How to define a cryptosystem when the encryption-decyrption scheme is based on Shamir's secret sharing scheme?

Question

I would like to make a parallelism between Shamir's secret sharing scheme and how to define a cryptosystem where the encryption scheme is based on secret sharing. To begin with I do not know if there can be such an analogue.

Suppose that we have a standard cryptosystem. Mathematically, a cryptosystem or encryption scheme can be defined as a tuple $(\mathcal {P},\mathcal {C},\mathcal {K},\mathcal {E},\mathcal {D})$. Also, I provide some details about the Shamir's secret sharing scheme starting with the next theorem that determines the intuition of the whole theorem.

$\textbf{Theorem:}$ Let $p$ be a prime, and let $\{(x_1,y_1), . . . ,(x_{t+1},y{t+1})\}\subseteq\mathbb{Z}_p$ to be a set of points whose $x_i$ values are all distinct. Then there is a unique degree-$t$ polynomial $f$ with coefficients from $\mathbb{Z}_p$ that satisfies $y_i \equiv_p f(x_i)$ for all $i$ (I would add to the theorem where $s=f(0)$).

As we already know in a $k$ out of $n$ secret sharing scheme, each agent splits the secret in $n$ parts however only $k=t+1$ parts (of a polynomial of degree $t$) are needed if we want to compute the secret. Suppose that $f$ is the polynomial function such that

$$f(x)=a_tx^t+a_{t-1}x^{t-1}+\cdots+a_1x+a_0=s+\sum_{i=1}^ta_ix^i,\quad\text{such that $y_i \equiv_p f(x_i)$ and $s=f(0)$}\quad (1)$$

I have the following questions:

1. Does $y_i \equiv_p f(x_i)$ mean $y_i\equiv f(x_i)(mod{p})$? Can we make calculations with the $y_i'$s like $y_1+...+y_{t+1}\equiv_{p}(f(x_1)+...f(x_{t+1})$? And if we can sum all $y_i$ does this mean that we obtain $s$?
2. If we want to make a parallelism with the classic cryptosystem what could we define as the cipher-text $\mathcal{C}$ the keys $\mathcal{K}$, the encryption-decryption functions?

Let me put it simply. What is supposed to be the encrytoion-decrytpion scheme here? For example in a simple cryptosystem the agent needs the key to decrypt the message. In this case we have a $t$ out of $n$ scheme. What could we define as encryption and what as decryption process here?

Answer 1

A secret sharing scheme is not a classical encryption scheme. So I don't think one can cast it in any meaningful way into that framework.

As for your other questions if you are operating in the finite field with $p$ elements where $p$ is a prime all computations are modulo $p$.

That $\equiv_p$ notation is a horrible notation but I presume it stands for equality modulo $p.$

The equation below $$y_1+...+y_{t+1}\equiv_{p}f(x_1)+...f(x_{t+1})=s$$ will not hold in general, certainly will not hold for a randomly chosen polynomial $f$ which is the whole point of Shamir secret sharing.