Two Results on Common-Mask Unclonability

This note records two results about common-mask challenges in unclonable encryption (UE). I believe both statements below are correct.

1. A Goldreich-Levin theorem for UE

Fix any finite-dimensional UE token ensemble. Split one token between Bob and Charlie before revealing the key \(k\) and a uniformly random common mask \(r\in\lbrace 0,1\rbrace^n\). Let \(p_D\) be the probability that both parties correctly output \(\langle r,m\rangle\).

Theorem. If \(p_D\geq1/2\), there exist local one-copy measurements, with no post-split communication, such that both parties recover \(m\) with probability

\[ p_S\geq(2p_D-1)^4. \]

Equivalently,

\[ p_D=\frac12+\varepsilon \quad\Longrightarrow\quad p_S\geq16\varepsilon^4. \]

This holds for arbitrarily entangled post-split states and does not require commuting decoders. It is an information-theoretic, possibly inefficient and nonuniform reduction.

2. An exponential bound for the two-basis UE parity game

Sample \(x,\theta,r\) independently and uniformly from \(\lbrace 0,1\rbrace^n\). Split one BB84 state

\[ \lvert x_\theta\rangle =\bigotimes_{j=1}^n H^{\theta_j}\lvert x_j\rangle \]

before revealing the common question \((\theta,r)\). Both recipients must output \(\langle r,x\rangle\) without communicating. Let \(\omega_n\) be the supremum of their joint success probability over all finite-dimensional splitting channels and local decoders.

Theorem. For every \(n\geq1\),

\[ \omega_n\leq \frac12\left[1+\left(\frac{\sqrt3}{2}\right)^n\right]. \]

Thus

\[ \omega_n-\frac12 \leq\frac12\left(\frac{\sqrt3}{2}\right)^n =2^{-\Omega(n)}. \]

An explicit product Breidbart strategy gives the lower bound

\[ \omega_n\geq \frac12+ \frac12\left(\frac{1+1/\sqrt2}{2}\right)^n. \]

The bounds are not equal. The proof does not establish the exact value, tightness, a product theorem \(\omega_n=\omega_1^n\), or a bound for infinite-dimensional or commuting-operator strategies.

Public timestamp and provenance

The timestamp binds the exact approved statement and the SHA-256 digests of both source manuscripts, rather than a manually entered page date.