Better-performing “25519” elliptic-curve cryptography

Automated reasoning and optimizations specific to CPU microarchitectures improve both performance and assurance of correct implementation.

Cryptographic algorithms are essential to online security, and at Amazon Web Services (AWS), we implement cryptographic algorithms in our open-source cryptographic library, AWS LibCrypto (AWS-LC), based on code from Google’s BoringSSL project. AWS-LC offers AWS customers implementations of cryptographic algorithms that are secure and optimized for AWS hardware.

Two cryptographic algorithms that have become increasingly popular are x25519 and Ed25519, both based on an elliptic curve known as curve25519. To improve the customer experience when using these algorithms, we recently took a deeper look at their implementations in AWS-LC. Henceforth, we use x/Ed25519 as shorthand for “x25519 and Ed25519”.

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In 2023, AWS released multiple assembly-level implementations of x/Ed25519 in AWS-LC. By combining automated reasoning and state-of-the-art optimization techniques, these implementations improved performance over the existing AWS-LC implementations and also increased assurance of their correctness.

In particular, we prove functional correctness using automated reasoning and employ optimizations targeted to specific CPU microarchitectures for the instruction set architectures x86_64 and Arm64. We also do our best to execute the algorithms in constant time, to thwart side-channel attacks that infer secret information from the durations of computations.

In this post, we explore different aspects of our work, including the process for proving correctness via automated reasoning, microarchitecture (μarch) optimization techniques, the special considerations for constant-time code, and the quantification of performance gains.

Elliptic-curve cryptography

Elliptic-curve cryptography is a method for doing public-key cryptography, which uses a pair of keys, one public and one private. One of the best-known public-key cryptographic schemes is RSA, in which the public key is a very large integer, and the corresponding private key is prime factors of the integer. The RSA scheme can be used both to encrypt/decrypt data and also to sign/verify data. (Members of our team recently blogged on Amazon Science about how we used automated reasoning to make the RSA implementation on Amazon’s Graviton2 chips faster and easier to deploy.)

Elliptic curve.png
Example of an elliptic curve.

Elliptic curves offer an alternate way to mathematically relate public and private keys; sometimes, this means we can implement schemes more efficiently. While the mathematical theory of elliptic curves is both broad and deep, the elliptic curves used in cryptography are typically defined by an equation of the form y2 = x3 + ax2 + bx + c, where a, b, and c are constants. You can plot the points that satisfy the equation on a 2-D graph.

An elliptic curve has the property that a line that intersects it at two points intersects it at at most one other point. This property is used to define operations on the curve. For instance, the addition of two points on the curve can be defined not, indeed, as the third point on the curve collinear with the first two but as that third point’s reflection around the axis of symmetry.

Elliptic-curve addition.gif
Addition on an elliptic curve.

Now, if the coordinates of points on the curve are taken modulo some integer, the curve becomes a scatter of points in the plane, but a scatter that still exhibits symmetry, so the addition operation remains well defined. Curve25519 is named after a large prime integer — specifically, 2255 – 19. The set of numbers modulo the curve25519 prime, together with basic arithmetic operations such as multiplication of two numbers modulo the same prime, define the field in which our elliptic-curve operations take place.

Successive execution of elliptic-curve additions is called scalar multiplication, where the scalar is the number of additions. With the elliptic curves used in cryptography, if you know only the result of the scalar multiplication, it is intractable to recover the scalar, if the scalar is sufficiently large. The result of the scalar multiplication becomes the basis of a public key, the original scalar the basis of a private key.

The x25519 and Ed25519 cryptographic algorithms

The x/Ed25519 algorithms have distinct purposes. The x25519 algorithm is a key agreement algorithm, used to securely establish a shared secret between two peers; Ed25519 is a digital-signature algorithm, used to sign and verify data.

The x/Ed25519 algorithms have been adopted in transport layer protocols such as TLS and SSH. In 2023, NIST announced an update to its FIPS185-6 Digital Signature Standard that included the addition of Ed25519. The x25519 algorithm also plays a role in post-quantum safe cryptographic solutions, having been included as the classical algorithm in the TLS 1.3 and SSH hybrid scheme specifications for post-quantum key agreement.

Microarchitecture optimizations

When we write assembly code for a specific CPU architecture, we use its instruction set architecture (ISA). The ISA defines resources such as the available assembly instructions, their semantics, and the CPU registers accessible to the programmer. Importantly, the ISA defines the CPU in abstract terms; it doesn’t specify how the CPU should be realized in hardware.

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The detailed implementation of the CPU is called the microarchitecture, and every μarch has unique characteristics. For example, while the AWS Graviton 2 CPU and AWS Graviton 3 CPU are both based on the Arm64 ISA, their μarch implementations are different. We hypothesized that if we could take advantage of the μarch differences, we could create x/Ed25519 implementations that were even faster than the existing implementations in AWS-LC. It turns out that this intuition was correct.

Let us look closer at how we took advantage of μarch differences. Different arithmetic operations can be defined on curve25519, and different combinations of those operations are used to construct the x/Ed25519 algorithms. Logically, the necessary arithmetic operations can be considered at three levels:

  1. Field operations: Operations within the field defined by the curve25519 prime 2255 – 19.
  2. Elliptic-curve group operations: Operations that apply to elements of the curve itself, such as the addition of two points, P1 and P2.
  3. Top-level operations: Operations implemented by iterative application of elliptic-curve group operations, such as scalar multiplication.
Levels of operations.png
Examples of operations at different levels. Arrows indicate dependency relationships between levels.

Each level has its own avenues for optimization. We focused our μarch-dependent optimizations on the level-one operations, while for levels two and three our implementations employ known state-of-the-art techniques and are largely the same for different μarchs. Below, we give a summary of the different μarch-dependent choices we made in our implementations of x/Ed25519.

  • For modern x86_64 μarchs, we use the instructions MULX, ADCX, and ADOX, which are variations of the standard assembly instructions MUL (multiply) and ADC (add with carry) found in the instruction set extensions commonly called BMI and ADX. These instructions are special because, when used in combination, they can maintain two carry chains in parallel, which has been observed to boost performance up to 30%. For older x86_64 μarchs that don’t support the instruction set extensions, we use more traditional single-carry chains.
  • For Arm64 μarchs, such as AWS Graviton 3 with improved integer multipliers, we use relatively straightforward schoolbook multiplication, which turns out to give good performance. AWS Graviton 2 has smaller multipliers. For this Arm64 μarch, we use subtractive forms of Karatsuba multiplication, which breaks down multiplications recursively. The reason is that, on these μarchs, 64x64-bit multiplication producing a 128-bit result has substantially lower throughput relative to other operations, making the number size at which Karatsuba optimization becomes worthwhile much smaller.

We also optimized level-one operations that are the same for all μarchs. One example concerns the use of the binary greatest-common-divisor (GCD) algorithm to compute modular inverses. We use the “divstep” form of binary GCD, which lends itself to efficient implementation, but it also complicates the second goal we had: formally proving correctness.

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Binary GCD is an iterative algorithm with two arguments, whose initial values are the numbers whose greatest common divisor we seek. The arguments are successively reduced in a well-defined way, until the value of one of them reaches zero. With two n-bit numbers, the standard implementation of the algorithm removes at least one bit total per iteration, so 2n iterations suffice.

With divstep, however, determining the number of iterations needed to get down to the base case seems analytically difficult. The most tractable proof of the bound uses an elaborate inductive argument based on an intricate “stable hull” provably overapproximating the region in two-dimensional space containing the points corresponding to the argument values. Daniel Bernstein, one of the inventors of x25519 and Ed25519, proved the formal correctness of the bound using HOL Light, a proof assistant that one of us (John) created. (For more on HOL Light, see, again, our earlier RSA post.)

Performance results

In this section, we will highlight improvements in performance. For the sake of simplicity, we focus on only three μarchs: AWS Graviton 3, AWS Graviton 2, and Intel Ice Lake. To gather performance data, we used EC2 instances with matching CPU μarchs — c6g.4xlarge, c7g.4xlarge, and c6i.4xlarge, respectively; to measure each algorithm, we used the AWS-LC speed tool.

In the graphs below, all units are operations per second (ops/sec). The “before” columns represent the performance of the existing x/Ed25519 implementations in AWS-LC. The “after” columns represent the performance of the new implementations.

Signing new.png
For the Ed25519 signing operation, the number of operations per second, over the three μarchs, is, on average, 108% higher with the new implementations.
Verification.png
For the Ed25519 verification operation, we increased the number of operations per second, over the three μarchs, by an average of 37%.

We observed the biggest improvement for the x25519 algorithm. Note that an x25519 operation in the graph below includes the two major operations needed for an x25519 key exchange agreement: base-point multiplication and variable-point multiplication.

Ops:sec new.png
With x25519, the new implementation increases the number of operations per second, over the three μarchs, by an average of 113%.

On average, over the AWS Graviton 2, AWS Graviton 3, and Intel Ice Lake microarchitectures, we saw an 86% improvement in performance.

Proving correctness

We develop the core parts of the x/Ed25519 implementations in AWS-LC in s2n-bignum, an AWS-owned library of integer arithmetic routines designed for cryptographic applications. The s2n-bignum library is also where we prove the functional correctness of the implementations using HOL Light. HOL Light is an interactive theorem prover for higher-order logic (hence HOL), and it is designed to have a particularly simple (hence light) “correct by construction” approach to proof. This simplicity offers assurance that anything “proved” has really been proved rigorously and is not the artifact of a prover bug.

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We follow the same principle of simplicity when we write our implementations in assembly. Writing in assembly is more challenging, but it offers a distinct advantage when proving correctness: our proofs become independent of any compiler.

The diagram below shows the process we use to prove x/Ed25519 correct. The process requires two different sets of inputs: first is the algorithm implementation we’re evaluating; second is a proof script that models both the correct mathematical behavior of the algorithm and the behavior of the CPU. The proof is a sequence of functions specific to HOL Light that represent proof strategies and the order in which they should be applied. Writing the proof is not automated and requires developer ingenuity.

From the algorithm implementation and the proof script, HOL Light either determines that the implementation is correct or, if unable to do so, fails. HOL Light views the algorithm implementation as a sequence of machine code bytes. Using the supplied specification of CPU instructions and the developer-written strategies in the proof script, HOL Light reasons about the correctness of the execution.

CI integration.png
CI integration provides assurance that no changes to the algorithm implementation code can be committed to s2n-bignum’s code repository without successfully passing a formal proof of correctness.

This part of the correctness proof is automated, and we even implement it inside s2n-bignum’s continuous-integration (CI) workflow. The workflow covered in the CI is highlighted by the red dotted line in the diagram below. CI integration provides assurance that no changes to the algorithm implementation code can be committed to s2n-bignum’s code repository without successfully passing a formal proof of correctness.

The CPU instruction specification is one of the most critical ingredients in our correctness proofs. For the proofs to be true in practice, the specification must capture the real-world semantics of each instruction. To improve assurance on this point, we apply randomized testing against the instruction specifications on real hardware, “fuzzing out” inaccuracies.

Constant time

We designed our implementations and optimizations with security as priority number one. Cryptographic code must strive to be free of side channels that could allow an unauthorized user to extract private information. For example, if the execution time of cryptographic code depends on secret values, then it might be possible to infer those values from execution times. Similarly, if CPU cache behavior depends on secret values, an unauthorized user who shares the cache could infer those values.

Our implementations of x/Ed25519 are designed with constant time in mind. They perform exactly the same sequence of basic CPU instructions regardless of the input values, and they avoid any CPU instructions that might have data-dependent timing.

Using x/Ed25519 optimizations in applications

AWS uses AWS-LC extensively to power cryptographic operations in a diverse set of AWS service subsystems. You can take advantage of the x/Ed25519 optimizations presented in this blog by using AWS-LC in your application(s). Visit AWS-LC on Github to learn more about how you can integrate AWS-LC into your application.

To allow easier integration for developers, AWS has created bindings from AWS-LC to multiple programming languages. These bindings expose cryptographic functionality from AWS-LC through well-defined APIs, removing the need to reimplement cryptographic algorithms in higher-level programming languages. At present, AWS has open-sourced bindings for Java and Rust — the Amazon Corretto Cryptographic Provider (ACCP) for Java, and AWS-LC for Rust (aws-lc-rs). Furthermore, we have contributed patches allowing CPython to build against AWS-LC and use it for all cryptography in the Python standard library. Below we highlight some of the open-source projects that are already using AWS-LC to meet their cryptographic needs.

Open-source projects.png
Open-source projects using AWS-LC to meet their cryptographic needs.

We are not done yet. We continue our efforts to improve x/Ed25519 performance as well as pursuing optimizations for other cryptographic algorithms supported by s2n-bignum and AWS-LC. Follow the s2n-bignum and AWS-LC repositories for updates.

Research areas

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Shape the Future of Cloud Computing Are you a graduate student passionate about Automated Reasoning and its real-world applications? Join our team of innovators and embark on a journey to revolutionize cloud computing through cutting-edge automated reasoning techniques.Our tools are called billions of times daily, powering the backbone of Amazon's products and services. We are changing the way computer systems are developed and operated, raising the bar for security, durability, availability, and quality. As an Applied Science Intern, you'll have the opportunity to work alongside our brilliant scientists and contribute to groundbreaking projects. From distributed proof search and SAT/SMT solvers to program analysis, synthesis, and verification, you'll tackle complex challenges at the intersection of theory and practice, driving innovation and delivering tangible value to our customers. This internship is not just about executing tasks – you'll explore novel approaches to solving intricate automated reasoning problems. You'll dive deep into cutting-edge research, leveraging your expertise to develop innovative solutions. You'll work on deploying your solutions into production, witnessing the real-world impact of your contributions. Throughout your journey, you'll have access to unparalleled resources, including state-of-the-art computing infrastructure, cutting-edge research papers, and mentorship from industry luminaries. This immersive experience will not only sharpen your technical skills but also cultivate your ability to think critically, communicate effectively, and thrive in a fast-paced, innovative environment. Join us and be part of a team that is shaping the future of cloud computing through the power of Automated Reasoning. Apply now and unlock your potential! Amazon has positions available for Automated Reasoning Applied Science Internships in, but not limited to, Arlington, VA; Boston, MA; Cupertino, CA; Minneapolis, MN; New York, NY; Portland, OR; Santa Clara, CA; Seattle, WA; Bellevue, WA; Santa Clara, CA; Sunnyvale, CA. Key job responsibilities We are particularly interested in candidates with expertise in: Theorem Proving, Boolean Satisfiability Solvers, Bounded Model Checking, Deductive Verification, Programming/Scripting Languages, Abstract Interpretation, Automated Reasoning, Static/Program Analysis, Program Synthesis In this role, you will work alongside global experts to develop and implement novel, scalable algorithms and modeling techniques that advance the state-of-the-art in areas at the intersection of Natural Language Processing and Speech Technologies. You will tackle challenging, groundbreaking research problems on production-scale data, with a focus on natural language processing, speech recognition, text-to-speech (TTS), text recognition, question answering, NLP models (e.g., LSTM, transformer-based models), signal processing, information extraction, conversational modeling, audio processing, speaker detection, large language models, multilingual modeling, and more. The ideal candidate should possess the ability to work collaboratively with diverse groups and cross-functional teams to solve complex business problems. A successful candidate will be a self-starter, comfortable with ambiguity, with strong attention to detail and the ability to thrive in a fast-paced, ever-changing environment. Key job responsibilities We are particularly interested in candidates with expertise in: Theorem Proving, Boolean Satisfiability Solvers, Bounded Model Checking, Deductive Verification, Programming/Scripting Languages, Abstract Interpretation, Automated Reasoning, Static/Program Analysis, Program Synthesis In this role, you will work alongside global experts to develop and implement novel, scalable algorithms and modeling techniques that advance the state-of-the-art in areas at the intersection of Natural Language Processing and Speech Technologies. You will tackle challenging, groundbreaking research problems on production-scale data, with a focus on natural language processing, speech recognition, text-to-speech (TTS), text recognition, question answering, NLP models (e.g., LSTM, transformer-based models), signal processing, information extraction, conversational modeling, audio processing, speaker detection, large language models, multilingual modeling, and more. The ideal candidate should possess the ability to work collaboratively with diverse groups and cross-functional teams to solve complex business problems. A successful candidate will be a self-starter, comfortable with ambiguity, with strong attention to detail and the ability to thrive in a fast-paced, ever-changing environment.
US, WA, Seattle
Unleash Your Potential as an AI Trailblazer At Amazon, we're on a mission to revolutionize the way people discover and access information. Our Applied Science team is at the forefront of this endeavor, pushing the boundaries of recommender systems and information retrieval. We're seeking brilliant minds to join us as interns and contribute to the development of cutting-edge AI solutions that will shape the future of personalized experiences. As an Applied Science Intern focused on Recommender Systems and Information Retrieval in Machine Learning, you'll have the opportunity to work alongside renowned scientists and engineers, tackling complex challenges in areas such as deep learning, natural language processing, and large-scale distributed systems. Your contributions will directly impact the products and services used by millions of Amazon customers worldwide. Imagine a role where you immerse yourself in groundbreaking research, exploring novel machine learning models for product recommendations, personalized search, and information retrieval tasks. You'll leverage natural language processing and information retrieval techniques to unlock insights from vast repositories of unstructured data, fueling the next generation of AI applications. Throughout your journey, you'll have access to unparalleled resources, including state-of-the-art computing infrastructure, cutting-edge research papers, and mentorship from industry luminaries. This immersive experience will not only sharpen your technical skills but also cultivate your ability to think critically, communicate effectively, and thrive in a fast-paced, innovative environment where bold ideas are celebrated. Join us at the forefront of applied science, where your contributions will shape the future of AI and propel humanity forward. Seize this extraordinary opportunity to learn, grow, and leave an indelible mark on the world of technology. Amazon has positions available for Machine Learning Applied Science Internships in, but not limited to Arlington, VA; Bellevue, WA; Boston, MA; New York, NY; Palo Alto, CA; San Diego, CA; Santa Clara, CA; Seattle, WA. Key job responsibilities We are particularly interested in candidates with expertise in: Knowledge Graphs and Extraction, Programming/Scripting Languages, Time Series, Machine Learning, Natural Language Processing, Deep Learning,Neural Networks/GNNs, Large Language Models, Data Structures and Algorithms, Graph Modeling, Collaborative Filtering, Learning to Rank, Recommender Systems In this role, you'll collaborate with brilliant minds to develop innovative frameworks and tools that streamline the lifecycle of machine learning assets, from data to deployed models in areas at the intersection of Knowledge Management within Machine Learning. You will conduct groundbreaking research into emerging best practices and innovations in the field of ML operations, knowledge engineering, and information management, proposing novel approaches that could further enhance Amazon's machine learning capabilities. The ideal candidate should possess the ability to work collaboratively with diverse groups and cross-functional teams to solve complex business problems. A successful candidate will be a self-starter, comfortable with ambiguity, with strong attention to detail and the ability to thrive in a fast-paced, ever-changing environment. A day in the life - Design, implement, and experimentally evaluate new recommendation and search algorithms using large-scale datasets - Develop scalable data processing pipelines to ingest, clean, and featurize diverse data sources for model training - Conduct research into the latest advancements in recommender systems, information retrieval, and related machine learning domains - Collaborate with cross-functional teams to integrate your innovative solutions into production systems, impacting millions of Amazon customers worldwide - Communicate your findings through captivating presentations, technical documentation, and potential publications, sharing your knowledge with the global AI community
US, WA, Seattle
Do you have a strong science background and want to help build new technologies? Do you have a physics background and want to help build and test superconducting circuits? Would you love to help develop the algorithms and models that power computer vision services at Amazon, such as Amazon Rekognition, Amazon Go, Visual Search, etc? Join the quantum revolution at Amazon and be part of a team that's pushing the boundaries of what's possible in quantum computing and quantum technologies. As a Research Science Intern focused on Quantum Technologies, you'll have the opportunity to work alongside leading experts in the field, contributing to cutting-edge research and driving innovation in areas such as quantum algorithms, quantum simulation, superconducting qubits, quantum key distribution, and quantum optics. We are looking for builders, innovators, and entrepreneurs who want to bring their ideas to reality and improve the lives of millions of customers. Research interns at Amazon work passionately to apply cutting-edge advances in technology to solve real-world problems. As an intern, you will be challenged to apply theory into practice through experimentation and invention, develop new algorithms using modeling software and programming techniques for complex problems, implement prototypes and work with massive datasets. Throughout your journey, you'll have access to unparalleled resources, including state-of-the-art computing infrastructure, cutting-edge research papers, and mentorship from industry luminaries. This immersive experience will not only sharpen your technical skills but also cultivate your ability to think critically, communicate effectively, and thrive in a fast-paced, innovative environment where bold ideas are celebrated. Amazon has positions available for Operations Research Science Internships in, but not limited to, Bellevue, WA; Boston, MA; Cambridge, MA; New York, NY; Santa Clara, CA; Seattle, WA; Sunnyvale, CA. Key job responsibilities We are particularly interested in candidates with the following skills: Quantum Algorithms, Quantum Simulators, Superconducting Qubits, Quantum Key Distribution , Optics In this role, you ain hands-on experience in applying cutting-edge analytical techniques to tackle complex business challenges at scale. If you are passionate about using data-driven insights to drive operational excellence, we encourage you to apply. The ideal candidate should possess the ability to work collaboratively with diverse groups and cross-functional teams to solve complex business problems. A successful candidate will be a self-starter, comfortable with ambiguity, with strong attention to detail and the ability to thrive in a fast-paced, ever-changing environment. A day in the life - Conduct research and develop new quantum algorithms to solve complex computational problems - Design and implement quantum simulation models to study the behavior of quantum systems - Investigate the properties and performance of superconducting qubits, a promising platform for building large-scale quantum computers - Explore the application of quantum key distribution protocols for secure communication and data encryption, ensuring the privacy and integrity of sensitive information - Explore the application of quantum optics principles to develop novel quantum sensing and communication technologies