## Overview#

In Oxford2, each instance of a baker’s misconduct results in the slashing of a fixed percentage of their staked funds. However, this approach does not distinguish between innocent mistakes and deliberate attacks. The rule conservatively punishes every misconduct with strong penalties. The rationale for this model is its simplicity and the observation that the cost of avoiding these mistakes is low anyway. Nonetheless, it overlooks the impact of double signing of attestations on the entire block committee, treating all cases uniformly and ignoring collusion among bakers.

To better reflect this distinction, protocols starting with Paris adjusts the amount of slashing based on the fraction of double-attestation in a single block. A low fraction of misconduct incurs moderate penalties, while a high fraction of misconduct is deemed to be critical and faces more serious repercussions.

This document presents the definition of the adaptive slashing function implementing this idea, as well as the new forbidden period.

### Informal presentation#

For a given block, we call “fraction of double-attestations” the ratio between the total weight of attestations for which a valid denunciation has been included and the total weight of possible attestations in a block.

The shape of the slashing function for double attestations is the following:

f: fraction of double attestations

S: slashing ratio

1.00%

0.09%

5.00%

2.25%

10.00%

9.00%

20.00%

36.00%

23.57%

50.00%

33.34%

100.00%

100.00%

Instead of using a constant function as in Oxford2, we use a convex function that saturates at 100% when a critical fraction of doubled attestations are issued. Accidental double attestations are unlikely to cause a large amount of slashing to be applied, but concerted attacks result in severe penalties.

Remark 1. Even though the baker does not attest to a weight exactly equivalent to its staked funds at each block, the slashing function is applied to bakers’ total staked funds for simplicity.

Remark 2. As in Oxford2, slashing happens at the end of each cycle. Since the denunciation period for a block ranges over its cycle and the next one, a baker can be punished for misbehaving during the previous cycle, not only the one that has just ended.

Remark 3. If a baker wants to get back their at-stake funds, it takes more than 2 cycles to complete the unstaking process. This ensures that the baker can’t decrease their at-stake funds after being denunciated and before facing penalties.

### A formal definition of slashing function for double-attestations#

• $$\mathcal{W}$$ denotes the maximal possible weight of attestations in a block, that is, the fixed number of available slots in any block. It is also known as CONSENSUS_COMMITTEE_SIZE.

• $$f(B)$$ is the fraction of double attestations for block $$B$$, that is, the ratio of the total weight of double attestations in $$B$$, over $$\mathcal{W}$$.

• $$T$$ is the threshold for the fraction of double attestations to be considered critical. A typical value for $$T$$ is $${1 \over 3} \mathcal{W}$$, which is the difference between $$\mathcal{W}$$ and the CONSENSUS_THRESHOLD which is set to $${2 \over 3} \mathcal{W}$$.

We define $$S(B)$$ the percentage of slashed funds for all misbehaving bakers at the block $$B$$ as follows:

$$S(B) = \text{min} (100\%, {1 \over T^2} \cdot f(B)^2 \cdot 100\%)$$

Then, the percentage of slashed funds $$S(b,C)$$ for a baker $$b$$ at the end of the cycle $$C$$ is defined as follows:

$$S(b, C) = \text{min} (100\%, \sum_{(b, B) \in C} S(B))$$

where $$(b, B) \in C$$ means that:

• the double attestation by baker $$b$$ of block $$B$$ has been denounced before the end of cycle $$C$$, and

• $$C$$ is the last cycle of the denunciation period for $$B$$.

## New definition for the forbidden period#

Given that slashing occurs with a delay, immediate action at denunciation time is necessary upon clear evidence of a baker’s misbehavior to prevent further misconduct, or to protect the baker against their own faulty setup. Any double-signing denunciation immediately triggers the beginning of a forbidden period that lasts at least 2 cycles, to make sure the slashing occurs before accepting new attestations or blocks from the baker.

Note that it is still possible for one baker to commit multiple double signings, but only if they all happen before any corresponding denunciation gets included in a block.

This forbidding is lifted as soon as both following conditions are met:

• all pending slashings for the delegate have occurred, and

• the current total frozen stake for the delegate (sum of the staking balances of the delegate itself and its stakers) is at least as high as the active stake that was used CONSENSUS_RIGHTS_DELAY cycles ago to compute the consensus rights for the next cycle.

The second condition may be fulfilled when the delegate and/or stakers stake additional funds so that the total frozen stake grows back to its pre-slashing value, thus matching the rights computed before the slashing. Or it may be fulfilled CONSENSUS_RIGHTS_DELAY cycles after the slashing, when the rights for the next cycle are finally based on the post-slashing stake.