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

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?

• ok maybe my question is not so clear.... Jan 16, 2022 at 11:15

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.