libdpf/doc/pages/multiparty.md

Multiparty & 3-server

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ELI5. make_dpf3 is two ordinary spines per party and a Shamir payload, so any two parties open and the third key is independent of the value. make_it_dpf3 shares the whole truth table instead, and all three shares are required. Doerner–Shelat and geneval are the two-party, no-dealer alternatives.

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Two-party keys are the default. The library also builds three-evaluator (2,3) keys, information-theoretic three-server DPFs, and dealer-free two-party keygen when the index is already shared.

Construction	Parties	Header
[dpf::make_dpf3](@ref dpf/dpf3.hpp)	Any two of three open (Shamir / dual spine, [ePrint 2024/1658](@ref bib_dpf3))	dpf3.hpp
[dpf::make_dpf3_doerner_shelat](@ref dpf/dpf3_ds.hpp)	Same keys from XOR shares of alpha	dpf3_ds.hpp
make_dpf3_cmp / make_dpf3_ic / make_multipoint3	Comparisons, intervals, multipoint	dpf3_cmp.hpp, dpf3_multipoint.hpp
[dpf::make_it_dpf3](@ref dpf/it_dpf3.hpp)	Information-theoretic 3-server DPF ([ePrint 2023/028](@ref bib_itdpf))	it_dpf3.hpp
[dpf::make_dpf_doerner_shelat](@ref dpf/doerner_shelat.hpp)	Two parties, shared index, no dealer for the point	doerner_shelat.hpp
[dpf::geneval_*](@ref dpf/geneval.hpp)	Answer shares for one query, no reusable key	geneval.hpp
[dpf::shamir::deal](@ref dpf/shamir.hpp)	(K,N) Shamir. (2,3) is the make_dpf3 payload	shamir.hpp

The payload inside make_dpf3 is degree-1 Shamir on the points 1, 2, and 3. That access structure is shamir::two_of_three, the type shamir_share. The same split takes other thresholds: shamir::deal<T, K, N> and shamir::share_secret. make_dpf3 stays (2,3). See [secret shares](@ref secret_shares) and examples/mwe/shamir.cpp.

auto [k1, k2, k3] = dpf::make_dpf3(std::uint8_t{42}, dpf::fp61{7});
auto opened = dpf::reconstruct(
    dpf::as_share(k1, dpf::eval_point(k1, std::uint8_t{42})),
    dpf::as_share(k2, dpf::eval_point(k2, std::uint8_t{42})),
    dpf::as_share(k3, dpf::eval_point(k3, std::uint8_t{42})));

Party meshes and socket walks live under [trio.hpp](@ref dpf/net/trio.hpp) and the party/ harnesses. Application-shaped sketches include [3-party Duoram](@ref app_duoram) and [three-server PIR](@ref app_pir3).

Go deeper: [guided tour — three evaluators](@ref tour_dpf3), [Doerner–Shelat](@ref tour_ds), [evaluating DPFs](@ref evaluation), [bibliography](@ref bibliography).