Randomized Compilation Protocol

• Randomized Compilation Protocol is a method that converts deterministic computations into processes with controlled randomness for certifiable sampling and verifiable computing.
• It employs histogram bucketing, random hash filtering, and adaptive challenge-response to achieve efficient interaction and precise probabilistic guarantees.
• The protocol enables the transformation of private-coin interactive proofs to public-coin models, reducing round complexity while maintaining robust soundness and completeness.

A randomized compilation protocol is a methodology for transforming deterministic computational processes—classically or quantumly described—into procedures with finely controlled randomness, often with the dual aims of certifiable sampling or verifiable randomness properties and robust, auditable execution. In interactive and cryptographic settings, randomized compilation is harnessed for sampling according to distributions held only by untrusted parties, simulating hidden random choices (private coins), or transforming complexity-theoretic proof systems to require only public randomness. Recent frameworks have formalized these techniques with precise probabilistic, complexity, and information-theoretic guarantees, enabling new applications in complexity theory, cryptography, verifiable computation, and distributed computing.

1. Protocol Architecture: Interactive Sampling with Annotated Probabilities

The foundational protocol assumes a two-party setting where a prover holds a discrete probability distribution $P$ over $n$-bit strings. The verifier—who lacks direct access to $P$—interacts with the prover to ultimately output a pair $(x, p)$, where $x$ purports to be sampled from $P$ and $p$ quantifies $P(x)$ or its tight approximation.

Key steps:

1. Histogram Bucketing. The prover divides the support of $P$ into “buckets” indexed by $j$, such that

$$\mathcal{B}_j = \{ x : P(x) \in (2^{-(j+1)}, 2^{-j}] \}$$

and reports the total weights $h_j = \sum_{x \in \mathcal{B}_j} P(x)$, forming the histogram.

1. Interval and Gap Partitioning. The verifier randomly selects a bucket interval $I_k$—from a partition of the histogram indices—with probability proportional to the total mass in those intervals, mitigating skewness or overconcentration by adaptive selection.
2. Random Hash Filtering. For each index $j$ in $I_k$, the prover constructs a filtered subset

$$\mathcal{X}_j = \{ x \in \mathcal{B}_j : f(x) = 0^m \}$$

using a random 3-wise independent hash $f$ with codomain $\{0,1\}^m$. The verifier checks that $|\mathcal{X}_j|$ is close to its expectation.

1. Final Sampling and Annotation. The verifier selects one $\mathcal{X}_j$ proportionally to $h_j$, then picks $x$ uniformly from $\mathcal{X}_j$, outputting $(x, p)$, where $p$ is either $P(x)$ (in the honest case, if the prover supplies it) or $2^{-j}$ as an upper-bound annotation.

This architecture leverages hashing, bucketization, and challenge-response subprotocols to compress the support and control the granularity of the sampling process. It achieves efficient interaction—requiring only a polynomial number of rounds and reducing verification overhead.

2. Probabilistic Soundness and Completeness Guarantees

The protocol achieves two strong (though structurally distinct) forms of guarantee:

• Completeness (Honest Prover): For almost all $x$ (excluding a negligible “bad” set), the output distribution satisfies

$$P_{(X,P)}(x, p) \in (1 \pm \varepsilon) P(x) \quad\text{with }p = P(x)$$

and $\varepsilon$ negligible (e.g., polynomially or exponentially small in $n$).

• Soundness (Potentially Dishonest Prover): It is proven impossible to demand that $p$ always lower-bounds the actual sampling probability when the prover may cheat. Instead, an averaged upper-bound guarantee holds:

$$\sum_{p} \frac{\Pr((X,P) = (x,p))}{p} \leq 1 + \delta$$

for arbitrarily small $\delta$. This can be interpreted as bounding the expected “inverse probability” and ensuring the aggregate risk of underrepresented $x$ is always controlled.

The impossibility of per-instance lower-bound soundness is established via explicit counterexamples. The presented guarantee is optimal in the sense that, on average, the prover cannot make the verifier unjustifiably believe an output is rare.

3. Transformation of Private-Coin Interactive Proofs

A principal application is the conversion of private-coin interactive proofs into public-coin protocols. In private-coin proofs, the verifier's random choices are hidden and must be faithfully simulated to construct a public-coin (Arthur-Merlin) protocol without loss of security or completeness.

The protocol is used as follows:

• For each round, the verifier samples the message $m_i$ (or random coins $r$) via the sampling protocol, obtaining both the value and its probability annotation.
• The prover responds to these public values as in the original protocol.
• The final transcript $(m_0,a_0,\dots,m_{k-1},a_{k-1},r)$ is distributed almost identically to one generated by the original private-coin verifier.

Efficiency comparison: Unlike the canonical Goldwasser-Sipser transformation (GS86), which requires simulating many independent runs of the private-coin protocol, this approach only calls the private-coin verifier once per simulated coin toss or message. The completeness and soundness degradation is an arbitrarily small constant per round, and overall verifier runtime is reduced.

4. Computational Efficiency and Scaling

All verification steps—including histogram computation, hash verification, support set selection, and output sampling—are performed in time polynomial in $n$ and parameter-dependent subpolynomial factors.

For constant-round protocols, the reduction incurs only a negligible loss in error bounds. The verifier’s work scales as

$\operatorname{poly}(n, (1/\varepsilon)^{1/\delta})$

where $\varepsilon$ is the desired error tolerance and $\delta$ governs soundness slack. This efficient scaling represents a strict improvement over transformations that amplify error probabilities by both repeated simulation and hashing.

Error accumulation across rounds is at most a small constant per step, with soundness maintained at standard interactive proof system thresholds after parallel repetition.

5. Applications in Randomized Compilation and Proof Complexity

The general protocol enables a range of applications:

• Complexity Theory: By enabling efficient public-coin transformations, the construction strengthens connections between $IP$, $AM$, and related complexity classes, supporting results like $IP = PSPACE$ and improved inclusions for proof systems.
• Randomized Compilation: In verifiable computations where the randomness may be private or only partially accessible (as in cryptographic protocols or outsourced computing), the protocol allows certifiable “unfolding” of the random choices, thus holding the prover publicly accountable for probabilistic outcomes.
• Auditing and Verifiability: For distributed simulations or randomized tasks in environments where outcome veracity is critical, this protocol can serve as a correctness and efficiency layer, ensuring that outputs are drawn as claimed and that manipulation or biasing is detectable.

A plausible implication is broader deployability in privacy-preserving systems, multiparty computation, and distributed verifiable randomness generation, where the ability to jointly audit and simulate probabilistic behavior is essential.

6. Comparison with Prior Approaches and Limitations

The protocol supersedes Goldwasser-Sipser’s method in both round complexity and verifier efficiency, with a strictly smaller error “tax” per round and nearly optimal completeness/soundness trade-off for constant-round proofs.

However, the per-instance guarantee remains unachievable; only average-case soundness is enforceable. The protocol relies on the existence of efficient three-wise independent hash families and efficient support set enumeration, but for all efficiently samplable $P$ these requirements are met.

The protocol's randomized bucketing technique is robust and generic, but, as indicated, cannot substitute for actual probabilistic coin tosses in certain cheating scenarios—a limitation intrinsic to interactive cryptographic sampling.

7. Summary Table: Completeness and Soundness Properties

Guarantee	Statement	Parameter
Completeness	$P_{(X,P)}(x, p) \in (1 \pm \varepsilon) P(x)$, $p = P(x)$ (for $x$ in $M$)	$\varepsilon \ll 1$
Soundness (Average)	$$\sum_{p} \frac{\Pr((X,P) = (x,p))}{p} \leq 1+\delta$$ (for all $x$)	$\delta \ll 1$
Per-Instance Bound	Not possible in general; only average bound holds	N/A

The completeness and soundness properties ensure the protocol outputs honest samples with correct probabilities (up to negligible error), and that on average the verifier cannot be misled about the likelihood of the outputs.

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Randomized compilation protocols, as instantiated by this framework, deliver efficient, robust sampling and verification from arbitrary distributions held by untrusted parties in interactive settings. The methodology extends to the compilation of private randomness into public-coin protocols, yielding efficiency and round complexity benefits in interactive proof systems, cryptographic delegation, and randomized computations with auditability and verifiability requirements (Holenstein et al., 2013).