This is a trivial question, but I had to ask:
since each generated share in a Shamir's secret sharing scheme initially consists of a pair of values (representing the coordinates of a point on the plane), how do I reduce the two values to one, so that the share that I eventually hand out will consist of only one value?
The way this is normally done is:
Share #1: (1, 65428)
Share #2: (2, 935747)
Share #3: (3, 3524)
But this does not seem very satisfactory to me, because the share still consists of two values, as the share's number (which could be seen as a metadata) is still one of two values. This way of doing is also inconvenient in the sense that if a share is marked as Share #7, this normally implies that there are at least 7 shares, possibly more.
The solution I am looking for is a way to have shares with totally random x values, such as
and no serial number on a share (e.g. Share #8), so that by intercepting one share, and adversary cannot even try to guess how many shares there are. Of course, the share generating mechanism is exactly the same. The only difference is that the x values, instead of being 1, 2, 3, 4… are random numbers.
So, now the question is: assuming I decide to use random x numbers, how can I conflate the two values of each share into one value, also assuming that no external (metadata) markings of the share (e.g. share #3) will be used?