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.

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This Applied Scientist is also expected to play a key role in raising the AI/ML skillset within the organization, so a passion for teaching/mentoring is important. This involves bringing deep AI/ML expertise to collaborations with other scientists, leading lunch & learn session, conducting tech talks, setting up regular office hours, evangelizing lessons learned & industry best practices, and more. The ideal candidate is one who possesses notable pedigree in ML-related academic work, along with deep, real-world, hands-on experience in executing/shipping AI-based products in a fast-paced setting. You will be looked up to as an ML expert that our organization’s leadership can rely on for guidance on AI/ML topics. Mentorship & Career Growth: Our team is dedicated to supporting new members. We have a broad mix of experience levels, functions and tenures, and we’re building an environment that celebrates knowledge sharing and mentorship. We care about your career growth and strive to assign projects based on what will help each team member develop into a better-rounded professional and enable them to take on more complex tasks in the future. Learn and Be Curious: We have a formal mentor search application that lets you find a mentor that works best for you based on location, job family, job level etc. Your manager can also help you find a mentor or two, because two is better than one. In addition to formal mentors, we work and train together so that we are always learning from one another, and we celebrate and support the career progression of our team members. Inclusion and Diversity. Our team is diverse! We drive towards an inclusive culture and work environment. We are intentional about attracting, developing, and retaining amazing talent from diverse backgrounds. Team members are active in Amazon’s 10+ affinity groups, sometimes known as employee resource groups, which bring employees together across businesses and locations around the world. These range from groups such as the Black Employee Network, Latinos at Amazon, Indigenous at Amazon, Families at Amazon, LGBTQ+, Warriors at Amazon (Military), Amazon People With Disabilities, and more. Learn more about Amazon on our Day 1 Blog: https://blog.aboutamazon.com About AWS Diverse Experiences AWS values diverse experiences. Even if you do not meet all of the preferred qualifications and skills listed in the job description, we encourage candidates to apply. If your career is just starting, hasn’t followed a traditional path, or includes alternative experiences, don’t let it stop you from applying. Why AWS? Amazon Web Services (AWS) is the world’s most comprehensive and broadly adopted cloud platform. We pioneered cloud computing and never stopped innovating — that’s why customers from the most successful startups to Global 500 companies trust our robust suite of products and services to power their businesses. Inclusive Team Culture Here at AWS, it’s in our nature to learn and be curious. Our employee-led affinity groups foster a culture of inclusion that empower us to be proud of our differences. Ongoing events and learning experiences, including our Conversations on Race and Ethnicity (CORE) and AmazeCon conferences, inspire us to never stop embracing our uniqueness. Mentorship & Career Growth We’re continuously raising our performance bar as we strive to become Earth’s Best Employer. That’s why you’ll find endless knowledge-sharing, mentorship and other career-advancing resources here to help you develop into a better-rounded professional. Work/Life Balance We value work-life harmony. Achieving success at work should never come at the expense of sacrifices at home, which is why we strive for flexibility as part of our working culture. When we feel supported in the workplace and at home, there’s nothing we can’t achieve in the cloud. About the team Utility Computing (UC) AWS Utility Computing (UC) provides product innovations — from foundational services such as Amazon’s Simple Storage Service (S3) and Amazon Elastic Compute Cloud (EC2), to consistently released new product innovations that continue to set AWS’s services and features apart in the industry. As a member of the UC organization, you’ll support the development and management of Compute, Database, Storage, Internet of Things (Iot), Platform, and Productivity Apps services in AWS, including support for customers who require specialized security solutions for their cloud services.
IL, Tel Aviv
Come build the future of entertainment with us. Are you interested in helping shape the future of movies and television? Do you want to help define the next generation of how and what Amazon customers are watching? Prime Video is a premium streaming service that offers customers a vast collection of TV shows and movies - all with the ease of finding what they love to watch in one place. We offer customers thousands of popular movies and TV shows from Originals and Exclusive content to exciting live sports events. We also offer our members the opportunity to subscribe to add-on channels which they can cancel at any time and to rent or buy new release movies and TV box sets on the Prime Video Store. Prime Video is a fast-paced, growth business - available in over 240 countries and territories worldwide. The team works in a dynamic environment where innovating on behalf of our customers is at the heart of everything we do. If this sounds exciting to you, please read on We are seeking an exceptional Applied Scientist to join our Prime Video Sports tech team in Israel. Our team is dedicated to developing state-of-the-art science to allow for personalizing the customers’ experience and customers to seamlessly find any live event in our selection. You will have the opportunity to work on innovative, large-scale projects that push the boundaries of what's possible in sports content delivery and engagement. Your expertise will be crucial in tackling complex challenges such as temporal information retrieval, leveraging Generative AI and Large Language Models (LLMs), and building state-of-the-art recommender systems. Key job responsibilities We are looking for an Applied Scientist with domain expertise in Personalization, Information Retrieval, and Recommender Systems, or general ML to lead the development of new algorithms and end-to-end solutions. As part of our team of applied scientists and software development engineers, you will be responsible for researching, designing, developing, and deploying algorithms into production pipelines. Your role will involve working with cutting-edge technologies such as Gen AI/LLMs to enhance content discovery and search capabilities. You'll also tackle unique challenges like temporal information retrieval to improve real-time sports content recommendations. As a technologist, you will drive the publication of original work in top-tier conferences in Machine Learning and Information Retrieval. We expect you to thrive in ambiguous situations, demonstrating outstanding analytical abilities and comfort in collaborating with cross-functional teams and systems. The ideal candidate is a self-starter with the ability to learn and adapt quickly in our fast-paced environment. About the team We are the Prime Video Sports team. In September 2018 Prime Video launched its first full-scale live streaming experience to world-wide Prime customers with NFL Thursday Night Football. That was just the start. Now Amazon has exclusive broadcasting rights to major leagues like NFL Thursday Night Football, Tennis major like Roland-Garros and English Premium League to list few and are broadcasting live events across 30+ sports world-wide. Prime Video is expanding not just the breadth of live content that it offers, but the depth of the experience. This is a transformative opportunity, the chance to be at the vanguard of a program that will revolutionize Prime Video, and the live streaming experience of customers everywhere.