MVKs for Individual Umbra Addresses

Protocol Description

The MVK for the $i$-th Umbra address, $k_{\text{MVK}, U_i}$, is derived using the same PRF construction but with the index $i$ for domain separation.

$$k_{\text{MVK}, U_i} \triangleq \text{Poseidon}( S_M, \text{EncodeAsFieldElement}(\text{"MVK-Umbra-Address"}), i )$$

Key Derivation Process

1. Input: Master seed $S_M$ (32 bytes)
2. Domain Separation: String $"MVK-Umbra-Address"$ encoded as a field element
3. Index Parameter: Address index $i$ for uniqueness
4. Hash Function: Poseidon hash function acting as a PRF
5. Output: Field element $k_{\text{MVK}, U_i}$ in $\mathbb{F}_p$

Index-Based Uniqueness

Each Umbra address has its own unique MVK derived using:

• The same master seed $S_M$
• The same domain separator $"MVK-Umbra-Address"$
• A unique index $i$ corresponding to the address number

This ensures that:

• Each Umbra address has a distinct MVK
• The same master seed generates different MVKs for different addresses
• MVKs are computationally independent

On-Chain Registration

Each $k_{\text{MVK}, U_i}$ must be registered on-chain before claims can be made from the corresponding Umbra address. This registration:

• Enables Claims: Required before claiming from the $i$-th Umbra address
• Address-Specific: Each Umbra address has its own MVK registration
• Immutable: Once registered, cannot be changed
• Public Commitment: Creates a public commitment without revealing the MVK

Security Analysis

Security Proof: Independence of Umbra MVKs

Goal: Each $k_{\text{MVK}, U_i}$ must be pseudorandom and computationally independent from all other MVKs.

Assumption: Poseidon-as-a-PRF.

Argument: The input to the PRF for each key is unique due to the index $i$ and the distinct domain separator strings. For any two distinct indices $i \neq j$, the inputs to the PRF are different. Therefore, the outputs $k_{\text{MVK}, U_i}$ and $k_{\text{MVK}, U_j}$ are computationally independent.

Properties

• Independent: Each MVK is computationally independent from others
• Pseudorandom: Each appears as a random field element
• Index-Based: Unique derivation per Umbra address index
• Unlinkable: Cannot be traced back to the master seed
• Address-Specific: Each Umbra address requires its own MVK registration