# How to Update Non-Membership Witness upon deletion in RSA Dynamic Universal Accumulator

The Universal Accumulator paper mentions the following for updating non-membership witness upon member deletion.

Choose $$r$$ such that $$a\hat x − rx ∈ Z$$$$2^l$$

What is Z$$2^l$$ and how can we efficiently evaluate $$r$$?

I suppose we need to brute force search by guessing values of r, is there any room for optimization here? Perhaps by making an educated guess about the value of r.

Update:

Can it be that $$a\hat x − rx$$ can be construed as $$a\hat x$$ $$mod x$$? If yes, then we can find

$$\hat a = a\hat x$$ $$modx$$

And,

$$r = (a\hat x - \hat a)/x$$

Am I right in assuming this?

• Does this answer help? Jul 12, 2020 at 22:05
• thanks for your response @Occams_Trimmer, I looked at it before but that answer mentions about Update during addition only. I was able to follow that successfully but the update on deletion seems trickier. Jul 12, 2020 at 22:38

The updated Non-Membership Witness ($$\hat a, \hat d$$) upon deleting a member $$\hat x$$ can be calculated as follows:
$$\hat a = a\hat x\mod x$$
$$r = (a\hat x - \hat a)/x$$
$$\hat d = (d\hat c$$ -r )$$\mod n$$