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The parties that answer are not the process that combines, and neither is where the ciphertext was written. All three are separate machines, so both types cross a wire and the encoding belongs to them rather than to whichever transport happens to carry them this week. Decoding takes the group, the way EmptyConfig does, because a point is bytes plus the curve they are read on. A record decoded under the wrong group fails at the decode instead of becoming a valid value on a curve nobody meant, and an answer naming no party is refused before it can occupy a seat.