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This is the part that makes Atrum possible. A prediction market needs a pool total — you cannot divide winnings without one. But a total computed from visible stakes is a total that reveals every stake. Atrum computes it from stakes nobody can read.

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Exponential ElGamal

Each stake is encrypted before it ever reaches the chain. The scheme is ElGamal on the BabyJubJub curve, in its exponential form:
where m is your stake, r is fresh randomness you pick per bet, G is the curve’s generator and H is the committee’s public key. The ciphertext is two curve points. Nothing about m is recoverable from them without the committee’s secret.
Fresh randomness per bet matters. Two bets of the same size encrypted with the same r would produce identical ciphertexts, and identical ciphertexts on-chain would say “these two stakes are equal” — a leak with no cryptography involved. The circuit refuses r = 0 and the client draws a new one each time.

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Why it adds

The useful property is that ciphertexts add:
Add the two points componentwise and you get an encryption of the sum. The contract performs that addition on every bet, and after a thousand bets it holds Enc(total) — having never seen a single stake. The accumulator contract holds ciphertext and has no decryption capability whatsoever. It is not that it declines to decrypt — it does not possess the key, and the key is not on the chain.

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The proof that ties it together

A ciphertext on its own proves nothing. You could encrypt 1,000,000 while spending a note worth 10. So the bet proof establishes, in zero knowledge, that:
  • you own an unspent note in the tree, and
  • the ciphertext you published is an encryption of exactly that note’s value
Both facts, without revealing the note or the value. The contract accepts the ciphertext only because the proof binds it to a real, owned, unspent note.
This is the step where privacy would quietly break in an obvious implementation. Making the amount private removes it from every check the contract used to perform, so each of those checks has to move into the circuit. A guard that silently stops applying is worse than one that was never written.

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Recovering the number

Decryption gives back [m]G — the message multiplied by the generator — not m itself. Recovering m from [m]G is the discrete logarithm problem, which is hard in general. It is tractable here only because stakes are bounded. The publisher uses baby-step giant-step, which costs about √bound operations: a pool of tens of millions is recovered in a few thousand steps, in milliseconds. That bound is not an accident. Every circuit range-checks values to 40 bits precisely so that decryption stays feasible. An unbounded plaintext would be encrypted perfectly well and then be unrecoverable forever.

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Who can decrypt, and what they publish

The committee key decrypts the total, and only the total — individual stakes were never separable to begin with, since the accumulator holds their sum. Mid-market odds come from that total, published deliberately coarse and infrequent. The reason is in Privacy model: a sequence of precise ratios, combined with publicly visible bet sides, becomes a system of equations that settlement’s exact totals can anchor. At settlement the final totals are published in the clear along with a Chaum-Pedersen proof — a zero-knowledge proof that the decryption is honest with respect to the published committee key. Not a claim to be trusted; a proof the contract verifies. That proof is why payouts are safe. A dishonest publisher could post any mid-market ratio it liked, and it would not matter, because no payout ever reads that number. Money is divided only by totals that came with a proof.