In normal base system (say binary), you'll do P = [1,2,4,8, ...] then a value can be represented as coefficients $\{a_i\}$ where $$N=\sum_{i=0}^n a_i\cdot P_i$$
However, I am given a problem where
The base is $P=\{\sqrt{2},\sqrt{3},\sqrt{5},\cdots, \sqrt{101}, \sqrt{103}\}$.
The value $N<12000$ rounded to $2^{-256}$
$a_i < 128,\forall a_i$
$n=27$, meaning it only uses the first $27$ primes.
I am to recover the coefficients $a$.
Can someone give some insights to how this can be approached? An idea I have right now is that I can multiply by say $2^{256}$ and work with integers, then I have to find an array of $a=\{a_1,a_2,\cdots,a_{27}\}$ with every element below 128 that "approximately" makes $n$. I also notice that since $N<12000$ and $\sum P\approx 170.48$, there is a bound 21821 which is pretty close..? Another way might be to use some property about gaps between $P_i$ and $P_{i+1}$?
I think it is similar to the Merkle-Hellman knapsack cryptosystem which can be cracked with lattice reduction, but it is not super-increasing here and I can't seem to find a way to make it so.
I can also model it as a knapsack problem but it either explodes in time complexity (IIRC most algorithms depend on $V$ which is large when you convert to integers) or is approximate and might not find the solution.
Hope someone can provide some insights~
Thank you! Gareth