Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Discrete Mathematics → Cryptography ↓


Post-Quantum Cryptography


In the ever-evolving field of cryptography, a looming threat is reshaping our standards: quantum computing. While classical computers leverage bits to perform computations, quantum computers use qubits. This change promises unprecedented computation capabilities, posing a threat to many widely used cryptographic systems, particularly those based on public-key infrastructures. Enter the field of post-quantum cryptography (PQC), a discipline in cryptographic science designed to future-proof our digital world against quantum threats.

Understanding cryptography

Cryptography, the science of securing communications, is always evolving to combat emerging threats. It has its roots in simple techniques like Caesar's cipher and has since evolved into complex algorithms that play an integral role in digital security.

Traditionally, cryptographic systems use mathematical challenges that are considered difficult for computers to solve quickly. Popular methods include techniques such as the RSA algorithm, which relies on the difficulty of factoring large prime numbers, and elliptic curve cryptography, which involves complex algebraic structures. However, quantum computers, taking advantage of their unique properties, threaten to solve these challenges in feasible timeframes.

Public-key cryptography: A quick recap

Public-key cryptography, or asymmetric cryptography, uses a pair of keys — a public key for encryption and a private key for decryption. This setup enables secure communications over insecure channels, forming the backbone of data security on the Internet.

  • RSA algorithm: Based on the difficulty of factoring large composite numbers. A public key typically consists of the product of two large prime numbers, while the private key consists of their factors.
  • Elliptic Curve Cryptography (ECC): Uses the algebraic structure of elliptic curves over finite fields. It is more efficient than RSA, providing similar security with smaller key sizes.

Both of these systems are effective against traditional attacks, yet become vulnerable to quantum computing architectures.

What is quantum computing?

Traditional computers operate on bits, which are binary units that have two possible values: 0 and 1. Quantum computers, on the other hand, use qubits, which can represent both 0 and 1 simultaneously due to the phenomenon of superposition.

The power of quantum computers lies not just in superposition but also in entanglement and quantum interference, properties that enable them to process complex calculations much more efficiently than classical computers. For some problems, where classical methods can take millennia, quantum computers can provide solutions in practical timeframes.

Why is this a threat to cryptography?

Algorithms designed to exploit these quantum properties can outperform classical methods in specific domains. An example of this is Shor's algorithm, which can factor large integers faster than the best-known classical algorithms. This ability directly threatens RSA and ECC, which are currently the basis of most secure communications on the Internet.

// Quantum algorithm (eg, Shor's algorithm) essentials in pseudocode
initialize quantum state |ψ⟩ 
apply superposition to |ψ⟩ 
perform quantum Fourier transform on |ψ⟩ 
measure |ψ⟩ to collapse state to target factors

The need for post-quantum cryptography

Realizing the imminent threat posed by quantum computing, researchers are working on cryptographic algorithms that can resist quantum attacks. Post-quantum cryptography (PQC) not only relies on the same mathematical challenges as traditional cryptographic systems, but also explores problems that are still computationally intensive, even for quantum computers.

Principles of post-quantum cryptography

PQC focuses on creating difficult algorithms for both quantum and classical computers. Here are some approaches:

  • Lattice-based cryptography: It relies on problems posed by lattices in multi-dimensional space, which are complex structures with challenging mathematical properties.
  • Hash-based cryptography: A time-tested method relying on the collision resistance of hash functions. Creation of signatures using hashes usually involves digital one-time signatures.
  • Code-based cryptography: It uses the hardness of decoding random linear codes as its basis.

Exploration of lattice-based cryptography

Lattice-based cryptography stands out among post-quantum technologies due to its versatility and efficiency. At its core, it relies on the difficulty of finding the "shortest vector" among a set of vectors spread across a lattice in n-dimensional space.

The shortest vector

These vectors, though simple in two or three dimensions, become increasingly complex as dimensions increase. This complexity is what provides security.

Changes in post-quantum cryptography

The transition to post-quantum cryptographic systems is an extensive and careful task. It involves examining existing systems, developing quantum-resistant replacements, and integrating them into the existing infrastructure without disrupting functionality.

Role of standardisation bodies

Organizations like the National Institute of Standards and Technology (NIST) play a key role in this transformation. NIST runs competitions and evaluation rounds to assess potential post-quantum algorithms for standardization. This collaborative effort ensures that any algorithm adopted undergoes rigorous scrutiny, proving its resilience in a number of scenarios.

Challenges in post-quantum cryptography

Despite the promise of PQC, there are challenges to its adoption, such as:

  • Computational overhead: Many PQC algorithms require larger key sizes than traditional counterparts, which impacts communication speed in bandwidth-restricted environments.
  • Integration complexity: The transition of cryptographic systems across widespread infrastructures, from banking systems to IoT devices, demands careful planning and execution.
  • Cutting-edge research: The field is still active, and new discoveries are shaping its future. This variability requires constant adaptation and learning.

Visualization of algorithms and concepts

To illustrate PQC, consider the example of a treasure buried beneath layers in a multidimensional maze.

key concept

Just as adventurers face multiple decision points on their way to the treasure, the PQC system presents computational challenges at multiple turns, making it extremely challenging to bypass. While a classical route may suffice in the current scenario, quantum threats require these additional protective layers to be taken seriously.

Conclusion: A glimpse of the future

Post-quantum cryptography is more than just a technological upgrade; it's a necessary shift in how we think about digital security amid unprecedented computational growth. As quantum technologies advance, so must our security measures, so that we can keep our data safe in an era where traditional barriers are no longer sufficient.

The collaborative effort to develop and integrate quantum-resistant technologies is not just a precaution, but a fundamental effort to protect our digital lives from constantly evolving threats and ensure trust and privacy in the digital age.


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