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3

Secret sharing captures a really wide set of techniques, and "not using secret sharing" is a bit ill-defined. Let me take an example: the seminal solution for two-party computation uses Yao's garbled circuits. It works as follows (very roughly): The sender garbles the circuit of the target function, which gives the following guarantees: given an appropriate ...

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I think you are mis-understanding secret sharing and what it can and cannot do. In a $(t,n)$ secret sharing scheme, the server/trusted party creates $n$ shares of the secret. At least $t$ shares must be present in order to recreate the secret from the shares. Now, if after reconstruction, the secret itself is revealed to the $t$ share holders requesting re-...

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By definition, $\max(a,b)$ is the largest of $a$ and $b$. That's also the meaning in many computer languages, e.g. Python's max. Thus $p>\max(S,n)$ means: $p>S$ and $p>n$.

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XOR secret sharing used directly is unconditionally secure, if the randomly generated first share $S_1$ is a uniformly distributed vector in $\{0,1\}^{128}$. Let the secret be $X \in \{0,1\}^{128}$ and the shares be $S_2=X+S_1,$ and $S_1,$ where $+$ denotes bitwise XOR. For a third party to learn the secret, they need to brute force $O(2^{128})$ guesses in ...

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