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Title. This is probably a remarkably stupid question, but I digress. Now, I understand the point of rangeproofs, but I don't understand why they're actually necessary.

So, rangeproofs prevent a malicious user from committing to a negative number in the Pedersen Commitment. But going by that protocol (example implementation), successfully committing to a negative number should be impossible (or at least unreasonably expensive) if you choose a prime for g. Roots (which is what a negative exponent is) of any prime number are always going to be irrational, which means that you can't convert them back into an integer unless you multiply by g-n which brings them back to 1. Unless g2 < q (which can be prevented by simply choosing a g which doesn't fit that rule), then it's barely even possible to commit to -1 because mod q only allows for 1 valid number, let alone anything higher which requires exponentially higher numbers to reach a valid integer. This means that you can simply reject any commitment that equals 1.

Not only that, but the discrete logarithm problem should mean that the chances of successfully getting 1 at all are essentially non-existent in the first place.

So, in short, I don't understand why rangeproofs are necessary. By my understanding, the commitment system in and of itself should be resistant against committing to negative numbers

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The example you cite is not using elliptic curve operations.

A Pedersen commitment in Monero takes the form C = xG + aH (alternatively notated C = gx ⋅ ha) and here G and H (or g and h) are elliptic curve points, hence the operation is repeated addition (or subtraction) of the respective point. Thus a (or x) can indeed be a negative. So -3H (or h-3) translates as "subtract the point H from itself 3 times (i.e. H - H - H - H)".

With this understanding, it is indeed possible to commit to a value that is negative, and thus a range proof is needed to ensure the committed value is in the range [0..2^64-1].

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  • Ah, that makes sense, I didn't know that Monero doesn't use the traditional scheme. That does bring up another question for me though, what's the reasoning for using the EC variant? I know that Monero of course makes use of ECC in other areas, but is there some benefit of using it here? Oct 26, 2021 at 2:03
  • "I didn't know that Monero doesn't use the traditional scheme" <- Monero does use the "traditional scheme". Confidential Transactions uses ECC, and thus the values used in the Pedersen commitments are scalars and elliptic curve points.
    – jtgrassie
    Oct 26, 2021 at 2:11
  • Well okay, but then why does CT use ECC? Oct 26, 2021 at 2:31
  • Why does anything use ECC: en.wikipedia.org/wiki/Elliptic-curve_cryptography#Rationale
    – jtgrassie
    Oct 26, 2021 at 2:34

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