# Beaver triples {#beaver_triples}

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<div class="eli5"><b>ELI5.</b> A product opens as d = x − a and e = y − b, with a and b the preprocessing blinds. The product share is de plus the blinded cross terms, all local once d and e are public. The session keeps blinds that were already opened and only samples monomials it has not seen.</div>
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ABY2.0-style sessions open masked wires once (Patra, Schneider, Suresh,
and Yalame, USENIX Security 2021 / [ePrint 2020/1225](@ref bib_aby2)).
A fresh triple follows Beaver, [CRYPTO 1991](@ref bib_beaver): both masked factors are
reconstructed, and the product share is a local correction.
Optional MAC tags are the Shark/SPDZ check.

Constant-round word arithmetic is a separate gadget. [Ball, Malkin, and
Rosulek, CCS 2016](@ref bib_garble_gadgets) give free addition, free scaling
by a public constant, and a unary projection of `m − 1` ciphertexts.
[arith_garble.hpp](@ref dpf/arith_garble.hpp) is that circuit. A session does
not become one: it still opens δ once per wire. A public table on masked
bits is [FLUTE](@ref bib_flute) in [flute.hpp](@ref dpf/flute.hpp): the table
is a multi-fan-in inner product, and the online exchange is two bits per
output bit. `eval_trio` is the same product on three XOR shares of each mask.

One call that samples a list of formulae opens the new wires in one
round. Communication is one masked value per newly opened wire, plus a
tag share of the same width when MACs are on. Preprocessing is one blind
per wire and one product share per monomial.

`schedule_objective::prep` (default) peels shared factors for Appendix-E
prep savings and may add interactive rounds.
`schedule_objective::rounds` emits the polynomial in one online round
(Pika / online Grotto). Composer-owned sessions use `rounds`.

Compose FSS walks, ABY products, and RSS refreshes on one sink with
[protocol composition](@ref protocol_compose).

**Go deeper:** [a small word](@ref arith_garble_word),
[a public table](@ref flute_lut), [beaver.hpp](@ref dpf/beaver.hpp),
[compose.hpp](@ref dpf/compose.hpp),
[F_Beaver](@ref beaver.hpp), [F_BeaverAuth](@ref beaver.hpp),
and the cost notes in the [guided tour](@ref tour_beaver).