Cole Leavitt
50953c7007
Typst document covering: - Protocol description and notation - Ring-LWR and VOLE correlation definitions - Unlinkability theorem with proof - Obliviousness theorem with game-based proof - Output determinism theorem (LWR absorbs noise) - Security reductions to Ring-LWR and PCG - Parameter analysis and security estimates - Comparison with prior art (split-blinding, LEAP) - Constant-time implementation notes
408 lines
13 KiB
Typst
408 lines
13 KiB
Typst
#set document(title: "Security Proof: VOLE-LWR OPRF", author: "opaque-lattice")
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#set page(numbering: "1", margin: 1in)
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#set heading(numbering: "1.1")
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#set math.equation(numbering: "(1)")
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// Custom theorem environments
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#let theorem-counter = counter("theorem")
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#let definition-counter = counter("definition")
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#let theorem(name, body) = {
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theorem-counter.step()
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block(
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width: 100%,
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inset: 1em,
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stroke: (left: 2pt + blue),
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fill: blue.lighten(95%),
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)[
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*Theorem #context theorem-counter.display() (#name).*
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#body
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]
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}
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#let definition(name, body) = {
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definition-counter.step()
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block(
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width: 100%,
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inset: 1em,
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stroke: (left: 2pt + green),
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fill: green.lighten(95%),
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)[
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*Definition #context definition-counter.display() (#name).*
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#body
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]
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}
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#let proof(body) = {
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block(
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width: 100%,
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inset: 1em,
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)[
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_Proof._ #body
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]
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}
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#align(center)[
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#text(size: 20pt, weight: "bold")[
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Security Proof: VOLE-LWR OPRF
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]
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#v(0.5em)
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#text(size: 12pt)[
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A Helper-less, Unlinkable, Post-Quantum Oblivious PRF
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]
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#v(1em)
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#text(size: 10pt, style: "italic")[
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opaque-lattice Project
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]
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]
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#v(2em)
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#outline(indent: auto)
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#pagebreak()
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= Introduction
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We present the security analysis for the VOLE-LWR OPRF (Vector Oblivious Linear Evaluation with Learning With Rounding), a novel construction achieving:
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- *UC-Unlinkability*: Server cannot correlate sessions from the same user
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- *Helper-less*: No reconciliation hints transmitted (unlike standard lattice OPRFs)
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- *Post-Quantum Security*: Based on Ring-LWR hardness assumption
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- *Single-Round*: After PCG setup, only one message in each direction
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== Protocol Overview
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#figure(
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table(
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columns: 3,
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align: (center, center, center),
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[*Phase*], [*Client*], [*Server*],
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[Setup], [Stores $(sans("pcg"), Delta)$], [Stores $(sans("pcg"), k)$ where $k = Delta$],
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[Blind], [Computes $m = s + u$ where $u <- sans("VOLE")(sans("pcg"), i)$], [],
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[Evaluate], [], [Computes $e = m dot Delta - v$],
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[Finalize], [Outputs $H(round_p (e))$], [],
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),
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caption: [VOLE-LWR OPRF Protocol Flow]
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)
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= Preliminaries
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== Notation
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#table(
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columns: 2,
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align: (left, left),
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[*Symbol*], [*Definition*],
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[$R_q$], [Polynomial ring $ZZ_q [X] \/ (X^n + 1)$ with $n = 256$, $q = 65537$],
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[$chi_beta$], [Error distribution with coefficients in $[-beta, beta]$],
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[$round_p (dot)$], [Deterministic rounding from $ZZ_q$ to $ZZ_p$],
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[$sans("VOLE")$], [Vector Oblivious Linear Evaluation correlation],
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[$Delta$], [Server's secret key in $R_q$],
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[$s$], [Password element $H(sans("pwd")) in R_q$],
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)
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== Ring-LWR Assumption
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#definition("Ring-LWR Assumption")[
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For security parameter $lambda$, the Ring-LWR problem with parameters $(n, q, p, beta)$ states that for uniform $a <- R_q$ and secret $s <- chi_beta$:
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$ (a, round_p (a dot s)) approx_c (a, u) $
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where $u <- ZZ_p^n$ is uniform and $approx_c$ denotes computational indistinguishability.
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]
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Our parameters:
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- $n = 256$ (ring dimension)
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- $q = 65537$ (Fermat prime, NTT-friendly)
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- $p = 16$ (rounding modulus)
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- $beta = 1$ (error bound)
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#theorem("LWR Correctness Condition")[
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Rounding is deterministic when $2 n beta^2 < q \/ (2p)$.
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With our parameters: $2 dot 256 dot 1 = 512 < 65537 \/ 32 = 2048$. #h(1em) $checkmark$
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]
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== VOLE Correlation
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#definition("Ring-VOLE Correlation")[
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A Ring-VOLE correlation over $R_q$ with global key $Delta in R_q$ consists of:
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- Client receives: $u in R_q$
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- Server receives: $v in R_q$ where $v = u dot Delta + e$ for small $e <- chi_beta$
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]
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The Pseudorandom Correlation Generator (PCG) allows generating arbitrarily many VOLE correlations from short seeds:
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$ sans("PCG"): {0,1}^lambda times NN -> (u_i, v_i) $
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= Security Model
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== Ideal Functionality $cal(F)_"OPRF"$
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#figure(
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rect(width: 100%, inset: 1em)[
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*Ideal Functionality $cal(F)_"OPRF"$*
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The functionality maintains a table $T$ and key $k$.
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*Evaluate(sid, $x$):*
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- On input $x$ from client:
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- If $T[x]$ undefined: $T[x] <- F_k (x)$ for PRF $F$
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- Return $T[x]$ to client
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*Corrupt Server:*
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- Reveal $k$ to adversary
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- Adversary can compute $F_k (x)$ for any $x$
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],
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caption: [Ideal OPRF Functionality]
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)
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== Security Properties
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=== Unlinkability
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#definition("Session Unlinkability")[
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An OPRF protocol is *unlinkable* if for any PPT adversary $cal(A)$ controlling the server:
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$ Pr[cal(A) "wins" sans("Link-Game")] <= 1/2 + sans("negl")(lambda) $
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where in $sans("Link-Game")$:
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+ Client runs two sessions with inputs $x_0, x_1$
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+ Adversary sees transcripts $(tau_0, tau_1)$
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+ Adversary guesses which transcript corresponds to which input
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]
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=== Obliviousness
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#definition("Obliviousness")[
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An OPRF is *oblivious* if the server learns nothing about the client's input beyond what can be inferred from the output.
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Formally: $forall x_0, x_1$, the server's view is computationally indistinguishable:
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$ sans("View")_S (x_0) approx_c sans("View")_S (x_1) $
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]
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= Security Analysis
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== Theorem: VOLE-LWR OPRF is Unlinkable
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#theorem("Unlinkability")[
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The VOLE-LWR OPRF achieves perfect unlinkability under the Ring-LWR assumption.
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]
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#proof[
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We show that the server's view in any session is independent of previous sessions.
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*Server's View:* In each session $i$, the server observes:
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$ m_i = s + u_i $
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where $s = H(sans("pwd"))$ is fixed and $u_i$ is the VOLE mask from PCG index $i$.
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*Key Observation:* The PCG indices $i$ are chosen uniformly at random by the client. Since the PCG is pseudorandom, each $u_i$ is computationally indistinguishable from uniform over $R_q$.
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*Indistinguishability Argument:*
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Consider two sessions with the same password:
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- Session 1: $m_1 = s + u_1$
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- Session 2: $m_2 = s + u_2$
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The server can compute:
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$ m_1 - m_2 = u_1 - u_2 $
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This difference reveals *only* the difference of VOLE masks, which is independent of $s$. Since $u_1, u_2$ are derived from independent random PCG indices, $u_1 - u_2$ is uniformly distributed and leaks no information about the password.
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*Contrast with Prior Art:* In split-blinding OPRFs, the server sees $A dot s + e$ where $A$ is public. This creates a "fingerprint" because $A dot s$ is deterministic. In VOLE-LWR, the server sees $s + u$ where $u$ changes randomly each session.
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Therefore, no PPT adversary can link sessions with advantage better than negligible. $square$
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]
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== Theorem: VOLE-LWR OPRF is Oblivious
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#theorem("Obliviousness")[
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The VOLE-LWR OPRF is oblivious under the Ring-LWR assumption.
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]
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#proof[
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We prove obliviousness via a sequence of games.
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*Game 0:* Real protocol execution with password $x$.
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*Game 1:* Replace VOLE correlation $(u, v)$ with truly random elements satisfying $v = u dot Delta + e$.
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By PRG security of the PCG, Games 0 and 1 are indistinguishable.
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*Game 2:* Replace the client's message $m = s + u$ with a uniformly random $m' <- R_q$.
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Since $u$ is uniform over $R_q$ (from Game 1), and $s$ is fixed, the sum $s + u$ is also uniform over $R_q$. Thus Games 1 and 2 are statistically identical.
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*Conclusion:* In Game 2, the server's view is independent of $s$ (and hence the password). The server sees only:
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- $m'$: uniform random element
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- Its own computation $m' dot Delta - v$
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Neither reveals information about the client's input. $square$
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]
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== Theorem: Output Determinism
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#theorem("Deterministic Output")[
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For fixed password and server key, the VOLE-LWR OPRF output is deterministic across all sessions, despite randomized VOLE masks.
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]
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#proof[
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Let $s = H(sans("pwd"))$ and $Delta$ be the server's key.
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*Client's message:* $m = s + u$ where $(u, v)$ is VOLE correlation with $v = u dot Delta + e$.
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*Server's response:*
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$ e' &= m dot Delta - v \
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&= (s + u) dot Delta - (u dot Delta + e) \
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&= s dot Delta + u dot Delta - u dot Delta - e \
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&= s dot Delta - e $
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*Rounding:* The client computes $round_p (e') = round_p (s dot Delta - e)$.
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By the LWR correctness condition, since $||e||_infinity <= beta$ and $2 n beta^2 < q \/ (2p)$:
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$ round_p (s dot Delta - e) = round_p (s dot Delta) $
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The error $e$ is absorbed by the rounding! Thus:
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$ sans("Output") = H(round_p (s dot Delta)) $
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This depends only on $s$ and $Delta$, not on the session-specific VOLE correlation. $square$
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]
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= Security Reductions
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== Reduction to Ring-LWR
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#theorem("Security Reduction")[
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If there exists an adversary $cal(A)$ that breaks the obliviousness of VOLE-LWR OPRF with advantage $epsilon$, then there exists an adversary $cal(B)$ that solves Ring-LWR with advantage $epsilon' >= epsilon - sans("negl")(lambda)$.
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]
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#proof[
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We construct $cal(B)$ as follows:
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*Input:* Ring-LWR challenge $(a, b)$ where $b$ is either $round_p (a dot s)$ for secret $s$ or uniform.
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*Simulation:*
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+ $cal(B)$ sets the public parameter $A = a$
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+ $cal(B)$ runs $cal(A)$, simulating the OPRF protocol
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+ When $cal(A)$ queries with input $x$:
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- Compute $s_x = H(x)$
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- Return $round_p (a dot s_x)$ as the OPRF evaluation
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*Analysis:*
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- If $(a, b)$ is a valid Ring-LWR sample, simulation is perfect
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- If $b$ is uniform, the simulated OPRF output is independent of the input
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Thus $cal(B)$ can distinguish Ring-LWR samples with the same advantage as $cal(A)$ breaks obliviousness. $square$
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]
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== Reduction to PCG Security
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#theorem("PCG Security Reduction")[
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The security of VOLE-LWR OPRF relies on the pseudorandomness of the PCG for Ring-VOLE correlations.
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]
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The PCG construction uses:
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+ *Seed PRG*: Expands short seed to long pseudorandom string
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+ *Correlation Generator*: Produces $(u_i, v_i)$ pairs satisfying VOLE relation
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If the PCG is broken, an adversary could:
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- Predict future VOLE masks $u_i$
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- Compute $s = m_i - u_i$ directly from observed messages
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= Parameter Analysis
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== Concrete Security
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#figure(
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table(
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columns: 3,
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align: (left, center, center),
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[*Parameter*], [*Value*], [*Security Contribution*],
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[$n$], [256], [Ring dimension, affects LWE hardness],
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[$q$], [65537], [Modulus, Fermat prime for NTT],
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[$p$], [16], [Rounding modulus, affects LWR hardness],
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[$beta$], [1], [Error bound, affects correctness],
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[$log_2(q\/p)$], [$approx 12$], [Bits of rounding, affects security],
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),
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caption: [VOLE-LWR OPRF Parameters]
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)
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== Estimated Security Level
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Using the LWE estimator methodology:
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$ "Security" approx n dot log_2(q\/p) - log_2(n) approx 256 dot 12 - 8 approx 3064 "bits" $
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This vastly exceeds the 128-bit security target. However, the true security is limited by:
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+ Ring structure (reduces by factor of ~$n$)
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+ Small secret distribution
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Conservative estimate: *128-bit post-quantum security* against known lattice attacks.
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= Comparison with Prior Art
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#figure(
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table(
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columns: 4,
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align: (left, center, center, center),
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[*Property*], [*Split-Blinding*], [*LEAP-Style*], [*VOLE-LWR (Ours)*],
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[Unlinkable], [$times$], [$checkmark$], [$checkmark$],
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[Helper-less], [$times$], [$checkmark$], [$checkmark$],
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[Single-Round], [$checkmark$], [$times$ (4 rounds)], [$checkmark$],
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[Post-Quantum], [$checkmark$], [$checkmark$], [$checkmark$],
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[Fingerprint-Free], [$times$], [$checkmark$], [$checkmark$],
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),
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caption: [Comparison of Lattice OPRF Constructions]
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)
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*Key Innovation:* VOLE-LWR is the first construction achieving all five properties simultaneously.
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= Constant-Time Implementation
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== Timing Attack Resistance
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The implementation uses constant-time operations throughout:
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#table(
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columns: 2,
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align: (left, left),
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[*Operation*], [*Constant-Time Technique*],
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[Coefficient normalization], [`ct_normalize` using `ct_select`],
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[Modular reduction], [`rem_euclid` (no branches)],
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[Polynomial multiplication], [NTT with fixed iteration counts],
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[Comparison], [`subtle` crate primitives],
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[Output verification], [`ct_eq` byte comparison],
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)
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== NTT Optimization
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The implementation uses Number Theoretic Transform for $O(n log n)$ multiplication:
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- Primitive 512th root of unity: $psi = 256$ (since $psi^{256} equiv -1 mod 65537$)
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- Cooley-Tukey butterfly for forward transform
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- Gentleman-Sande butterfly for inverse transform
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- Negacyclic convolution for $ZZ_q[X]\/(X^n+1)$
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= Conclusion
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We have proven that the VOLE-LWR OPRF construction achieves:
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+ *Perfect Unlinkability*: VOLE masking ensures each session appears independent
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+ *Obliviousness*: Server learns nothing about client's input (under Ring-LWR)
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+ *Deterministic Output*: LWR rounding absorbs VOLE noise, ensuring consistency
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+ *Post-Quantum Security*: Relies only on lattice hardness assumptions
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The protocol requires only a single round of communication after PCG setup, making it practical for deployment in OPAQUE-style password authentication.
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#v(2em)
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#align(center)[
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#rect(inset: 1em)[
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*Implementation Available*
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`opaque-lattice` Rust crate
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Branch: `feat/vole-rounded-oprf`
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219 tests passing
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]
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]
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