How to convert from small field to large field If I have number $2$, which is additively shared into $3+4 \bmod 5$.
Now I need to change the filed to a large one, like $500$.
How to convert the shares, so that I can have a new number or shares $a+b \bmod 500 = 3+4 \bmod 5$.
I think that what the OP is thinking about is a $(2,2)$ additive secret-sharing scheme in which a secret $S$ is split into two shares $X\neq 0$ and $S-X\neq S$. Here, $X$ is an arbitrarily chosen nonzero integer. Neither share holder can determine the value of $S$, but when both shares are submitted to the trusted reconstructor, the reconstructor can recover $S$ as the sum of the two shares.  In the OP's scheme, $S=2$ is split into shares $3$ and $4$ where right-handed folks can assume that $X=3, S-X=4$ while left-handed folks can assume that $X=4, S-X=3$ and all of these elements are assumed to be elements of $\mathbb F_5$. Thus, the secret $S = (X) + (S-X)$ (which has value $7$ in ordinary integer arithmetic) is also  an element of $\mathbb F_5$ and works out to be the correct value $2 \in \mathbb F_5$.
The OP then wonders

Assuming that the secret $S$ continues to have value $2$, what should the share values $a$ and $b$ be so that $a+b = 2\bmod 500$?

Note that we do not necessarily have to have a field of order $500$ (as the OP mistakenly assumes exists) for this simple additive secret-sharing scheme; the additive group $\mathbb Z_{500}$ will work just fine. So, the answer to the OP's query is that the share values can be any two integers whose sum is $2$ modulo $500$. For example, the shares could be $(a,b) = (3,499)$ or $(a,b) = (498,4)$ if we choose to keep one of the two share values the &quot;same&quot; as before; with quotation marks around same to remind folks that $3,4,498, 499$ are elements of  $\mathbb Z_{500}$, not $\mathbb F_5$. It is, of course, an added bonus that $a+b\mod 500 = a+b \mod 5$. But other choices would work equally well; $(a,b) = (251,251)$ will stop juvenile bragging rights arguments among share holders (&quot;neh, neh, ni, neh, neh, my share is bigger than yours!&quot;).