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CS-436 · computer science · mathematics · Information and coding theory

Information-Theoretic Additive Randomized Encodings

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

An additive randomized encoding represents $f(x_1,\ldots,x_n)$ through $\sum_i g_i(x_i)$ in a finite Abelian group. Correctness recovers $f$ from the sum; information-theoretic privacy requires a simulator using only the intended output, under the stated randomness convention.

In the LOCAL model vertices communicate synchronously along graph edges; messages and local computation are unrestricted and complexity is the number of rounds. After $t$ rounds a vertex can depend only on its radius-$t$ neighborhood and any supplied identifiers or randomness.

A code $C\subseteq\Sigma^N$ is a set of permitted length-$N$ words. Its Hamming minimum distance is the minimum number of unequal coordinates of two distinct words. For a linear $[N,K,D]_q$ code, rate is $K/N$ and relative distance is $D/N$; decoding recovers a codeword from corrupted observations.

A cryptographic security requirement compares an efficient adversary's advantage with a negligible function of the security parameter. A negligible function is smaller than every inverse polynomial eventually; computational and information-theoretic security are different requirements.

Statement recorded in the supplied source.
Characterize the functions $f(x_1,\ldots,x_n)$ that admit information-theoretically secure additive randomized encodings: local randomized maps $g_i(x_i)$ into a finite abelian group whose sum reveals $f$ and essentially nothing else about the inputs. [S2]
Scope qualification.
The meaning of essentially nothing else, shared versus private randomness, and exact versus approximate correctness are not specified.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Characterize functions admitting perfectly or statistically private local additive encodings.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006182 and source code OWR-14299084-004 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-436-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-436

Information-Theoretic Additive Randomized Encodings

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

An additive randomized encoding represents $f(x_1,\ldots,x_n)$ through $\sum_i g_i(x_i)$ in a finite Abelian group. Correctness recovers $f$ from the sum; information-theoretic privacy requires a simulator using only the intended output, under the stated randomness convention.

In the LOCAL model vertices communicate synchronously along graph edges; messages and local computation are unrestricted and complexity is the number of rounds. After $t$ rounds a vertex can depend only on its radius-$t$ neighborhood and any supplied identifiers or randomness.

A code $C\subseteq\Sigma^N$ is a set of permitted length-$N$ words. Its Hamming minimum distance is the minimum number of unequal coordinates of two distinct words. For a linear $[N,K,D]_q$ code, rate is $K/N$ and relative distance is $D/N$; decoding recovers a codeword from corrupted observations.

A cryptographic security requirement compares an efficient adversary's advantage with a negligible function of the security parameter. A negligible function is smaller than every inverse polynomial eventually; computational and information-theoretic security are different requirements.

Statement recorded in the supplied source.
Characterize the functions $f(x_1,\ldots,x_n)$ that admit information-theoretically secure additive randomized encodings: local randomized maps $g_i(x_i)$ into a finite abelian group whose sum reveals $f$ and essentially nothing else about the inputs. [S2]
Scope qualification.
The meaning of essentially nothing else, shared versus private randomness, and exact versus approximate correctness are not specified.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Characterize functions admitting perfectly or statistically private local additive encodings.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006182 and source code OWR-14299084-004 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems. Codex (GPT-6; exact serving revision not exposed) supplied the auxiliary proof route used here. OpenAI gpt-6-astra, reasoning ultra, independently worked through its mathematical steps, wrote this exposition, and performed the finite checks. This is model collaboration, not independent human verification.

The report discusses positive and negative information-theoretic ARE results without giving one complete security experiment in the extract. We specify independent private local randomness, a finite Abelian output group, perfect privacy of the sum, and zero-error decoding. The root supplied the Fourier obstruction below; its algebra and scope were independently checked here. This does not cover correlated hidden setup or arbitrary approximate-security definitions.

A perfect-privacy obstruction for AND with independent coins

Let the two parties have bits $x,y$ and independently output group-valued random variables with laws $\mu_x,\nu_y$ on a finite Abelian group $G$. The aggregate has law $\mu_x*\nu_y$. Perfect privacy for AND requires \[ D_0=\mu_0*\nu_0=\mu_0*\nu_1=\mu_1*\nu_0,\qquad D_1=\mu_1*\nu_1. \]
Proposition: These conditions imply $\sum_{g\in G}D_0(g)D_1(g)=\sum_{g\in G}D_0(g)^2>0$. Hence no decoder of the aggregate can recover AND with zero error.
Proof: For a character $\chi$ of $G$, put $\widehat\mu(\chi)=\sum_g\mu(g)\chi(g)$. Expanding the convolution sum shows $\widehat{\mu*\nu}=\widehat\mu\widehat\nu$. Write $a_x=\widehat\mu_x(\chi)$, $b_y=\widehat\nu_y(\chi)$ and $c_i=\widehat D_i(\chi)$. Then \[ a_0b_0=a_0b_1=a_1b_0=c_0. \] If $c_0\ne0$, both $a_0,b_0$ are nonzero, so division gives $a_1=a_0,b_1=b_0$ and $c_1=c_0$. If $c_0=0$, the product $c_0\overline{c_1}$ is zero. Thus in all cases $c_0\overline{c_1}=|c_0|^2$.

For completeness, character orthogonality gives \[ \sum_g U(g)\overline{V(g)}=|G|^{-1}\sum_\chi\widehat U(\chi)\overline{\widehat V(\chi)}. \] One can verify it by writing a finite Abelian group as a product of cyclic groups and using the characters $\exp(2\pi i\sum_j r_j g_j/q_j)$. The finite geometric sum over each $r_j$ is $q_j$ when the corresponding coordinates agree and zero otherwise. Expanding the right side therefore leaves exactly the left side. Apply this identity to $(U,V)=(D_0,D_1)$ and $(D_0,D_0)$ and use the previous characterwise equality. The squared norm is positive because $D_0$ is a probability distribution.

Hence some aggregate value has positive probability under both output cases. A zero-error decoder would have to output both zero and one on that same value, which is impossible, even for a randomized decoder required to be always correct. ∎

The precise limitation of the masking route

For functions of the form $f(x_1,\ldots,x_k)=\sum_i\phi_i(x_i)$ valued in $G$, deterministic local outputs $\phi_i(x_i)$ already have aggregate exactly $f$, and so satisfy aggregate-only privacy. This observation does not extend to AND; the proved overlap is the obstruction. Shared correlated randomness invalidates the product-convolution premise, while statistical rather than perfect privacy invalidates the exact equalities. Neither change is ruled out by this proof.

Verification and remaining gaps

Exact exhaustive checks over selected small rational distributions on $\mathbb Z/2\mathbb Z$ verify the overlap identity whenever all three zero-output convolutions agree. The argument is a known elementary special-model impossibility, not a classification of information-theoretic ARE or a contradiction of source constructions under different guarantees.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2025/3. Relevant passages read: OWR 2025/3, printed pages150--151, Ishai ARE Part1.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed the written mathematical arguments, independently reran the finite checks, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

A separate OpenAI gpt-6-astra, reasoning ultra, invocation also read this written argument and independently performed the finite checks specified in the local peer-review audit. The exact serving revision is not exposed. This is model peer review, not independent human verification.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch026/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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