Benefit of using random key in Shamir's Secret Sharing I am implementing Shamir's Secret Sharing, and I find that in (t,n)-threshold scheme, the shares are just using 1,2,3...,n as the key to form the shares (i,f(i)).

For example, I want share "Hello", and the shares look like this:

01-AAAA1,AAAA2,AAAA3,AAAA4,AAAA5
02-BBBB1,BBBB2,BBBB3,BBBB4,BBBB5
03-CCCC1,CCCC2,CCCC3,CCCC4,CCCC5
04-DDDD1,DDDD2,DDDD3,DDDD4,DDDD5


where 01,02,03,04 are the keys and XXXX are the values.

But I find in some place, people are using random numbers as the key. And I want to further extend it by using different key for each character.

For example, the secret is still "Hello", but the share are become:

x1-AAAA1, x2-AAAA2, x3-AAAA3, x4-AAAA4, x5-AAAA5
y1-BBBB1, y2-BBBB2, y3-BBBB3, y4-BBBB4, y5-BBBB5
z1-CCCC1, z2-CCCC2, z3-CCCC3, z4-CCCC4, z5-CCCC5
u1-DDDD1, u2-DDDD2, u3-DDDD3, u4-DDDD4, u5-DDDD5


where x1,x2,x3,x4,x5 are random and distinct.

In this way the only disadvantage is that we need more store space for keys, and it may takes more time to generate random number.

But I wonder whether there is some advantages? Say, Safety issue?

I am new with computer security, hope to get answer. Thanks!
There are no security advantages to evaluating the polynomial at random places instead of sequential. The information theoretic security proof of Shamir secret sharing does not depend on the evaluation points being chosen in any specific manner.