Cipher Maps: When Category Theory Meets Oblivious Computing

Source: Dev.to

Introduction

What if we could compute on encrypted data while preserving algebraic structure?
Not through expensive homomorphic encryption, but via a principled mathematical framework that unifies:

• Oblivious computing
• Bernoulli types (statistical approximation)
• Categorical abstractions

My recent work on cipher maps and algebraic cipher types explores exactly this idea.

Core Idea: Cipher Functors

A cipher functor cA lifts a monoid

[ (S,\ *,\ e) ]

into a cipher monoid

[ (cA(S),\ \oplus,\ \hat e) ]

Latent space (real monoid)   : (S, *, e)
          ↓ cA (cipher functor)
Cipher space (encrypted)    : (cA(S), ⊕, ê)

Key Properties

These properties give us a type system for oblivious computing: instead of ad‑hoc encryption schemes, we obtain principled constructions that preserve algebraic structure.

Why This Matters

• Formal privacy guarantees are baked into the type.
• Error bounds are derived from Bernoulli‑type theory.
• Composable abstractions let us build complex oblivious pipelines from verified components.

Illustrative Examples

(Links: Algebraic Cipher Types, Cipher Maps)

Unified Framework

The paper unifies three major threads:

1. Algebraic cipher types – categorical foundations
2. Bernoulli types – statistical approximation theory
3. Oblivious function approximation – practical algorithms

Cipher Maps as Morphisms

• Objects – cipher types (encrypted/oblivious data structures)
• Morphisms – functions that preserve both algebraic structure and statistical properties

Composition of morphisms yields principled ways to build complex oblivious computations while tracking error propagation.

Key Contribution: Oblivious Bernoulli Maps

An Oblivious Bernoulli map is a function

[ f : cA(S) \to cA(T) ]

that:

• Operates on encrypted/oblivious inputs
• Produces encrypted/oblivious outputs
• Has provable error bounds (from Bernoulli‑type theory)
• Preserves categorical structure (from cipher functors)

Practical Result

1. Start with a regular function f : S → T.
2. Lift it to a cipher map cA(f) : cA(S) → cA(T).
3. Obtain automatic privacy guarantees and formal error bounds.
4. Compose multiple cipher maps while tracking error propagation.

Why the Abstract Mathematics?

Without these abstractions, each new oblivious algorithm would require a bespoke security proof.

Concrete Instantiation: Cipher‑Trapdoor‑Sets

The cipher‑trapdoor‑sets paper presents a concrete example:

• Base monoid – powerset ((\mathcal{P}(U), \cup, \varnothing))
• Cipher functor – hash‑based Bernoulli maps

Result: Hash‑Based Oblivious Sets (HBOS), i.e., Bloom filters equipped with:

• Formal privacy semantics
• Compositional error bounds
• Type‑safe set operations (union, intersection, etc.)

Applications in the Wild

• Database operators – join, project, select on oblivious data (hiding access patterns)
• Aggregation queries – SUM, COUNT, AVG lifted to appropriate monoids
• Gradient aggregation – oblivious types enable differential privacy without a trusted aggregator
• Streaming analytics – cipher maps over streaming monoids give constant‑memory, provably private pipelines

Implementation Sketch (C++)

// Pseudocode sketch
template <typename T>
class CipherFunctor {
public:
    using Cipher = ObliviousBernoulli;

    // Lift a clear value into the cipher domain
    Cipher lift(const T& value);

    // Approximate inverse (decrypt/approximate)
    T approximate_inverse(const Cipher& cipher) const;

    // Monoidal operation on ciphertexts
    Cipher combine(const Cipher& a, const Cipher& b) const;
};

Implemented with modern template metaprogramming, the design aims for a zero‑cost abstraction.

Research Directions

Outlook

We are at an inflection point:

• Privacy regulations demand oblivious computing.
• Scale pushes us toward approximate algorithms.
• Complexity calls for compositional abstractions.

Cipher maps provide the mathematical foundation to build these systems correctly—turning “ad‑hoc crypto + hope” into principled constructions, automatic privacy, and compositional security.

Related Papers

• Algebraic Cipher Types
• Cipher Maps: Unified Framework
• Hash‑Based Oblivious Sets
• Bernoulli Types Series
• Encrypted Search with Oblivious Types

(All links to be added.)

Privacy‑preserving systems with formal guarantees