On Oct 3, 2009, at 2:42 AM, Kevin W. Wall wrote:

Hi list...I have a question about Shamir's secret sharing.According to the _Handbook of Applied Cryptography_ Shamir’s secret sharing (t,n) threshold scheme works as follows:SUMMARY: a trusted party distributes shares of a secret S to nusers.RESULT: any group of t users which pool their shares can recover S. The trusted party T begins with a secret integer S ≥ 0 it wishes to distribute among n users. (a) T chooses a prime p > max(S, n), and defines a0 = S.(b) T selects t−1 random, independent coefficients definingthe randompolynomial over Zp.(c) T computes Si = f(i) mod p, 1 ≤ i ≤ n (or for any ndistinctpoints i, 1 ≤ i ≤ p − 1), and securely transfers theshare Sito user Pi , along with public index i. The secret S can then be computed by finding f(0) more or less by using Lagrangian interpolation on the t shares, the points (i, Si). The question that a colleague and I have is there any cryptographic purpose of computing the independent coefficients over the finite field, Zp ? The only reason that we can see to doing this is to keep the sizes of the shares Si bounded within some reasonable range and it seems as though one could just do something like allowing T choose random coefficients from a sufficient # of bytes and just do all the calculations without the 'mod p' stuff. We thought perhapsShamir did the calculations of Zp because things like Java'sBigIntegeror BigDecimal weren't widely available when came up with this scheme back in 1979. So other than perhaps compatibility with other implementations (which we are not really too concerned about) is there any reason to continue to do the calculations over Zp ???

`It's nice to be able to give a size limit for the shares. They're`

`going to need to be transmitted and stored. Since there are many`

`primes around, working over Zp ensures that shares about about the`

`same size as the secret.`

`However, there's also a more fundamental problem: In step (b), how do`

`you choose your coefficients randomly over all of Z? There is no`

`uniform probability distribution over Z to work with. Any realistic`

`implementation will choose from some finite subset. But then the`

`scheme may not be completely secure: If you have the value of f() at`

`t-1 points, the fact that the coefficients are limited to some finite`

`set also constrains the possible values at the remaining point - and`

`you don't know exactly how. Working over Zp's group structure ensures`

`that if you have t-1 values, all p-1 possible remaining values are`

`equally likely, so you've learned nothing.`

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