How to generate and solve finite fields and insure integers instead of decimals for coefficients and random points

Question

Background: I have tried to follow this tutorial on secret sharing: https://medium.com/@apogiatzis/shamirs-secret-sharing-a-numeric-example-walkthrough-a59b288c34c4. I have managed to use Shamir's secret sharing successfully and by hand I am able to work very simple problems out. I'm trying to convert this into code (using javascript and as few dependencies as possible). However I didn't understand why the author of the post used a finite field ? In Shamir's paper he used normal polynomial interpolation as an example.

Issue: I'm trying to use a large prime number value for L(0) and I had made the conscious decision of avoiding finite fields (as I don't really understand why they are being used, or how to use them properly). The issue is that when I select random points to generate the polynomial I end up with large decimal values and likewise when I generate the secret shards (each shard is one of the l(i)*y_i of the lagrange polynomial interpolation) to distribute they are actually large numbers with many decimal values. Unfortunately I'm failing to successfully decrypt the end result and I believe this is due to the accuracy of big numbers and the limitation of decimals in Javascript (or any language for that matter).

Question: How did the author of the above post generate such clean numbers (no decimals, just really large integer numbers)? Is this related to the use of a finite field ? How can I do this same thing efficiently ?

Detailed Step By Step:

1. Generate large prime
2. Pick k-1 random points
3. Use the large prime and the k-1 random points to generate the coefficients for the polynomial
4. Randomly select k number of x values between 1 to phi (where phi=(p-1)(p-q), where p & q are 2 large primes such that pq=n and n is a key size X bytes long)
5. Evaluate k number of x values on the polynomial for the y value
6. use the X,Y values to create the lagrange polynomial interpolation.

Thanks for this one !!!!!!!

Answer 1

Roundoff error is likely to be bad news if you're trying to do one of these computations with floating-point numbers. It means that floating-point arithmetic doesn't obey the standard laws of algebra: e.g. in general $(a+b)+c$ is not the same as $a + (b+c)$ when roundoff is involved. For any of these computations to work reliably you'll want to use exact integer arithmetic, either with a "bignum" package or with a language that has support for that built-in.