# Lagrange Interpolation

This Julia code demonstrates the implementation of Lagrange interpolation over a modular field, a crucial component in cryptographic schemes such as Shamir's Secret Sharing. The implementation showcases the flexibility and composability of Julia's ecosystem by seamlessly integrating the external `Polynomials`

package with the custom modular field arithmetic provided by `CryptoGroups`

.

`CryptoGroups`

handles the finite field arithmetic, while `Polynomials`

manages the polynomial operations, without creating a direct dependency between the two. The result is a concise yet powerful implementation that can be easily adapted for various cryptographic applications. The example includes a test case that reconstructs a secret (the constant term of the polynomial) using Lagrange interpolation, illustrating its practical application in secret sharing schemes.

```
using Test
using CryptoGroups.Fields
using Polynomials
function lagrange_interpolation(x::Vector{T}, y::Vector{T}) where T
n = length(x)
result = Polynomial{T}([0])
for i in 1:n
term = Polynomial{T}([1])
for j in 1:n
if i != j
term *= Polynomial{T}([-x[j], 1]) / (x[i] - x[j])
end
end
result += y[i] * term
end
return result
end
p = 23
secret = 3
poly = Polynomial{FP{p}}([secret, 5, 2, 5])
x = FP{p}[2, 4, 6, 8]
y = poly.(x)
interp_poly = lagrange_interpolation(x, y)
@test value(interp_poly(0)) == secret
```

`Test Passed`

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