The Role of Mathematics in Cryptography: What Executives and Decision-Makers Need to Know

Cryptography now supports almost every digital business. It protects private data, secures messages, verifies identity, and helps meet legal rules. Without it, modern systems would not work.

More data now passes through cloud systems, phones, and connected devices. That data needs protection.

At the core of cryptography is math. Math in cryptography decides how encryption works and how long it stays secure. The safety of these systems depends on math problems that are hard to solve without the correct key.

The goal of this article is to explain how mathematical foundations shape modern cryptography and why that matters for decision-makers. It shows how these ideas affect cloud services, finance, healthcare, and future quantum-safe systems.

The Mathematical Foundations of Classical Cryptography

Most of today’s encryption relies on basic math ideas that existed long before quantum computing became mainstream. These methods still protect emails, payments, logins, and stored data.

Number Theory and Modular Arithmetic

Number theory works with whole numbers. Modular arithmetic means numbers loop back after reaching a fixed limit, like a clock.

This math enables secure online communication, even over public networks:

• Protects data sent over the internet
• Allows secure key exchange between systems
• Supports encrypted emails, logins, and transactions
• Prevents unauthorized access during data transfer

For more information on math’s use in securing communication, please see the work by (Rescorla, E. 2018).

Core Mathematical “Hardness”

Some math problems are easy to create but very hard to undo. Breaking large numbers into primes or guessing secret values would take too long for normal computers to finish. That difficulty keeps encrypted data safe (National Institute of Standards and Technology, 2019).

Business Value (for Executives)

This math gives companies confidence that private data stays private. For businesses handling payments, user accounts, or sensitive records, trust in encryption protects customers and reduces risk.

Finite Fields and Group Theory: Diving into Abstract Algebra:

Abstract algebra works with number systems that follow strict rules, such as how values are added, multiplied, or reversed. Finite fields are small, fixed sets of numbers, which makes calculations predictable and efficient for computers.

Many encryption systems rely on operations from finite fields and other algebraic structures. These operations support both everyday symmetric encryption and more advanced public-key methods (mathematical foundations of cryptography, 2014).

Algebraic structures:

• Are used inside most modern symmetric-key algorithms
• Form the foundation of block ciphers
• Support Elliptic Curve Cryptography (ECC)
• Enable fast, reliable encryption on hardware

Security Basis (Hard Problems)

ECC security relies on a math problem that is easy to compute one way and extremely hard to reverse. Given a public point on a curve, finding the hidden private key would take too long for classical computers to solve.

Business Value

ECC delivers strong security with shorter keys than older systems like RSA. Shorter keys mean faster performance, lower bandwidth use, and reduced computing cost. For mobile apps, IoT devices, embedded systems, and blockchain platforms, this efficiency directly supports scale and reliability (National Institute of Standards and Technology, 2020).

Information Theory, Randomness, and Hash Functions

Modern cryptography depends on randomness. Keys, nonces, salts, and many protocol steps rely on numbers that attackers cannot predict. If randomness is weak, even strong algorithms can fail.

This is why probability, entropy, and math matter in cryptography, shaping how secure systems behave in practice, as explained in the math behind cryptography. Why randomness matters:

• Generates secure encryption keys
• Prevents attackers from guessing values
• Protects against replay and pattern-based attacks
• Keeps cryptographic protocols unpredictable

Hash functions play a different but related role. They turn any input into a fixed-size output in a way that cannot be reversed. Good hash functions also make it extremely hard to find two inputs that produce the same result.

These properties come from careful mathematical and statistical design, central to the mathematics used in real cryptographic systems.

Use cases

These tools are used anywhere data must stay accurate and tamper-proof.

• Password hashing for secure storage
• Message integrity checks
• Digital signatures by signing a hash, not the full file
• Data integrity checks during transfer or storage
• Random salt generation to prevent pattern attacks

Business value

These functions protect trust, accountability, and compliance and

• Confirm data has not been altered
• Support user authentication and access control
• Enable non-repudiation in signed transactions
• Meet integrity requirements in finance and healthcare
• Support secure protocols like TLS/SSL and secure APIs

Summary: What classical cryptography math gives us and its limits

Classical cryptography is built on math that has been tested for many years. It supports widely used standards and allows secure systems to run at scale. Its security comes from math problems that are hard for normal computers to solve, which is why it protects data, payments, and communications today.

But this approach has limits. Advances in quantum computing threaten the math behind many current systems. As quantum machines improve, encryption that works now may not protect data in the future. That risk makes a shift toward quantum-safe cryptography necessary.

The Quantum Threat: Why Advanced Mathematics (Again) Matters

Quantum computers do not work like normal computers. They can try many possibilities at the same time, which changes what “hard to break” really means in cryptography.

This matters because many current systems depend on math problems that quantum computers can solve faster. A known quantum method, often called Shor’s algorithm, can break encryption used by RSA, ECC, and Diffie–Hellman (National Institute of Standards and Technology ,2016)

Research discussed by the Association for Computing Machinery explains why these systems may not stay secure in a quantum future.

The risk is not theoretical. Encrypted data can be copied today and decrypted later when quantum tools improve. Financial records, healthcare data, and intellectual property often need long-term protection. That is why planning for quantum-safe cryptography needs to start now.

Post-Quantum Cryptography (PQC): New Mathematical Foundations

Post-quantum cryptography looks for encryption methods that quantum computers cannot break. Instead of relying on older math problems, these systems use different ones that are still believed to be hard for both normal and quantum machines.

Mathematics sits at the center of this work. New schemes must be built, tested, and stress-checked using math before they are trusted. This includes proving that the problems behind them stay hard, checking performance, and making sure they can run efficiently in real systems.

Advanced Mathematics Behind Post-Quantum Cryptography

Post-quantum cryptography relies on new math that holds up even against quantum computers. These methods are being tested now because older systems may not survive future attacks. For decision-makers, this math explains why some algorithms are safer, slower, or harder to deploy than others.

Lattice-Based Cryptography

Lattice-based cryptography is one of the leading approaches in post-quantum security. Many new standards being reviewed today are built on it (Moody, D., 2022).

What is a Lattice (simple view)

A lattice is a grid of points spread across many dimensions. These points follow strict rules based on math but become very hard to analyze as dimensions increase. Certain problems in lattices are extremely hard to solve, even for quantum computers.

• Shortest Vector Problem (SVP): Finding the shortest non-zero path in a large lattice is computationally impractical at scale.
• Learning With Errors (LWE): Small amounts of noise are added to equations, making it nearly impossible to recover the original secret values.

These hard problems form the security base of many post-quantum systems.

How Lattice Math is Used in Cryptography

Many algorithms being evaluated by the National Institute of Standards and Technology (NIST) use lattice-based math (National Institute of Standards and Technology, 2022, July 5).

• Key exchange and encryption schemes
• Digital signature systems
• Quantum-resistant replacements for RSA and ECC

These algorithms are secure because breaking them would require efficiently solving the Learning With Errors (LWE) problem (or closely related lattice problems) in high dimensions. This is a task that is believed to be computationally infeasible for both classical and quantum computers.

Mathematical Optimizations For Efficiency

Lattice systems involve heavy math operations, especially with polynomials. To keep them fast enough for real use, optimized algorithms are required.

Lattice-based cryptography:

• Uses fast polynomial math
• Relies on methods like the Number Theoretic Transform
• Improves speed for encryption, decryption, and key handling

Why Business Leaders Should Care

Lattice-based post-quantum cryptography offers a realistic path to long-term security, but it is not plug-and-play.

• Requires careful parameter choices
• Has performance and size trade-offs
• Needs math and cryptography expertise to implement safely

For companies building security products, this expertise is a competitive edge. For industries like cloud services, finance, and healthcare, adopting the right post-quantum systems is a long-term risk decision, not just a technical upgrade.

Code-Based Cryptography and Error-Correcting Codes

Code-based cryptography comes from the same math used to fix errors in data transmission. The idea is simple: add extra structure so data can be recovered even if parts are damaged (National Institute of Standards and Technology, 2023).

How Error-Correcting Codes Work

Error-correcting codes add redundancy to data. If some bits change or get lost, the original data can still be recovered. In cryptography, this same idea is used in reverse. Why the math is hard:

• Decoding a random linear code without a secret key is extremely difficult
• The problem grows fast as data size increases
• No efficient classical or quantum method is known to break it

Use In Post-Quantum Cryptography

Some post-quantum encryption and key-exchange schemes are built on code-based math:

• Used in encryption and key encapsulation
• Offers an alternative to lattice-based systems
• Studied as part of post-quantum research and standards efforts, including work described in post-quantum cryptography research

Business Value

Code-based cryptography adds flexibility and risk control:

• Reduces reliance on a single cryptographic approach
• Supports algorithm diversity and future switching
• Helps manage long-term security risk if other methods fail

Multivariate Polynomial Cryptography

Multivariate cryptography uses math problems built from many equations with many variables. When these equations grow large, solving them becomes extremely difficult.

The system hides a secret inside a large set of polynomial equations. Without the secret structure, solving those equations takes too much time to be practical. This idea is explained clearly in this article on multivariate cryptography. Some post-quantum systems use multivariate math, mainly for signatures:

• Used in digital signature schemes
• Can support fast signing and verification
• Often produces small signature sizes

Trade-Offs And Business Impact

Multivariate schemes are not free of cost or risk.

• May require larger public keys
• Need careful parameter selection
• Can increase setup and integration complexity

For companies choosing post-quantum tools, these trade-offs affect performance, storage, compliance, and system compatibility. Understanding them helps align cryptography choices with real business limits rather than theory alone.

Hash-Based and Hybrid Schemes

Hash-based schemes use cryptographic hash functions instead of complex algebra or number theory.

Their security comes from properties like one-way behavior and resistance to collisions, which are well understood and widely used today. Because of this, hash-based signatures are often simpler to reason about, as explained in this article on cryptographic hash functions.

The trade-off is size and usage limits, since some schemes produce larger signatures or can only be used a limited number of times.

Hybrid approaches combine more than one type of math in a single system. A design might pair lattice-based encryption with hash-based signatures, or mix code-based methods with hashing. This spreads risk across different security assumptions and helps balance speed, storage size, and long-term safety.

For businesses, hash-based and hybrid schemes can offer a cautious path to quantum-safe security. They may be easier to test, explain, or certify, especially in regulated environments.

For industries like healthcare, government, and critical infrastructure, that caution can reduce long-term risk while new standards continue to mature.

Hash-Based Signatures

Hash-based signatures use hash functions instead of complex math problems. A hash turns data into a fixed result that cannot be reversed or easily matched with another input. This approach is well understood and explained in work on cryptographic hash functions.

The trade-off is practical. These signatures are often larger and may only be used a limited number of times. That makes them safer in some cases, but less flexible in systems that need frequent signing.

Hybrid/Composite Approaches

Hybrid approaches combine more than one type of cryptography in a single system. For example, a system may use lattice-based encryption together with hash-based signatures. This spreads security across different math foundations and helps balance speed, size, and long-term safety.

Business Value

For businesses, hybrid schemes can reduce risk. They offer a more cautious path to quantum-safe security and may be easier to review, test, or certify.

For industries like healthcare, government, and critical infrastructure, this conservative approach can make adoption safer and more practical.

Underlying Themes: Computational Complexity & “Hard Problems”

Cryptography works because some math problems take too long to solve. Security does not come from secrecy, but from the assumption that certain problems cannot be solved efficiently, even with powerful computers.

Why “hard problems” matter

• Encryption depends on problems that are slow to break
• Examples include factoring numbers, solving discrete logs, lattice problems, and decoding random codes
• These problems protect keys, messages, and signatures
• If a faster method is found, the security breaks

Math research does not stop. New ideas continue to change what is considered safe, both in building stronger systems and in finding weaknesses.

What this means for businesses

• Cryptography is not permanent
• Algorithms need regular review and updates
• Migration planning is part of long-term security
• Staying current reduces future risk

Efficiency & Implementation Math (Performance Considerations)

Strong encryption must also run fast. Security and performance must work together. If the math is too slow or heavy, systems fail in real use.

Many modern systems rely on fast math operations to stay practical. In lattice-based cryptography, polynomial arithmetic must be efficient to avoid delays and high resource use (National Institute of Standards and Technology, 2023).

Techniques like fast transforms are used to speed up these operations, as discussed in research on efficient cryptographic implementations. Why performance math matters:

• Keeps encryption fast enough for real systems
• Reduces memory and processing load
• Supports large-scale use in cloud and mobile environments
• Makes quantum-safe systems usable, not theoretical

Choosing the right parameters is just as important. Values like size, limits, and error margins control both security and speed. Poor choices can weaken protection or make systems too slow to deploy. Business impact:

1. Slow encryption hurts user experience
2. Large keys increase storage and network costs
3. Inefficient systems fail on mobile and IoT devices
4. Teams with strong crypto expertise deliver both safety and speed

Application Areas: How This Mathematics Translates into Real-World Cryptographic Business

Advanced math in cryptography supports real systems that companies rely on every day. It protects data, enables secure communication, and reduces long-term risk.

Enterprises like Google, Microsoft, Amazon all use cryptography to secure data across cloud platforms and internal systems. This includes secure storage, key management, encrypted backups, service-to-service communication, and zero-trust security designs.

Mathematical cryptography provides clear security guarantees. It protects confidentiality, confirms data integrity, and verifies authenticity.

Efficient key exchange and scalable performance, especially with systems like ECC and post-quantum cryptography, allow these protections to work at large scale and remain viable as threats change.

For executives, early investment in post-quantum or hybrid cryptography protects long-term business interests. It reduces future migration costs, supports regulatory compliance, and builds trust with customers and regulators by showing commitment to durable security.

Finance and Decentralized Finance (DeFi)

Financial systems rely on cryptography to secure money, identities, and digital assets across both traditional and decentralized platforms:

• Secures transactions and asset transfers
• Uses digital signatures for ownership and authorization
• Protects identities and access
• Supports blockchain and ledger security
• Enables custody services and secure communication

Advanced math matters because finance requires both security and speed. Systems like ECC are widely used today because they offer strong protection with smaller keys and high throughput (European Payments Council, 2023).

As quantum risks grow, lattice-based and code-based post-quantum cryptography may be needed to protect future assets. Cryptographic math also supports smart contracts, secure wallets, and identity systems.

For fintech and DeFi firms, early adoption of quantum-safe cryptography can reduce long-term risk. It helps prevent “harvest-now, decrypt-later” attacks, protects sensitive financial data, and supports compliance and regulatory readiness.

Healthcare, Pharma, and Sensitive Data Industries

These industries handle data that must stay private for many years.

Use Cases

Cryptography protects patient records, medical data, and research files. It secures drug formulas and other sensitive information. It also allows safe data sharing between hospitals, labs, and cloud systems (National Institute of Standards and Technology, 2022).

Why Math-Based Cryptography Matters

Medical and research data often need long-term protection. If this data is exposed later, the damage can be serious. Quantum-safe cryptography helps keep information secure as technology and threats change.

Business Value

Strong encryption supports privacy and compliance. It lowers the risk of future data breaches and legal issues. It also makes secure collaboration between institutions possible.

Quantum-Safe Cryptography Providers, PQC Vendors, and Crypto-Native Startups

Companies like Open Quantum Safe (OQS) and SandboxAQ among others sit closest to the future of security. They design and deliver the tools that organizations will rely on once current cryptography no longer holds.

Their work includes:

• Building quantum-safe encryption libraries
• Hardware security modules
• Secure key-management systems.

Many also help enterprises test, integrate, and migrate away from older cryptographic standards. For large organizations, these vendors often become long-term partners rather than one-time suppliers.

In cryptography, the product is the math. The real value comes from choosing the right mathematical problems, setting safe parameters, and making systems fast enough for real use. This includes work in lattice methods, coding techniques, and performance tuning.

Unlike traditional software, small mathematical mistakes can break security completely, which is why deep expertise matters more than branding or features.

Business value

Firms with strong math and cryptography teams can move faster and more safely as standards change. They are better positioned to influence post-quantum standards, advise regulated industries, and help enterprises avoid costly mistakes during migration.

This creates trust, long-term contracts, and a clear competitive advantage in a market where security failures carry high financial and reputational costs.

Challenges, Trade-offs, and the Importance of Cryptographic Expertise

Cryptography is difficult because small mistakes can have serious consequences. Security depends on correct math, correct implementation, and correct decisions over time. This is why cryptography is not a plug-and-play problem.

Complexity and Risk: More Math Means More Ways to Fail

Advanced cryptography increases risk when used incorrectly. Even systems that are secure on paper can fail in real deployments (Cintas Canto, A., Kaur, J., Mozaffari Kermani, M., & Azarderakhsh, R., 2023).

• Poor parameter choices can weaken security
• Weak randomness can expose keys
• Side-channel leaks can bypass encryption
• Unsafe performance optimizations can introduce new attacks

Strong security requires careful implementation, testing, and ongoing review.

Trade-offs: Performance, Size, and Interoperability

Post-quantum cryptographic systems often require more resources than older systems like ECC. Larger keys and signatures can increase storage use, network traffic, and processing time. These impacts affect user experience, system latency, and operating costs.

Migration also adds complexity. Existing protocols, hardware, certificates, and user systems may need redesign or replacement. For small and mid-sized companies, the cost of development, audits, and migration must be weighed against long-term security needs.

Talent and Expertise Shortage

The math behind post-quantum cryptography is complex and specialized. It requires people who understand both theory and real-world implementation. These skills are difficult to hire and take time to develop.

Companies without in-house expertise may rely on vendors or consultants. That creates additional risk since poor vendor choices can lead to weak designs or hidden flaws. Careful vetting and ongoing oversight become essential.

Uncertainty and Standardization Risks

Cryptographic standards are still evolving. Some algorithms may be replaced as new research or attacks emerge. What is considered safe today may change in the future.

For decision-makers, timing is difficult. Moving too early can lock a company into schemes that do not last. Moving too late can expose sensitive data that needs long-term protection. Balancing cost, risk, and readiness requires informed judgment, not guesswork.

Strategic Recommendations for Executives & Decision-Makers

Cryptography protects what keeps a business alive: data, money, and trust. Good decisions come from understanding where things can fail and planning before problems appear.

Audit Your Cryptographic Risk Today

Most companies already use cryptography, but many do not know where or how. Encryption is often spread across databases, cloud services, backups, APIs, and third-party tools. If one part is weak, the whole system can fail.

Leaders should know which data must stay secret for years, which data is regulated, and which systems would cause real damage if broken. You cannot protect what you have not mapped.

Plan for a Quantum-Safe Roadmap (Cryptographic Agility)

Cryptography should not be locked in place forever. Systems built today must expect change. Crypto-agile systems separate encryption from business logic, making it possible to swap algorithms later.

Even if quantum-safe cryptography is not required today, designing for change now prevents panic later. Planning early costs less than emergency fixes.

Invest in Cryptographic Expertise

Cryptography is easy to misuse. The math can be correct and still fail if applied the wrong way. Teams need people who understand both how encryption works and how it breaks in real systems. When companies rely on vendors, trust must be earned, not assumed.

Good providers explain their choices, show test results, and accept outside review. Silence or vague claims are warning signs.

Balance Security With Performance and Usability

Security that slows systems too much will be bypassed or removed. Every business has limits on speed, cost, and complexity. Mobile apps, cloud platforms, and connected devices feel performance problems quickly.

At the same time, data that must stay secret for many years needs stronger protection, even if it costs more. The right balance depends on what is being protected and for how long.

Monitor Standards, Research, and Regulation

Cryptography changes as math improves and attacks evolve. Some systems that are safe today may not be safe tomorrow. Standards bodies, researchers, and regulators shape what becomes required or banned.

Companies that watch these changes can plan calmly. Those who ignore them are forced to rush when rules or threats change.

Use Cryptography as a Differentiator, Not Just a Compliance Checkbox

Strong cryptography can support growth, not just risk reduction:

1. Signals long-term trust to customers
2. Appeals to security- and privacy-focused markets like finance, healthcare, and enterprise cloud
3. Reduces concerns about future data exposure
4. Shows readiness for post-quantum threats
5. Positions the company as careful, reliable, and future-aware

When customers must choose who to trust with sensitive data, a visible commitment to strong cryptography becomes a clear advantage.

Conclusion

Remember, cryptography works because of mathematics. The math is what makes data hard to break, fake, or steal. Without it, encryption would not protect messages, payments, or sensitive records. This is why math sits at the core of every secure digital system.

Today, data is more valuable, rules are stricter, and attacks are stronger.

Quantum computing adds long-term risk to many systems that businesses still rely on. Because of this, math-driven cryptography is not optional. It is a business asset that protects trust, compliance, and long-term stability.

The right response is proactive planning. Companies should know where cryptography is used, design systems that can change when needed, and invest in real expertise. Treat cryptography as a core part of technology and business strategy, not just a box to check, because security lasts only as long as the math holds.

References:

1. Hoffstein, J., Pipher, J., & Silverman, J. H. (n.d.). An introduction to mathematical cryptography. Brown University. Retrieved January 9, 2026, from https://www.math.brown.edu/johsilve/MathCryptoHome.html
2. Biolecta. (n.d.). Mathematics in cryptography. Biolecta. Retrieved January 9, 2026, from https://biolecta.com/articles/mathematics-in-cryptography/
3. Understand The Math. (n.d.). The math behind cryptography: Securing your digital world. https://www.understandthemath.com/blog/math-behind-cryptography
4. Thomas, R. (2025, February 19). Post-quantum cryptography. Plus Magazine. https://plus.maths.org/content/post-quantum-cryptography
5. Richter, M. (2022). A mathematical perspective on post-quantum cryptography. Mathematics, 10(15), 2579. https://www.mdpi.com/2227-7390/10/15/2579
6. Liang, Z., & Zhao, Y. (2022). Number theoretic transform and its applications in lattice-based cryptosystems: A survey. arXiv. https://arxiv.org/abs/2211.13546
7. Lyubashevsky, V., Peikert, C., & Regev, O. (2013). On ideal lattices and learning with errors over rings. Journal of the ACM, 60(6), Article 43, 1–35. https://dl.acm.org/doi/10.1145/2535925https://ju.acm.org/magazine/articles/cryptography-and-post-quantum.html
8. IBM Quantum Learning. (n.d.). Cryptographic hash functions. IBM Quantum Education. Retrieved January 9, 2026, from https://quantum.cloud.ibm.com/learning/en/courses/quantum-safe-cryptography/cryptographic-hash-functions
9. NCC Group. (n.d.). Demystifying multivariate cryptography. https://www.nccgroup.com/research-blog/demystifying-multivariate-cryptography/
10. National Institute of Standards and Technology (NIST). (2020). Recommendation for Key Management: Part 1 – General (SP 800-57 Part 1 Rev. 5). https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-57pt1r5.pdf
11. Rescorla, E. (2018). The Transport Layer Security (TLS) Protocol Version 1.3 (RFC 8446). Internet Engineering Task Force (IETF). https://www.rfc-editor.org/rfc/rfc8446.html
12. National Institute of Standards and Technology (NIST). (2019). Recommendation for Pair-Wise Key Establishment Using Integer Factorization Cryptography (SP 800-56BRev. 2). https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Br2.pdf
13. National Institute of Standards and Technology (NIST). (2016). Report on Post-Quantum Cryptography (NISTIR 8105). https://csrc.nist.gov/pubs/ir/8105/final
14. National Institute of Standards and Technology (NIST). (2024). What is Post-Quantum Cryptography? https://www.nist.gov/cybersecurity/what-post-quantum-cryptography
15. Moody, D. (2022). Cryptographic Standards in a Post-Quantum Era. National Institute of Standards and Technology (NIST). https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=934896
16. National Institute of Standards and Technology (NIST). (2023). Variants of the Syndrome Decoding Problem and algebraic attacks. https://csrc.nist.gov/presentations/2023/variants-of-the-syndrome-decoding-problem-and-alge
17. National Institute of Standards and Technology (NIST). (2023). FIPS 203 (Initial Public Draft): Module-Lattice-based Key-Encapsulation Mechanism Standard (ML-KEM). https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.203.ipd.pdf
18. European Payments Council. (2023). Guidelines on cryptographic algorithms usage and key management (EPC342-08 v12.0). https://www.europeanpaymentscouncil.eu/sites/default/files/kb/file/2023-03/EPC342-08%20v12.0%20Guidelines%20on%20Cryptographic%20Algorithms%20Usage%20and%20Key%20Management.pdf
19. National Institute of Standards and Technology (NIST). (2022). Implementing the HIPAA Security Rule: A Cybersecurity Resource Guide (SP 800-66 Rev. 2). https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-66r2.pdf
20. Cintas Canto, A., Kaur, J., Mozaffari Kermani, M., & Azarderakhsh, R. (2023). Algorithmic Security is Insufficient: A Comprehensive Survey on Implementation Attacks Haunting Post-Quantum Security. arXiv. https://arxiv.org/abs/2305.13544