# Multiparty & 3-server {#multiparty}

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<div class="eli5"><b>ELI5.</b> 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.</div>
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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`.

```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).