# RingVRF

**Overall**

  In this project, committee members need to be rotated to achieve time-limited security for private key shards. Therefore, a method is required to select different batches of committee members.

  Simultaneously, the identities of committee members must be kept confidential to prevent them from colluding with each other and launching attacks against signatures.

  Thus, a method is needed to identify committee members holding specific private keys without revealing their identities.

**Core Objectives**

1. **Secure Committee Rotation**\
   Enables periodic rotation of committee members holding private key shards for time-limited security.
2. **Identity Confidentiality**\
   Prevents committee member identification to avoid collusion attacks on signatures.
3. **Anonymous Key Ownership Proof**\
   Allows a member to prove their public key belongs to a specified set without revealing which one.

> **In essence**: Ring VRF lets a user prove their public key is in set {pk₀,...,pkₙ₋₁} and generate verifiable random output `v` using private key `sk`, while **hiding their specific public key index** `ℓ`.

so we introduced **Ring VRF algorithm.**  

**The purpose of the Ring VRF algorithm is:** To allow a user to prove that the public key corresponding to their private key exists within a set of public keys, **without revealing which specific public key they possess.**

RVRF =
 {((pk0, . . . , pkN−1
, r, v),sk, )|∈ {0, . . . , N − 1} s.t. (pk\` ,sk) ∈ Rpk ∧ v ← PRFsk(r)}

The inputs to the algorithm are a **public set of public keys** and a **random number**.

Pedersen commitment is used as the commitment scheme.

The algorithm achieves its core functionality **by utilizing** the one-out-of-many protocol. Additionally, a **sub-proof** guarantees that both the commitment value `c` (from the Pedersen commitment) and the PRF result `v` correspond to the same private key (`sk`).

**Algorithm Definition**

**Inputs/Outputs**

* **Input**:
  * Public key set {pk₀,...,pkₙ₋₁}
  * Random seed r ∈ {0,1}\*
* **Output**:
  * Random value `v = PRFₛₖ(r)`
  * Proof `π` verifying both:
    1. `pkₗ ∈ {pk₀,...,pkₙ₋₁}` for some `ℓ`
    2. `v` was correctly generated from `sk`

#### **Key Components**

**1. Pseudorandom Function (PRF)**

* **Definition**: `PRFₛₖ(r) := H(r)ˢᵏ`
* **Parameters**:
  * Cyclic group `G` of prime order `q` with generator `g`
  * Hash function `H: {0,1}* → G`
* **Security**: Computes verifiable random output bound to `sk`.

**2. Pedersen Commitment**

* **Role**: Hides prover's index `ℓ` and secret `sk`.
* **CRS**: Commitment key `ck = h` (public parameter).

**3. One-out-of-Many Protocol**

* **Core innovation**: Proves `∃ℓ` such that `pkₗ ∈ {pk₀,...,pkₙ₋₁}` without revealing `ℓ`.
* **Achieves**: Full anonymity within the public key set.

**4. Consistency Sub-Proof**

* **Critical guarantee**: Links Pedersen commitment `c` and PRF output `v` to the **same private key `sk`**.
* **Prevents**: Forgeries using mismatched keys.

#### **Protocol Workflow**

**Prover (sk holder)**:

1. Compute `v = H(r)ˢᵏ`
2. Generate Pedersen commitment to `sk`
3. Use **One-out-of-Many** to prove `pkₗ ∈ {pk₀,...,pkₙ₋₁}`
4. Run **sub-proof** linking commitment and `v` to `sk`
5. Output `(v, π)`

**Verifier**:

1. Validate One-out-of-Many proof (key membership)
2. Verify consistency between commitment and `v`
3. Confirm `v` is correctly derived from `r`


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