Question

Let \(F\) be a field, \(f \in F[x]\) of degree less than \(n\), and \(u_{0}, \ldots, u_{n-1} \in F \backslash\\{0\\}\) distinct. Determine the set of all interpolation polynomials \(g \in F[x]\) of degree less than \(n\) with \(g\left(u_{i}\right)=f\left(u_{i}\right)\) for \(0 \leq\) \(i \leq n-2\). (In the situation of Section 5.3, this represents the knowledge of all players minus player \(n-1\).) Let \(c \in F\). How many of these \(g\) have constant coefficient \(c\) ? (Your answer should imply that the secret sharing scheme is secure.)

Short answer

Each polynomial of degree less than \(n\) corresponding to any field element as constant coefficient maintains scheme security.

Step-by-step solution

Understanding the Problem

We have a field \(F\) and a polynomial \(f \in F[x]\) of degree less than \(n\). We also have distinct elements \(u_0, u_1, \ldots, u_{n-1}\) in \(F\). We want to find all interpolation polynomials \(g\) of degree less than \(n\) such that \(g(u_i) = f(u_i)\) for \(0 \le i \le n-2\). We also need to determine how many of these polynomials have a constant coefficient \(c\).

Setting Up the Interpolation Problem

The polynomials \(g\) satisfy \(g(u_i) = f(u_i)\) for \(0 \leq i \leq n-2\). This implies \(g - f\) has roots at \(u_0, u_1, \ldots, u_{n-2}\). Let \(h(x)\) be the polynomial such that \(h(x) = (x-u_0)(x-u_1)\cdots(x-u_{n-2})\). Thus, \(g(x) - f(x) = h(x) \cdot q(x)\) for some polynomial \(q(x)\).

Degree Constraints

Since \(g\) and \(f\) have degree less than \(n\) and \(h(x)\) has degree \(n-1\), the degree of \(q(x)\) must be zero, i.e., \(q(x)\) is a constant polynomial. Hence, \(g(x) = f(x) + c \, h(x)\) where \(c \in F\).

Finding Polynomials with Constant Coefficient \(c\)

The constant term of \(g(x) = f(x) + c \, h(x)\) is \(f(0) + c \, h(0)\). To have the constant coefficient \(c\), we need \(f(0) + c \, h(0) = c\). Rearranging gives \(c (1 - h(0)) = f(0)\). Since \(h(0) \) can be zero for multiple values of \(c\) because it could range over each element of the field, this equation can have more than one solution.

Counting the Solutions

The polynomial \(h(x)\) has degree \(n-1\), and since constants \(c\) can take any value in \(F\), we have \(|F|\) possibilities for \(c\). Thus, the number of polynomials \(g\) that satisfy the condition in the question is \(|F| = \text{the number of elements in the field } F\). This shows the secret sharing scheme is secure.

Key concepts

Field Theory

Field theory is a branch of mathematics that focuses on the properties and structures of fields. Fields are foundational to understanding many mathematical concepts, including polynomials.
Fields can be viewed as a set equipped with two operations: addition and multiplication.

• The field must have a set of elements where each pair of elements can be added together, and there is an "addition identity" (denoted 0) where adding it to any element yields that element itself.
• Similarly, every element (except 0) must have a "multiplicative inverse", meaning there is another element that you can multiply it by to get the "multiplicative identity" (denoted 1).

Fields are important in interpolation because they provide the necessary environment for polynomial functions to act over. The coefficients and roots of polynomials used in interpolation often belong to a specific field. In the context of interpolation polynomials, the elements of the field determine possible values for the points and coefficients involved in constructing the polynomials.

Polynomial Roots

Polynomial roots are central when dealing with the interpolation of polynomials. A root of a polynomial is a solution to the polynomial equation; when the polynomial is evaluated at this solution, the result is zero.
For interpolation, particularly in problems like finding a polynomial that fits a set of conditions, roots help in understanding the relationship between polynomials and their values at given points.

• Given a polynomial, roots can be used to create another polynomial that shares some characteristics. For instance, if two polynomials have the same roots, they may differ only by a constant multiplicative factor.
• For the exercise discussed, knowing the roots helps us construct the polynomial "h(x)" that serves as a factor in any interpolation polynomial "g(x)" minus the original polynomial "f(x)".

The roots of the polynomial give constraints on what the interpolation polynomial can be, which is crucial in problems where we interpolate based on known values at specific points.

Secret Sharing Schemes

Secret sharing schemes are methods used to distribute a secret amongst a group of participants, each of whom is allocated a share of the secret.
The secret cannot be reconstructed except by a sufficient number of shares. This concept is closely related to polynomials, where each piece of the secret is seen as a point on the polynomial.

• One common technique is Shamir's Secret Sharing, which uses polynomial interpolation to split a secret into pieces.
• In Shamir’s scheme, a specific polynomial is created using the secret as the constant term. The coefficients of the polynomial are chosen such that each piece of the secret corresponds to the polynomial evaluated at a different point.

The strength of this scheme relies on the properties of the field and the degree of the polynomial. As described in the problem, the knowledge of some values (e.g., every piece except one) does not reveal the secret itself, demonstrating the security of the scheme. This ensures that not all players can access the secret without the required number of pieces.