Verifying an R1CS using the Axe toolkit.
Kestrel Institute's Axe toolkit (see ACL2::axe) can be used to verify R1CS circuits. Axe is implemented as an ACL2 program. We use our formalization of the semantics of R1CSes in ACL2 (see r1cs). This formalization defines the syntax of an R1CS as an ACL2 object and defines the semantics of such objects by characterizing what it means for an R1CS to hold over a valuation (an assignment of values to its variables). The formalization is based on our ACL2 formalization of prime fields (see pfield::prime-fields).
The verification process takes as input an R1CS in our format (perhaps translated from other format, such as a JSON representation). We then lift the R1CS into an equivalent logical term. For this, we use Axe's R1CS Lifter. The Lifter first forms a mathematical term asserting that the R1CS holds over arbitrary symbolic variables. It then applies the Axe Rewriter to transform that term into an equivalent expression over the prime field operators. To do this, the Axe Rewriter opens up the functions that define the R1CS semantics, producing a logical term equivalent to the R1CS but no longer depending on the R1CS semantics functions. The resulting term is essentially the conjunction of the constraints in the R1CS, expressed in terms of the underlying prime field operations, but some simplifications may also have been applied.
To verify an R1CS, we need a specification to compare it to. For this, we form
an ACL2 term expressing the relationship that should hold over the inputs and
outputs of the R1CS (but not its intermediate variables). When possible, we
use our libraries of existing bit-vector and cryptographic functions. Typically,
the specification says that the output variables of the R1CS are equal to some
expression over the input variables. We then call the Axe R1CS Prover to
attempt to prove that, whenever the R1CS holds (over all of its input, output,
and intermediate variables), the specification relation also holds (over the
inputs and outputs). We call this the ``forward direction'' of the proof.
Sometimes we also prove the backward direction (the claim that, if
the specification holds over the inputs and outputs, then there exist values
for the intermediate variables such that the entire R1CS holds). This requires
exhibiting suitable expressions for the intermediate variables, but these can
often be obtained automatically (see community book
We always assume that all values are field elements. If necessary, additional assumptions on the input variables (e.g., claims that some of them are single bits) can be passed to the Prover.
The Axe Prover supports two main proof tactics, rewriting and variable substitution. Rewriting is used to solve the constraints, attempting to turn each one into an equality of a variable and some other term not involving that variable. This works because the variables in an R1CS can often be ordered, with each constraint defining one or more new variables in terms of previously defined variables. After solving for a variable, substitution can be used to replace it, everywhere in the proof, with the term that it is known to be equal to. The rewriting process applies rewrite rules from our large and growing library of verified rules. Crucially, all of these rules are proved as theorems in the ACL2 system, so additional rules can be added without compromising the soundness of the proof system. Our library contains many rules that recognize R1CS gadgets, covering a variety of common operations such as XORing, bit packing, bit rotations, etc. A typical rule converts some class of expressions involving prime field operations into simpler but equivalent expressions involving more standard operations. Often several rounds of rewriting and substitution are necessary, as the Prover turns the information in the R1CS into more usable forms. For example, an XOR constraint cannot be solved until its input values are known to be single bits, but this may not be known until the constraints that define those values are themselves solved. The Axe Prover typically applies a sequence of sets of rules, exhaustively applying rewriting and substitution with one rule set before moving on to the next one. This allows us to do the proof in stages, each of which handles one kind of operation (e.g., recognizing and transforming bit packing gadgets, or moving negated addends to the other sides of equalities). Once every intermediate variable has been eliminated by substitution, we are usually left with a single term for each output of the R1CS, representing it as a function of the input variables. Often these terms no longer contain any prime field operations, because rewrite rules have recognized all of the R1CS gadgets and replaced them with more traditional operations (e.g., bit-vector concatenation, addition, rotation, etc.). The final step of the proof is usually to allow the Prover to expand the functions in the specification, unrolling all of the (bounded) recursion to expose simpler operations from our libraries. If all goes well, each of the output bits of the specification will have an expression that exactly matches the expression derived from the R1CS (perhaps after some normalization, also done by rewriting). This completes the proof.
This rest of this topic describes the concrete steps for using Axe to verify an R1CS. It may be helpful to follow along with an example. There is one here.
To use the Axe R1CS Prover to verify an R1CS, follow these steps:
1. Write or obtain a specification for the R1CS. You may be able to use existing formalizations from the ACL2 libraries (e.g., formalizations of hash functions), but you may need to account for the fact that some R1CS variables are single bits that need to be packed into larger quantities. In general, you will need to create an ACL2 predicate that takes the inputs and outputs of the R1CS and determines whether the outputs are correct with respect to the inputs. There may be many input and output variables (often, hundreds). The intermediate variables of the R1CS are not mentioned at all by this spec function. Often, the spec function will do 3 things:
2. Create an ACL2 constant representing the R1CS, for use in Step
3 below. The precise method for doing this depends on where the R1CSes are
coming from. It might require instrumenting some external toolchain that
manipulates R1CSes to emit the R1CSes in some form that can be converted to
our format. For Circom R1CSes, one can call
3. Lift the R1CS into logic. This is done by calling
This is the hardest step. We suggest looking at some example proofs.
The proof will include assumptions that:
The conclusion will assert that the spec function (written in step 1) holds over the appropriate arguments.
Most of the work in this step involves providing lists of rewrite rules
that suffice to rewrite all the idioms in the R1CS into the more
traditional operations used in the spec (e.g., to convert bit packing
from using prime field operations into calls of our operator