How will the scheduled increase of ring size in the current hard fork v7 affect security/privacy, transaction size/fees and verification time?
Is there a reason why exactly 7 and not 10 or maybe 20?
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Privacy: Outputs are where amounts of Monero are stored on the blockchain. When you spend one of your outputs in a transaction, it cannot be known which of 7 different source outputs you're really spending. Therefore your anonymity set grows by 40%. Another advantage is that outputs that you receive will now more likely be used as decoys in other people's transactions, which adds to your plausible deniability.
Transaction size: A typical Monero transaction would be 13,792 bytes, up from 13,524 bytes. This is an increase of 2.0%.
Transaction fees: Monero uses KB boundaries when calculating fees. Since a typical transaction would still be counted as a 14KB transaction whether it has 5 or 7 ring members, in most cases the fee would be unchanged.
It is of course possible that, depending on your specific transaction (especially if you're spending many outputs in the same transaction), that the transaction fee would increase. For example, if you're spending 10 outputs in a single transaction, the fee would increase by 6%.
Transaction verification time: Ring signature verification time is directly proportional to the number of ring members. Verification of the MLSAG ring signatures (which obfuscate the source outputs) would therefore increase by 40%. However, MLSAGs are not the only part of the transaction that needs to be verified. There is also a range proof Borromean ring signature required for each output created.
By my estimations, a Monero transaction currently requires about 70ms to verify on a single core, of which about 60ms is spent verifying just the output range proofs. Therefore overall verification time would rise from approx 70ms to approx 74ms, which is a 6% increase.
The choice of 7 is a tradeoff between privacy and verification time.
Later this year, the Borromean ring signatures will be replaced with Bulletproof signatures. This may very significantly reduce transaction verification times and provide for the possibility of increasing ring sizes further.