From fb4ff718259b9966542072b407bc2e23c330e4af Mon Sep 17 00:00:00 2001 From: zac-williamson Date: Thu, 30 Nov 2023 16:26:09 +0000 Subject: [PATCH] added existing hackmd docs to public-vm --- yellow-paper/docs/public-vm/alu.md | 35 ++++ .../public-vm/bytecode-validation-circuit.md | 190 ++++++++++++++++++ yellow-paper/docs/public-vm/control-flow.md | 32 +++ .../docs/public-vm/gen/images/alu/alu.png | Bin 0 -> 476263 bytes .../images/control-flow/avm-control-flow.png | Bin 0 -> 569552 bytes .../gen/images/state-model/memory.png | Bin 0 -> 43597 bytes yellow-paper/docs/public-vm/state-model.md | 105 ++++++++++ yellow-paper/docs/public-vm/tagged-memory.md | 60 ++++++ 8 files changed, 422 insertions(+) create mode 100644 yellow-paper/docs/public-vm/alu.md create mode 100644 yellow-paper/docs/public-vm/bytecode-validation-circuit.md create mode 100644 yellow-paper/docs/public-vm/control-flow.md create mode 100644 yellow-paper/docs/public-vm/gen/images/alu/alu.png create mode 100644 yellow-paper/docs/public-vm/gen/images/control-flow/avm-control-flow.png create mode 100644 yellow-paper/docs/public-vm/gen/images/state-model/memory.png create mode 100644 yellow-paper/docs/public-vm/state-model.md create mode 100644 yellow-paper/docs/public-vm/tagged-memory.md diff --git a/yellow-paper/docs/public-vm/alu.md b/yellow-paper/docs/public-vm/alu.md new file mode 100644 index 00000000000..bd83dde6a23 --- /dev/null +++ b/yellow-paper/docs/public-vm/alu.md @@ -0,0 +1,35 @@ +# Algebraic Logic Unit + +The algebraic logic unit performs operations analogous to an arithmetic unit in a CPU. + +This component of the VM circuit evaluates both base-2 arithmetic operations and prime-field operation. It takes its input/output from the intermediate registers in the state controller. + +The following block diagram maps out an draft of the internal components of the "ALU" + +![](./gen/images/alu/alu.png) + +Notes: + +For logic operations (e.g. AND/OR) we use lookup tables. The max. size of each lookup table cannot grow too large as the Prover pays a constant cost linear with the size of the lookup table. + +To this end we use lookup tables for logic operations that take _8-bit input operands_ for a total table size of 2^16. + +i.e. the table contains the output for every possible 8-bit combination of 2 input operands. + +#### Slice registers + +We need to slice our inputs into 8-bit chunks for logic operations, in order to index the lookup tables. + +As a simplification, we can say that _any_ operation that requires range-constraints will split the input operands into 8-bit slices, as we can then apply consistent range-checking logic. + +#### Carry flag + +Used to test for overflows. If we want the high-level instruction set to have "add with carry" we need to expose the carry flag to the state controller. + +## Example operation: 32-bit ADD(a, b, c) + +Assume we start with `a` in intermediate register `R1`, `b` in intermediate register `R2`, and `c` in intermediate register `R3` + +1. Store the first 32 bits of `a + b` in slice registers `s1, s2, s3, s4`, with the carry bit in `carry` +2. Validate $a + b = s_1 + 2^8s_2 + 2^{16}s_3 + 2^{24}s_4 + 2^{32}\text{carry}$ +3. Validate $c = s_1 + 2^8s_2 + 2^{16}s_3 + 2^{24}s_4$ diff --git a/yellow-paper/docs/public-vm/bytecode-validation-circuit.md b/yellow-paper/docs/public-vm/bytecode-validation-circuit.md new file mode 100644 index 00000000000..ccdd88cadc5 --- /dev/null +++ b/yellow-paper/docs/public-vm/bytecode-validation-circuit.md @@ -0,0 +1,190 @@ +# Bytecode Validation Circuit + +Goal: Validate that a polynomial commitment to AVM program opcodes maps to the bytecode representation of an AVM program of maximum size $n$. + +# Definitions - Curves and Fields + +The bytecode validation circuit is implemented over the BN254 elliptic curve with group elements defined via $\mathbb{G}_{bn254}$. + +The field $\mathbb{F}$ represents the finite field whose characteristic equals the number of points on the BN254 curve. + +# Bytecode representation + +Each opcode in the AVM can be described by an integer in the range $[0, \ldots, 256^{31}]$ (i.e. 31 bytes of data). All opcodes excluding `SET` require much less data than 31 bytes. + +In the AVM circuit architecture, multiple columns are used to define a VM operation. These columns describe the following quantities: + +1. the opcode to be executed (1 byte of data) +2. three parameter columns that define either literal values or memory indexes +3. three "flag" columns that define metadata associated with each parameter (e.g. whether the parameter a should be interpreted as a literal value or a memory index, or an indirect memory index) + +To both minimize the amount of information used to _define_ a given AVM program, the AVM posesses an additional column that describes the _packed_ opcode. i.e. the integer concatenation of all 6 of the above column values. We define this column via the vector of field elements $\mathbf{op} \in \mathbb{F}^n$ (where $n$ is the number of opcodes in the program). $\mathbf{op}$ is defined as the _column representation_ of an AVM program. + +## Packed bytecode representation + +When _broadcasting_ the data for AVM programs, we desire an encoding that minimizes the raw number of bytes broadcast, we call this the _packed representation_ of the program. + +The number of bytes required to represent an element in $\mathbf{op}$ in the AVM can be derived from the value of the 1st byte (e.g `ADD` requires 7 bytes of data - the ADD opcode (1 byte) and three memory indices (each of size 2 bytes)). + +See (ref: TODO!) for a table that describes the amount of data required for each opcode. + +Each field element in a BN254 circuit can represent _31_ bytes of bytecode data. The packed representation of an AVM program $\mathbf{b} \in \mathbb{F}^n$ is defined as the concatenation of $\mathbf{op}$ into 31-byte chunks, represented as field elements. + +There exists a mapping function $g$ that, given the packed representation $\mathbf{b}$, will produce the column representation $\mathbf{op}$. + +$$ +g(\mathbf{b}) = \mathbf{op} +$$ + +A full description of $g$ is provided [further down in this document](#Definition-of-mapping-function-g). + +## Committed representation + +The committed representation of an AVM program is an elliptic curve polynomial commitment $[P] \in \mathbb{G}_{bn254}$, created via the KZG polynomial commitment scheme (ref). + +$[P]$ is a commitment to $P(X) \in \mathbb{F}[X]^n$ where $P(X) = \sum_{i=0}^{n-1} op_i X^i$ + +# Bytecode validation logic + +Given inputs $\mathbf{b} \in \mathbb{F}^n$ and $[P] \in \mathbb{G}_{bn254}$, we must validate that $[P] = \text{commit}_{KZG}(g(\mathbf{b}))$. + +This requires the following _high level_ steps: + +1. For all $i \in [0, \ldots, n - 1]$, validate that $b_i < 256^{31} - 1$ +2. Compute $\mathbf{op} = g(\mathbf{b})$ +3. Perform a _polynomial consistency check_ between $\mathbb{op}$ and $[P]$ + +# Polynomial Consistency Check + +> The most straightforward way of validating $\mathbb{op}, [P]$ would be to directly construct $[P]$ from $\mathbb{op}$. +> We do not do this, as this would require a large multiscalar multiplication over the BN254 curve. This could only be performed efficiently over a Grumpkin SNARK circuit, which would add downstream complexity to the Aztec architecture (currently the only Grumpkin proofs being accumulated are elliptic-curve-virtual-machine circuits). The rollup circuit architecture already supports efficient recursive aggregation of BN254 proofs - the desire is for the bytecode validation circuit to be a canonical Honk SNARK over the BN254 field. + +To perform a polynomial consistency check between $\mathbb{op}$ and $[P]$, we perform the following: + +1. Generate a challenge $z \in \mathbb{F}$ by computing the Poseidon hash of $H(op_0, \ldots, op_{n-1}, [P])$ +2. Compute $\sum_{i=0}^{n-1} op_i z^i = r \in \mathbb{F}$ +3. Validate via a KZG opening proof that $[P]$ commits to a polynomial $P(X)$ such that $P(z) = r$ + +In the same manner that Honk pairings can be deferred via aggregating pairing inputs into an accumulator, the pairing required to validate the KZG opening proof can also be deferred. + +## Evaluating the polynomial consistency check within a circuit + +The direct computation of $r = \sum_{i=0}^{n-1} op_i z^i$ is trivial as the field is native to a BN254 SNARK circuit, and will require approx. 2 constraints per opcode. + +Validating a KZG opening proof will require approx. 3 non-native elliptic curve scalar multiplications, which will have a cost of approx. 30,000 constraints if using `stdlib::biggroup` from the PLONK standard library. + +The major cost of the consistency check is the Poseidon hash of the packed bytecode vector $\mathbb{b}$ and the commitment $[P]$ - this will incur approx. 22 constraints per element in $\mathbb{b}$ + +# Definition of mapping function $g$ + +The following is a pseudocode description of $g$, which can efficiently be described in a Honk circuit (i.e. no branches). + +We define a function `slice(element, idx, length)`. `element` is a field element interpreted as a length-31 byte array. `slice` computes the byte array `element[idx] : element[idx + length]`, converts into a field element and returns it. + +We define a size-256 lookup table `c` that maps an avm instruction byte to the byte length required to represent its respective opcode. + +``` +g(b) { + let i := 0; // index into bytecode array `b` + let j := 0; // byte offset of current bytecode element + let op := []; // vector of opcode values we need to populate + for k in [0, n]: + { + let f := b[i]; + let instruction_byte := f.slice(j, 1); + let opcode_length := c[instruction_byte]; + let bytes_remaining_in_f := 30 - j; + let op_split := opcode_length > bytes_remaining_in_f; + let bytes_from_f := op_split ? bytes_remaining_in_f : opcode_length; + let op_hi := f.slice(j, bytes_from_f); + + let f' := b[i+1]; + let bytes_from_f' := opcode_length - bytes_from_f; + let op_lo := f'.slice(0, bytes_in_f'); + + op[k] := op_lo + (op_hi << (bytes_in_f' * 8)); + i := i + op_split; + j := op_split ? bytes_in_f' : j + opcode_length; + } + return op; +} +``` + +Pseudocode definition of `slice` function constraints: + +We define `pow(x)` to be a size-31 lookup table that maps an input $x \in [0, \ldots, 31]$ into the value $2^{8x}$ + +We require the Prover has computed witness field elements `f_lo`, `f_hi`, `result` that satisfy the following constraints: + +``` +slice(f, index, length) +{ + assert(f_hi < pow(index)); + assert(f_lo < pow(31 - index - length)); + assert(result < pow(length)); + assert(f == f_lo + result * pow(31 - index - length) + f_hi * pow(31 - index)); + return result; +} +``` + +## Evaluating `g` within a Honk circuit + +The `g` function requires the contents of $\mathbb{b}$ be present via a lookup table. We can achieve this by instantiating elements of $\mathbb{b}$ via the ROM abstraction present in the Plonk standard library (table initialisation costs 2 constraints per element, table reads cost 2 constraints per element) + +We can instantiate tables `c` , `pow` as lookup tables via the same mechanism. + +The `slice` function requires 3 variable-length range checks. In Honk circuits we only can support fixed-length range checks. + +The following pseudocode defines how a variable-length range check can be composed of fixed-length range checks. Here we assume we have previously constrained all inputs to be less than $2^{248} - 1$ + +``` +less_than(a, b) { + // this block is not constrained and defines witness gneeration + let a_lo := a & (2^{124} - 1) + let b_lo := b & (2^{124} - 1) + let a_hi := (a >> 124) + let b_hi := (b >> 124) + let borrow := b_lo < a_lo + let r_lo := b_lo - a_lo + borrow*2^124 + let r_hi := b_hi - a_hi - borrow + + // this block defines constraints + assert(a_lo < 2^124) + assert(a_hi < 2^124) + assert(b_lo < 2^124) + assert(b_hi < 2^124) + assert(r_lo < 2^124) + assert(r_hi < 2^124) + assert(borrow*borrow - borrow = 0) // bool check + assert(a_lo + 2^{124}a_hi = a) + assert(b_lo + 2^{124}b_hi = b) + assert(r_lo = b_lo - a_lo + borrow*2^124) + assert(r_hi = b_hi - a_hi - borrow) +} +``` + +Each `slice` call requires three `less_than` calls, and each iteration of `g` requires 3 `slice` calls. In total this produces 36 size-124 range checks per iteration of `g`. Each size-124 range check requires approx. 5 constraints, producing 180 constraints of range checks per opcode processed. + +A rough estimate of the total constraints per opcode processed by the `g` function would be 200 constraints per opcdoe. + +# Bytecode Validation Circuit Summary + +The bytecode validation circuit takes, as public inputs, the packed bytecode array $\mathbf{b} \in \mathbb{F}$ and the bytecode commitment $[P] \in \mathbb{G}_{bn254}$ (represented via field elements). + +The circuit evaluates the following: + +1. For all $i \in [0, \ldots, n - 1]$, validate that $b_i < 256^{31} - 1$ +2. Compute $\mathbf{op} = g(\mathbf{b})$ +3. Perform a _polynomial consistency check_ between $\mathbf{op}$ and $[P]$ + +### Summary of main circuit costs + +The polynomial consistency check requires a Poseidon hash that includes the packed bytecode array $\mathbb{b}$. This requires approx. 22 Honk constraints per 31 bytes of bytecode. + +The `g` function will cost approx. 200 constraints per opcode. + +For a given length `n` , the approx. number of constraints required will be approx `222n`. + +A circuit of size 2^21 (2 million constraints) will be able to process a program containing approximately $n = 9,400$ steps. In contrast, a Soldity program can contain a maximum of 24kb of bytecode. + +Note: unless the efficiency of the validation circuit can be improved by a factor of ~4x, it will not be possible to construct bytecode validation proofs client-side in a web browser. Delegating proof construction to a 3rd party would be acceptable in this context because the 3rd party is untrusted and no secret information is leaked. diff --git a/yellow-paper/docs/public-vm/control-flow.md b/yellow-paper/docs/public-vm/control-flow.md new file mode 100644 index 00000000000..707e78ecf8b --- /dev/null +++ b/yellow-paper/docs/public-vm/control-flow.md @@ -0,0 +1,32 @@ +# VM Control Flow + High-level architecture + +This document breaks down the VM into internal components and defines how these components interact with one another. + +This can be considered an intermediate abstraction layer between the high-level VM definition and the explicit cirucit architecture. The intention is for this abstraction to be analogous to how CPU chips are broken down into discrete components. + +# Sub-operations + +> Notation note: I use the term "clock cycle" in a way that is analogous to "row in the execution trace". + +We wish to define a set of sub-operations our VM can execute. Multiple sub-operations can be executed per clock cycle. Each instruction in the instruction set exposed by the VM is composed of 1 or more sub-operations. + +The intention is for sub-operations to be implementable as independent VM circuit relations. + +# Control flow + +![](./gen/images/control-flow/avm-control-flow.png) + +> Notation note: whenever the VM "sends a signal" to one or more VM components, this is analogous to defining a boolean column in the execution trace that toggles on/off specific functionality + +- The instruction controller uses a program counter to read the current opcode to be executed, and send signals to downstream components to execute the opcode +- The state controller reads data from memory into the intermediate registers, using the opcode parameter values +- The VM "algebraic logic unit" executes the opcode given the intermediate register values, and writes output into the intermediate registers +- The state controller writes the output from an intermediate register into memory + +## Chiplets + +This borrows the chiplet nomenclature from the Miden VM - these components encapsulate functionality that can be effectively defined via an independent sub-circuit (analog in real world = specific part of a CPU chip die) + +Chiplets can be developed iteratively i.e. first draft of the AVM only needs a barebones algebraic logic unit. + +> We want apply the following design heuristic to chiplets: they are _loosely coupled_ to other AVM components. 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