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**Example text**

Un−1 in a field F (say, F is Q or a finite field) such that the ui are distinct, and let f be the polynomial fn−1 xn−1 + · · · + f1 x + f0 , where f0 ∈ F is the secret, encoded in an appropriate way. Then we give vi = f (ui ) = fn−1 un−1 + · · · + f1 ui + f0 to player i. The reconstruction of i the polynomial f from its values v0 , . . , vn−1 at the n distinct points u0 , . . 2). The interpolating polynomial at n points of degree less than n is unique, and hence all n players together can recover f and the secret f0 , but one can show that any proper subset of them can obtain no information on the secret.

The final remainder is r = 15x + 8. The degree of q is deg a − deg b if q = 0. The following algorithm formalizes this familiar classical method for division with remainder by a polynomial whose leading coefficient is a unit. 5 Polynomial division with remainder. Input: a = ∑0≤i≤n ai xi , b = ∑0≤i≤m bi xi ∈ R[x], with all ai , bi ∈ R, where R is a ring (commutative, with 1), bm a unit, and n ≥ m ≥ 0. Output: q, r ∈ R[x] with a = qb + r and deg r < m. 1. 4. Division with remainder 2. for i = n − m, n − m − 1, .

Al−1 ) of a positive integer a, with a = ∑0≤i