docs/fhevm/solidity-guides/hcu.md

Homomorphic Complexity Units ("HCU") in FHEVM

This guide explains how to use Fully Homomorphic Encryption (FHE) operations in your smart contracts on FHEVM. Understanding HCU is critical for designing efficient confidential smart contracts.

Overview

FHE operations in FHEVM are computationally intensive compared to standard Ethereum operations, as they require complex mathematical computations to maintain privacy and security. To manage computational load and prevent potential denial-of-service attacks, FHEVM implements a metering system called Homomorphic Complexity Units ("HCU").

To represent this complexity, we introduced the Homomorphic Complexity Unit ("HCU"). In Solidity, each FHE operation consumes a set amount of HCU based on the operational computational complexity for hardware computation. Since FHE transactions are symbolic, this helps preventing resource exhaustion outside of the blockchain.

To do so, there is a contract named HCULimit, which monitors HCU consumption for each transaction and enforces two key limits:

• Sequential homomorphic operations depth limit per transaction: Controls HCU usage for operations that must be processed in order.
• Global homomorphic operations complexity per transaction: Controls HCU usage for operations that can be processed in parallel.

If either limit is exceeded, the transaction will revert.

HCU limit

The current devnet has an HCU limit of 20,000,000 per transaction and an HCU depth limit of 5,000,000 per transaction. If either HCU limit is exceeded, the transaction will revert.

To resolve this, you must do one of the following:

• Refactor your code to reduce the number of FHE operations in your transaction.
• Split your FHE operations across multiple independent transactions.

HCU costs for common operations

Boolean operations (ebool)

Function name	HCU (scalar)	HCU (non-scalar)
and	22,000	25,000
or	22,000	24,000
xor	2,000	22,000
not	-	2
select	-	55,000
randEbool	-	19,000

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Unsigned integer operations

HCU increase with the bit-width of the encrypted integer type. Below are the detailed costs for various operations on encrypted types.

8-bit Encrypted integers (euint8)

Function name	HCU (scalar)	HCU (non-scalar)
add	84,000	88,000
sub	84,000	91,000
mul	122,000	150,000
div	210,000	-
rem	440,000	-
and	31,000	31,000
or	30,000	30,000
xor	31,000	31,000
shr	32,000	91,000
shl	32,000	92,000
rotr	31,000	93,000
rotl	31,000	91,000
eq	55,000	55,000
ne	55,000	55,000
ge	52,000	63,000
gt	52,000	59,000
le	58,000	58,000
lt	52,000	59,000
min	84,000	119,000
max	89,000	121,000
neg	-	79,000
not	-	9
select	-	55,000
randEuint8	-	23,000

16-bit Encrypted integers (euint16)

Function name	HCU (scalar)	HCU (non-scalar)
add	93,000	93,000
sub	93,000	93,000
mul	193,000	222,000
div	302,000	-
rem	580,000	-
and	31,000	31,000
or	30,000	31,000
xor	31,000	31,000
shr	32,000	123,000
shl	32,000	125,000
rotr	31,000	125,000
rotl	31,000	125,000
eq	55,000	83,000
ne	55,000	83,000
ge	55,000	84,000
gt	55,000	84,000
le	58,000	83,000
lt	58,000	84,000
min	88,000	146,000
max	89,000	145,000
neg	-	93,000
not	-	16
select	-	55,000
randEuint16	-	23,000

32-bit Encrypted Integers (euint32)

Function name	HCU (scalar)	HCU (non-scalar)
add	95,000	125,000
sub	95,000	125,000
mul	265,000	328,000
div	438,000	-
rem	792,000	-
and	32,000	32,000
or	32,000	32,000
xor	32,000	32,000
shr	32,000	163,000
shl	32,000	162,000
rotr	32,000	160,000
rotl	32,000	163,000
eq	82,000	86,000
ne	83,000	85,000
ge	84,000	118,000
gt	84,000	118,000
le	84,000	117,000
lt	83,000	117,000
min	117,000	182,000
max	117,000	180,000
neg	-	131,000
not	-	32
select	-	55,000
randEuint32	-	24,000

64-bit Encrypted integers (euint64)

Function name	HCU (scalar)	HCU (non-scalar)
add	133,000	162,000
sub	133,000	162,000
mul	365,000	596,000
div	715,000	-
rem	1,153,000	-
and	34,000	34,000
or	34,000	34,000
xor	34,000	34,000
shr	34,000	209,000
shl	34,000	208,000
rotr	34,000	209,000
rotl	34,000	209,000
eq	83,000	120,000
ne	84,000	118,000
ge	116,000	152,000
gt	117,000	152,000
le	119,000	149,000
lt	118,000	146,000
min	150,000	219,000
max	149,000	218,000
neg	-	131,000
not	-	63
select	-	55,000
randEuint64	-	24,000

128-bit Encrypted integers (euint128)

Function name	HCU (scalar)	HCU (non-scalar)
add	172,000	259,000
sub	172,000	260,000
mul	696,000	1,686,000
div	1,225,000	-
rem	1,943,000	-
and	37,000	37,000
or	37,000	37,000
xor	37,000	37,000
shr	37,000	272,000
shl	37,000	272,000
rotr	37,000	283,000
rotl	37,000	278,000
eq	117,000	122,000
ne	117,000	122,000
ge	149,000	210,000
gt	150,000	218,000
le	150,000	218,000
lt	149,000	215,000
min	186,000	289,000
max	180,000	290,000
neg	-	168,000
not	-	130
select	-	57,000
randEuint128	-	25,000

256-bit Encrypted integers (euint256)

Function name	HCU (scalar)	HCU (non-scalar)
and	38,000	38,000
or	38,000	38,000
xor	39,000	39,000
shr	38,000	369,000
shl	39,000	378,000
rotr	40,000	375,000
rotl	38,000	378,000
eq	118,000	152,000
ne	117,000	150,000
neg	-	269,000
not	-	130
select	-	108,000
randEuint256	-	30,000

Encrypted addresses (euint160)

When using eaddress (internally represented as euint160), the HCU costs for equality and inequality checks and select are as follows:

Function name	HCU (scalar)	HCU (non-scalar)
eq	115,000	125,000
ne	115,000	124,000
select	-	83,000

Additional Operations

Function name	HCU
cast	32
trivialEncrypt	32
randBounded	23,000-30,000