Why randomness space must be significantly larger than the commitment space |R|>>|C| in order to generate a commitment string?

Sheldon

6/26/23, 11:05 AM

Why randomness space must be significantly larger than the commitment space |R|>|C|? the picture is from https://youtu.be/IkNZWJFcfcU?t=236

0 + 0

randomness

collision-resistance

commitments

Score:1

Crypto

Daniel S

6/26/23, 12:09 PM

For the hiding property we require that the commitment value provides no information about the message. In particular we hope that for any given message $m$ it is possible for $H(m,r)$ to take all possible values in $C$ (otherwise one could exclude some messages as corresponding to some commitments). If $H$ behaves like a random function, then it unlikely to have this surjective property unless $|R|>|C|\log|C|$.

More powerfully we probably want the values of $H(m,r)$ for any fixed $m$ to be distributed uniformly across the values of $C$.

Note that these are information theoretic "zero knowledge" requirements rather than complexity/computation bounded requirements.

0 + 0

Daniel S

6/26/23, 2:05 PM

For any secret value the random blinding of a Pedersen commitment covers the whole image space uniformly. It can do this because the random blinding is a random permutation rather than a random function.

poncho

6/26/23, 5:26 PM

"In particular we hope that for any given message $m$ it is possible for $H(m,r)$ to take all possible values in $C$"; while this would be sufficient, this is not actually necessary. What is necessary is that any plausible adversary be unable to obtain any information about $m$ from the commitment; however, because computationally unbounded adversaries are not plausible, we can consider lesser goals (such as deducing information about $m$ computationally infeasible..)

Daniel S

6/26/23, 5:47 PM

@poncho: There are three levels of hiding in commitment schemes [perfect, statistical, and computational](https://en.wikipedia.org/wiki/Commitment_scheme#Perfect,_statistical,_and_computational_hiding). The lecturer is tacitly requiring statistical hiding (hence my reference to information theory) and the requirement $|R|\gg |C|$ is necessary to meet this.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Why randomness space must be significantly larger than the commitment space |R|>>|C| in order to generate a commitment string?

TH: เหตุใดพื้นที่สุ่มจึงต้องใหญ่กว่าพื้นที่ผูกมัด |R|>>|C| อย่างมาก เพื่อสร้างสายผูกมัด?

RO: De ce spațiul aleatoriu trebuie să fie semnificativ mai mare decât spațiul de angajament |R|>>|C| pentru a genera un șir de angajament?

RU: Почему пространство случайности должно быть значительно больше, чем пространство обязательств |R|>>|C| для создания строки обязательства?

VI: Tại sao không gian ngẫu nhiên phải lớn hơn đáng kể so với không gian cam kết |R|>>|C| để tạo ra một chuỗi cam kết?

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.