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How to construct random divisor once the set of rational points over the Jacobian of a hyperelliptic curve is created?

I know how to deliberately create a divisor with knowledge of the mumford coordinates, but is there a way to generate a divisor randomly inside of X = J(FF) and extract its mumford coordinates as polynomials over FF?

Where divisors D = X([a,b]) are elements of X = J(FF), a and b are the mumford polynomials, and J is the Jacobian of a hyperelliptic curve.