Introduction
The Security of Bitcoin, and blockchains like LiquidThe use of digital signature algorithms such as ECDSA is crucial. Schnorr signatures. Both use a C-library called libsecp256k1, after the elliptic shape that it operates on. Bitcoin Core You can also find out more about the following: LiquidThese digital signature algorithms are provided by. These Algorithms use a mathematical computation known as a modular inverse. This is a relatively costly component of the computation.
In “Fast constant-time gcd computation You can also find out more about the following: modular inversion,” Daniel J. Bernstein and Bo-Yin Yang develop a new modular inversion algorithm. In 2021, this algorithm, referred to as “safegcd,” was implemented for libsecp256k1 by Peter Dettman. As As part of the testing process for this novel algorithm Blockstream Research was the first to complete a formal verification of the algorithm’s design by using the Coq Proof assistant to verify that the algorithm terminates with the correct modular inverse on 256-bit inputs.
The Gap The following are some examples of how to use Algorithm You can also find out more about the following: Implementation
The The formalization effort in 2020 only showed that the algorithm developed by Bernstein You can also find out more about the following: Yang It is important that you use the correct tool. HoweverUsing that algorithm in the libsecp256k1 language requires implementing the mathematical descriptions of the safegcd algorithms. For The mathematical description of an algorithm can perform matrix multiplications of vectors as wide as 256-bit signed integers. However, the C programming language only provides integers up to 64-bits (or 128-bits with some language extensions).
Implementing the safegcd algorithm requires programming the matrix multiplication and other computations using C’s 64 bit integers. AdditionallySeveral other optimizations have also been added to the implementation process. In There are four different implementations of libsecp256k1’s safegcd algorithm: two constant-time algorithms for signature generation (one optimized for 32 bit systems and another for 64 bit systems) and two variable-time algorithms for verification (one for 32 bit systems and the other for 64 bit systems).
Verifiable C
In In order to verify that the C program correctly implements safegcd, all the implementation detail must be checked. We Use Verifiable C, part of Verified Software Toolchain Use C code to understand the reasoning behind it Coq Theorem proving
Verification Then, for each function that is being verified, specify the preconditions (and postconditions) using separation logic. Separation This logic is used to reason about subroutines and memory allocations. It also deals with concurrency.
Once each function is given a specification, verification proceeds by starting from a function’s precondition, and establishing a new invariant after each statement in the body of the function, until finally establishing the post condition at the end of the function body or the end of each return statement. Most of the formalization effort is spent “between” the lines of code, using the invariants to translate the raw operations of each C expression into higher level statements about what the data structures being manipulated represent mathematically. For For example, the C language may consider an array of 64 bit integers to be a representation for a 256 bit integer.
The The final result is a proof that has been checked by the Coq proof assistant, that libsecp256k1’s 64-bit variable time implementation The cheapest way to buy a car is by using the safegcd modular inverse algorithm is functionally correct.
Limitations of the Verification
There There are some limitations with the functional correctness test. The Separation logic is used in Verifiable C implements a concept known as partial correction. That means it only proves the C code returns with the correct result if it returns, but it doesn’t prove termination itself. We You can mitigate this restriction by using our previous Coq Proof of the bounds for the safegcd to prove that in fact the loop counter value is never greater than 11 iterations.
Another The C language itself does not have a formal specification. Instead The following are some of the ways to get in touch with us. Verifiable The C project uses CompCert to provide a formal specification for a C language. This CompCert guarantees the assembly code generated from a C program that has been verified will meet its specifications (subject only to the limitation above). However this doesn’t guarantee that the code generated by GCC, clang, or any other compiler will necessarily work. For C compilers can, for instance, have different evaluation order for arguments within a call to a function. And even if the C language had a formal specification any compiler that isn’t itself formally verified could still miscompile programs. This Does it happen in practice?
Lastly, Verifiable C doesn’t support passing structures, returning structures or assigning structures. While In libsecp256k1, all structures are passed by pointer. Verifiable Structure assignment is used in a few cases. For The modular inverse-correctness proof required 3 assignments to be replaced by an specialized function that performs the assignment of the structure field by field.
Summary
Blockstream Research has formally verified the correctness of libsecp256k1’s modular inverse function. This The work shows that it is possible to verify C code in practice. Using A general purpose proof assistant enables us to verify complex mathematical arguments used in software.
Nothing This prevents the other functions implemented in libsecp256k1 being verified. Thus It is possible to achieve the highest possible software accuracy guarantees with libsecp256k1.
This This is a guest blog by Russell O'Connor You can also find out more about the following: Andrew Poelstra. Opinions BTC does not necessarily endorse the views or opinions of any third parties. Inc You can also find out more about Bitcoin Magazine.
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