91 lines
4 KiB
Markdown
91 lines
4 KiB
Markdown
# Point-programmable vector commitments {#ppvc_manual}
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A point-programmable vector commitment binds a vector
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`x` in `(Z/2^s Z)^n` and still lets one hidden coordinate be chosen
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after the commitment is published.
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`n` is a power of two, the bit length of the input type, and at most
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2^20, because evaluation stores one entry per domain point. `s` is
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the `Width` parameter, from 1 to 64.
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The manual construction is `dpf::ppvc`. `dpf::k_ppvc<K, ...>` is `K`
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independent copies of that object.
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The committer samples an index `i` and builds `s` aligned 1-bit DPF
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pairs there, the same point key as [DPF basics](@ref basics_body).
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Both roots of every pair are bound with a Naor commitment under a
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public matrix `A`. Opening releases one key from each pair, together
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with a shift `delta = xi - i`.
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Off `i`, the two keys of a pair evaluate to the same bit.
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At `i`, they evaluate to opposite bits.
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Choosing the side therefore writes an arbitrary value into that one
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coordinate and leaves every other coordinate fixed.
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The shift moves the written coordinate from `i` onto the public target
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`xi`. The opening carries `delta`, not `i` and not `xi`.
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`open(st, mu, tau, xi)` has two modes.
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- `mu = 0` programs the coordinate. After rotation, entry `xi` equals `tau`.
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- `mu = 1` programs the sum of every coordinate. That sum equals `tau`.
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Those two maps are bijections on `Z/2^s Z`. Programming one of them
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programs the other.
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```cpp
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using scheme = dpf::ppvc<std::uint8_t, 8>;
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const auto pp = scheme::setup();
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const auto [com, st] = scheme::commit(pp);
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const std::uint8_t xi = 40;
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const auto op = scheme::open(st, 0, 0x5a, xi);
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const auto x = scheme::eval(op); // hidden indexing
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const auto rotated = scheme::eval_rotated(op); // value 0x5a sits at xi
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const bool ok = scheme::accept(pp, com, op, x, xi);
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```
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`setup` samples `A`. `setup_from_seed` expands one 128-bit seed into the
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same matrix, which is the common random string when many sessions share
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it. `commit` samples `i`. `commit_at` uses an index the caller already
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chose. The shift hides `i` when that index was sampled independently of
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`xi`. `commit_from_seed` and `commit_at_from_seed` rerun key generation
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from a replica seed. Seed expansion keeps its counter in thread-local
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storage, so two expansions on one thread must not overlap.
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## What an opening proves {#ppvc_verify}
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`verify` checks each opened root against its Naor string.
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`accept` also checks the programmed statement: the rotated coordinate
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when `mu` is 0, the column sum when `mu` is 1.
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Correction words travel with the opened key. They are not inside the
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commitment. `verify` sees one side of each pair.
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`check_well_formed` is the check on a replica the committer still holds:
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shared correction words, party bits 0 and 1, both Naor openings, and
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exactly one place where the two keys disagree, at the recorded index,
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with payload 1.
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`audit` expands a seed and accepts when the published commitment matches
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that expansion and the replica is well formed.
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When many replica seeds sit as leaves of a GGM tree, the audit opening of
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the pool is a [`dpf::pprf_copath`](@ref dpf/pprf.hpp) built by
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`dpf::puncture(master, live…, /*program_hidden=*/false)`: every audited
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leaf re-expands with `dpf::pprf_eval`, and a live seed is never among the
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published nodes. Sampling the audit set and combining live copies stay in
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the protocol, not in this library.
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`k_ppvc` asks for the same checks on every copy, and for distinct hidden
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indices. `combine_rotated` adds the rotated vectors in `Z/2^s Z`.
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Reprogramming copy `r` changes coordinate `xi[r]` of that sum.
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The commitment is `2 * s * (3 * 128 + Sigma)` bits.
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`Sigma` defaults to 128 and must be a multiple of 8.
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The generator is `dpf::prg::aes128` unless another 128-bit PRG is named.
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**Defined in**\n
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@ref dpf/ppvc.hpp
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**Try**\n
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@ref mwe/ppvc.cpp
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Naor's string commitment is Moni Naor, [Bit Commitment Using Pseudorandomness](@ref bib_naor), Journal of Cryptology 1991.
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The point keys are the Boyle–Gilboa–Ishai construction named in
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[DPF basics](@ref point_functions).
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