gpt 6 astra pro attempt on CS-38

Indistinguishability Obfuscation from Constant-Degree Homomorphic Assumptions

Definitions and mathematical statement

An indistinguishability obfuscator is a randomized efficient transformation $C\mapsto\mathcal O(C)$ preserving the circuit's function and producing only polynomial overhead. For same-size equivalent circuits $C_0,C_1$, it requires \[ \mathcal O(C_0)\approx_c\mathcal O(C_1), \] meaning every probabilistic polynomial-time distinguisher has negligible advantage, in the usual security-parameter experiment. The goal is such a construction from an explicitly stated combination of standard polynomial-security assumptions of the indicated type, without subexponential security losses. "Standard assumptions" is not itself a formal predicate: a complete theorem must name the exact LWE/LPN/pairing variants, parameters, and reduction loss. No unrestricted high-degree multilinear-map assumption is silently added.

Formulation and concept references: [S1].

Short English statement

Can general indistinguishability obfuscation be obtained from the stated kinds of conventional assumptions without requiring subexponential security?

Research attempt

Current frontier and assumption audit.

The full version of the supplied STOC construction, Theorem 5.5, assumes subexponential hardness of SXDH, a specified large-modulus LWE problem, sparse-error LPN over ${\mathbb{Z}}_p$, and a polynomial-stretch Boolean $\mathsf{NC}^0$ PRG. Its polynomial-security conclusion for collusion-resistant functional encryption is not general iO. [S1][E1] Proof-sensitive obfuscation avoids input-by-input security loss for circuits with short proofs of equivalence; the 2025 quasi-linear construction still retains that hypothesis. [E2][E3] An ideal pseudorandom-oracle construction is in a different model, not an instantiation from the requested standard assumptions. [E4]

Attack route.

Replace exponentially many inputwise hybrids by a short sequence of semantics-preserving circuit rewrites, perhaps through proof compression. We investigate whether short paths suffice for security, whether every equivalent pair has one, and whether a reduction can actually access it. None of the latter two assertions follows from semantic equivalence alone.

Attempt: exact accounting of hybrid loss.

For distributions $H_0,\ldots,H_h$ and a test $D$, let $d_i=\Pr[D(H_i)=1]$. Telescoping gives \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-39-3a-68-79-62-72-69-64} |d_h-d_0|\le\sum_{i=1}^h|d_i-d_{i-1}|. \end{equation} If an edge experiment samples $I$ uniformly and supplies $H_{I-1}$ or $H_I$, using the same $D$ has signed gap $(d_h-d_0)/h$. An exponential range of indices needs only $\log_2h$ random bits, so an efficiently samplable indexed hybrid can save time; it does not remove the advantage loss.

For instance $2^{-\sqrt\lambda}$ is negligible but multiplying it by $2^\lambda$ does not give a vanishing bound. More generally, for any fixed polynomial rescaling $\kappa=\lambda^b$ and any $a>0$, the negligible function \[ \epsilon(\kappa)=2^{-(\log_2\kappa)^2} \] satisfies $2^{\lambda^a}\epsilon(\lambda^b)\to\infty$. Therefore no fixed polynomial parameter inflation makes an exponential hybrid loss a logical consequence of arbitrary polynomial security. This is an obstruction to that inference, not to every possible cryptographic reduction.

Suppose instead a uniform procedure supplies a path of at most polynomially many rewrites between a given equivalent padded circuit pair of size $s$. Suppose each indexed rewrite has a polynomial-time simulation and a polynomial-loss reduction to a fixed finite collection of named assumptions, uniformly in the path. For $s\le\operatorname{poly}(\lambda)$, polynomial security and (30006749:hybrid) give negligible endpoint advantage. This is a valid composition statement; neither an edge-secure encoding nor a universal path procedure is constructed here. Merely verifying a supplied path is not the same as producing it.

Why universal proof compression is a deep extra demand.

Let $\mathcal R$ be any sound, polynomial-time decidable circuit rewrite relation. Auxiliary wires and expansions are allowed but counted in the full certificate encoding. If one polynomial bounds the total encoding length of a path between every equivalent circuit pair, then \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-39-3a-63-6f-6c-6c-61-70-73-65} \mathsf{NP}=\mathsf{coNP}. \end{equation} Given a formula $\varphi$, compare its circuit with constant one, padding as needed. Guess a path of the promised polynomial size and check it. Soundness excludes nontautologies; the universal bound certifies every tautology. Thus $\mathsf{TAUT}\in\mathsf{NP}$, proving (30006749:collapse). This implication needs only existence of short paths, not a path-finding algorithm. A total polynomial-time path finder with this sound verifier would even decide equivalence and imply $\mathsf P=\mathsf{NP}$. Neither implication is an implication of iO itself; it is attached to this particular proposed method.

Similarly, a polynomial-time canonical representation satisfying \[ C_0\equiv C_1\quad\Longleftrightarrow\quad \operatorname{can}(C_0)=\operatorname{can}(C_1) \] would decide tautology by comparison with $\operatorname{can}(1)$ and imply $\mathsf P=\mathsf{NP}$. A truth table supplies perfect semantic canonicalization but uses $2^n$ output bits. It does not meet polynomial overhead. Randomness and computational indistinguishability cannot simply be replaced by canonicalization.

A restricted bilinear-encoding test.

Consider a generic asymmetric bilinear model whose source exponents are affine linear in hidden variables $x_1,\ldots,x_m$. Permit exponent addition, multiplication by known scalars independent of those hidden variables, and pairing two source elements; prohibit a target-to-source map. Induction shows that source exponents remain affine and target exponents have degree at most two. Thus this bare evaluator cannot produce exponent $x_1x_2x_3$ as a formal polynomial. The same holds as a function over a field of size larger than three, by uniqueness of reduced polynomials of individual degree below the field size. The script verifies the evaluation-matrix rank increase when a cubic is added to quadratic functions over ${\mathbb{F}}_5$.

This toy obstruction excludes auxiliary encodings, LWE ciphertexts, proofs and randomized representations. It does not rule out bilinear groups as a component of iO; the supplied construction uses additional structure. It only prevents an unlicensed iteration of a degree-two pairing from becoming an assumed multilinear operation.

Stress test.

A proof supplied by a circuit-generating adversary can support a proof-sensitive theorem. General iO quantifies over equivalent circuits without requiring that certificate. Switching to nonuniform reductions would likewise require an explicit assumption change, not cure the issue silently. The efficiency improvement in [E3] retains the proof hypothesis. Instantiating an ideal oracle is not justified by hybrid bookkeeping. No toy-model limitation above is claimed for the combined LWE/LPN/pairing constructions.

Outcome.

Outcome: Exploratory approach with unresolved gap. The attempt establishes a polynomial-loss composition criterion, a quantifier-level failure of polynomial parameter inflation for exponential hybrids, and a proof-complexity obstruction to universal short certificates. It constructs no new obfuscator. A successful continuation needs a uniform, polynomial-loss comparison for arbitrary equivalent circuits without first requiring polynomially checkable short proofs for all such pairs.

Sources

• S1 A. Jain, H. Lin, A. Sahai, STOC 2021: 60-73, 2021. https://doi.org/10.1145/3406325.3451093. Catalogue formulation source.
• E1 A. Jain, H. Lin and A. Sahai, Indistinguishability obfuscation from well-founded assumptions, arXiv:2008.09317v1, Section 5, Theorems 5.3--5.5; full version underlying [S1]. https://arxiv.org/pdf/2008.09317.
• E2 A. Jain and Z. Jin, Indistinguishability obfuscation via mathematical proofs of equivalence, FOCS 2022, 1023--1034; Cryptology ePrint 2022/1430. https://eprint.iacr.org/2022/1430.
• E3 Y. Ma, C. Dai and E. Shi, Quasi-linear indistinguishability obfuscation via mathematical proofs of equivalence and applications, EUROCRYPT 2025, LNCS 15603, 157--186; ePrint 2025/307, stated equivalence-proof hypothesis. https://eprint.iacr.org/2025/307.
• E4 A. Jain, H. Lin, J. Luo and D. Wichs, The pseudorandom oracle model and ideal obfuscation, CRYPTO 2023, 233--262; ePrint 2022/1204, abstract and model. https://eprint.iacr.org/2022/1204.