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Adi Shamir introduced a polynomial-interpolation construction for threshold secret sharing in 1979. In Shamir's secret sharing, a secret is the constant term of a random polynomial of degree less than $k$ over a finite field; shares are its values at distinct nonzero field elements. Any $k$ shares recover the polynomial and hence the secret.^[ref]

Fewer than $k$ shares leave every possible secret equally compatible when the other coefficients are uniformly random. This is an information-theoretic threshold guarantee, rather than a claim that recovering the secret merely takes a long time. The construction makes distributed authorization and resilience to missing shares compatible. It requires enough field elements for all distinct share positions.

In Shamir's secret sharing, observing any $t<r$ shares leaves the secret's prior distribution unchanged. In particular, the posterior distribution is uniform if the prior distribution was uniform; uniformity of the secret is not required for perfect secrecy.

To prove this, fix $t$ distinct nonzero evaluation points and a candidate secret $S=u$. The $t$ evaluations impose $t$ independent linear equations on the $r-1$ random coefficients. The evaluation matrix has rank $t$: its first $t$ columns form a Vandermonde matrix multiplied by a nonsingular diagonal matrix of nonzero evaluation points. Every observed share tuple therefore has exactly $q^{r-1-t}$ compatible coefficient tuples. Since the coefficients were independently uniform, its conditional probability isindependent of $u$. Thus the shares and the secret are independent random variables, which proves perfect secrecy.^[ref]

Shamir's secret sharing distributes a secret $S$ in a finite field ${\mathbb{F}}_q$ so that any $r$ of $s$ shares recover it, while fewer than $r$ shares reveal no information about it. Choose $1\le r\le s\le q-1$, distinct nonzero field elements $x_1,\dots ,x_s$, and independent uniformly distributed coefficients $a_1,\dots ,a_{r-1}$, independent of $S$. Setand give participant $i$ the pair $(x_i,f(x_i))$. Any $r$ shares determine $f$ by polynomial interpolation, hence determine $S=f(0)$. Fewer than $r$ shares have a distribution independent of $S$; this is Perfect secrecy of Shamir's threshold scheme.^[ref]

For example, over the finite field ${\mathbb{F}}_7$, a two-share threshold with secret $3$ and coefficient $a_1=2$ gives $f(X)=3+2X$. The shares $(1,5)$ and $(2,0)$ recover $a_1=2$ and $S=3$. Either share alone is compatible with every secret, and the uniform choice of $a_1$ makes its distribution independent of that secret.