the form:

z = a x + b y + c (mod p)

We first define the secret value of x, and then select values of a, b and c for the secrets. If we have three secret shares, we would have:

a1 x + b1 y — z = -c1 (mod p) 
a2 x + b2 y — z = -c2 (mod p)
a3 x + b3 y — z = -c3 (mod p)

and to recover the secret, we need to find x. Now we can have a matrix form of:

If we take this example:

z = 4 x + 19 y + 68 (mod 73)
z = 52 x + 27 y + 10 (mod 73)
z = 36 x +65 y + 18 (mod 73)

We can arrange in the form:

We can then solve this for x, y and z with:

import numpy as np
from sympy import Matrix
p=73
A = np.array([[4, 19, -1], [52, 27, -1], [36, 65, -1]]) 
B = np.array([-68, -10, -18]) 

A = Matrix(A) 
A = A.inv_mod(p)
print (A)
x = np.dot(A,B) % p
print (x)

A sample run shows that the secret is 42:

A= [[ 4 19 -1]
 [52 27 -1]
 [36 65 -1]]
B= [-68 -10 -18]
Inverse A (mod p) Matrix([[2, 36, 35], [70, 67, 9], [23, 30, 19]])
x= 42 y= 29 z= 57

A demo is defined here:

https://asecuritysite.com/shares/sss_bl?val1=97

References

[1] Blakley, G. R. (1979, June). Safeguarding cryptographic keys. In 1979 International Workshop on Managing Requirements Knowledge (MARK) (pp. 313–318). IEEE [here].