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5.1 Commitment Construction

In Abyss, deposits are not tracked as balances tied to addresses. Instead, each deposit is represented as a cryptographic commitment derived from a secret known only to the depositor. This commitment is what enters the anonymity pool. Formally, a commitment is constructed as: 
commitment C := H(secret_key || deposit_nonce || deposit_amount)
Where:
  • secret_key is generated client-side and never revealed
  • deposit_nonce ensures uniqueness across deposits
  • deposit_amount binds value to the commitment
The hash function H is collision-resistant and circuit-compatible. Once computed, C is inserted as a leaf into the global Merkle tree representing the anonymity pool. This design ensures that the protocol can verify the existence of a deposit without learning anything about who made it or how it will be spent.

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5.2 Merkle-Based Anonymity Set

All commitments exist within a single Merkle tree: 
AnonymitySet := { C₁, C₂, …, Cₙ }
The anonymity set size |AnonymitySet| = n directly bounds privacy. A withdrawal proof only asserts that one of these commitments is being spent, without revealing which one. Key property: 
Pr(link deposit → withdrawal) ≈ 1 / n
assuming uniform usage and no external leakage. Abyss intentionally avoids multiple pools or denominations to prevent entropy fragmentation. Every new deposit strictly increases the anonymity set for all participants.

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5.3 Nullifier Design

To prevent double-spending without revealing identity, Abyss uses nullifiers. For a given secret: 
nullifier N := H(secret_key || withdrawal_index)
Properties:
  • Deterministic per withdrawal
  • Unique per spend
  • Unlinkable to commitment without secret knowledge
On withdrawal, N is revealed as a public input and checked against an on-chain nullifier registry: 
require(!nullifierUsed[N], "DOUBLE_SPEND");
nullifierUsed[N] = true;
This ensures that no withdrawal can be repeated, while revealing nothing about which commitment generated it.

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5.4 Infinite Withdrawal Model

Unlike single-use note systems, Abyss supports multiple withdrawals from a single deposit. Internally, this is enforced through balance accounting inside the ZK circuit. The circuit asserts: 
Σ withdrawn_amounts ≤ deposit_amount
Each withdrawal consumes balance but does not exhaust the commitment unless fully spent. This enables:
  • Micropayments
  • Streaming payouts
  • Merchant payments
  • Partial liquidity exits
All without reducing anonymity.

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5.5 Anonymity Set Dynamics

Anonymity is not static. It evolves with protocol usage. Positive factors:
  • Frequent deposits
  • Overlapping withdrawals
  • Time separation between actions
Negative factors:
  • Immediate withdraw-after-deposit
  • Unique withdrawal patterns
  • Low pool usage
Abyss does not enforce behavior. It provides the cryptographic substrate. Privacy emerges from correct usage patterns, not protocol coercion.

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5.6 Security Implications

Commitments, nullifiers, and Merkle inclusion together ensure:
  • Ownership without identity
  • One-time spend guarantees
  • No balance inflation
  • No linkage across time
These primitives are the foundation upon which all higher-level privacy features in Abyss are built.