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In ECC, the proof showing that given $G$, $x$ and $y$ is in the range $[-z,z]$ is known as the range proof.

Related to: Proving that two points on elliptic curve are within range

So, if: $$H=xG−yG$$ it is possible to prove that $x−y$ is in the range $[−z,z]$.

Could you explain how to apply this theory? Do you have a reference, please?

Is z fixed by the method or is it configurable?

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Proving that $-z\leq x-y\leq z$ given $C=xG-yG=(x-y)G$ is the same as proving that $0\leq a<n$ where $a=z + x - y$ and $n=2z+1$ given $C'= aG = C + zG$.

$C'$ can be considered a Pedersen commitment of the form $aG+bH$, except that there is no blinding factor $b$ set.

To prove that $C'$ is a Pedersen commitment to a value $a$ less than $n$, first you need to generate a number of components, a selection of which together can sum to any natural number less than $n$, but cannot sum to any value equalling or exceeding $n$. See this answer for a method.

Then you create a Pedersen commitment for each of those components, and demonstrate that the Pedersen commitment $C'$ is constructed as the addition of some selection of those component Pedersen commitments. To see how, see this answer.

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  • $\begingroup$ Thank you for your answer. I've understood the firs step: How to split a natural number x into components? I didn't understand very well the second step: especially the bullet proofs. Is it a brute force approach? $\endgroup$ Jul 1 at 16:33
  • $\begingroup$ @PeterMacGonagan I don't know what you mean by brute force. Let's say you are proving a commitment is to a value less than 8, and you have components 1,2,4. When you declare each Pedersen commitment for each component, the commitment is blinded. But you use a ring signature to prove the first commitment is either to 0 or to 1. Then another ring signature to prove the second commitment is either to 0 or 2. Then another to prove the third commitment is to 0 or 4. Then the sum of these commitment can't equal or exceed 8. Search for "Borromean Ring Signature" to see how they are implemented. $\endgroup$
    – knaccc
    Jul 1 at 21:26
  • $\begingroup$ Ok, I read the reference about Borromean Ring Signature and I think I understand how it works. I had misunderstood what range proof was for. My apologies. I thought that it was possible thanks to this theory to know if a point of an elliptic curve was included in a certain interval like for example between 1G and 2^64G. or even to know if a point was negative. But I think that it is not possible to know this information. $\endgroup$ Jul 3 at 12:40
  • $\begingroup$ @PeterMacGonagan since $1-2^{64}$ is just about within the realm of where you would worry about the multiple of $G$ being brute-forced, normally you'd use a Pedersen commitment to represent the value so that the value is not brute-forceable. You can use the method I've discussed to prove that a curve point is between $1G$ and $2^{64}G$, but you would need to know the multiple of G prior to using a range proof to prove it is within the range. $\endgroup$
    – knaccc
    Jul 3 at 17:10
  • $\begingroup$ If I understand correctly: we just compute $1G$ to $2^{64}G$ elliptic points and we just compare our elliptic point with theses values to know if we are in this interval or is there a trick that I don't see? $\endgroup$ Jul 4 at 13:04

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