Card-Based Protocols with Mixed Encoding: Computable Functions and the Minimum Number of Cards

What it is

Preprint 2026/2704 on the IACR ePrint archive (received 31 October 2026, approved 4 November 2026) by Saeki Koizumi, Xiao-Nan Lu and Hidetomo Nabeshima. It studies card-based secure computation with mixed encoding: one player commits a bit with a single card, the other with two. The scope is perfectly secure protocols with finite runtime and public actions.

What the abstract adds beyond the summary

• Frozen card counted. A player giving a single-card commitment picks one of two cards of different types and keeps the other hidden. This "frozen card" is included in the card count, so Mizuki's five-card AND protocol (Theoretical Computer Science, vol. 622, 2016) is minimal under that accounting.
• Scope of the lower bound. No four-card AND protocol with mixed encoding exists, even with arbitrary shuffles and a non-committed output.
• Structural lemma. A secure protocol in committed format never turns over a single-card input. The authors use this to determine which functions are computable when all inputs are single-card commitments, and to characterize the mixed-encoding case, which includes AND but not XOR.
• Formal verification. The existence of a protocol is expressed as a quantified Boolean formula (QBF), with security as a quantifier alternation. QBF solvers confirm the four-card bound only for a class of protocols, not for all protocols.
• Further analysis. The paper discusses the number and type of shuffles, and analyzes Mizuki's protocol under injection attacks that change only the input.

Caveats

Only the abstract was available. The page gives no protocol details, solver names, formula sizes or the outcome of the injection-attack analysis.