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


Cryptography is an important field of study within discrete mathematics and plays a vital role in ensuring security on a variety of platforms today. At its core, cryptography is the practice of secure communications in the presence of adversaries. It has a long history, dating back thousands of years, but in contemporary settings, it is a fundamental part of computer science and information security.

Understanding cryptography

Cryptography involves creating methods to protect communications. It is used to protect information from unauthorized access, ensure data integrity, provide authentication, and even ensure that data cannot be denied once it has been created. The main concepts revolve around encryption and decryption:

Encryption and decryption

Encryption is a process in which plain text information is converted into an unreadable form called cipher text using an algorithm and a key. The key is a vital part of the encryption process, which determines how the text is transformed.

Decryption is the reverse process, where the cipher text is converted back into readable plain text. This process requires knowledge of the key used in encryption.

Example of basic encryption: Caesar cipher

A simple example of an encryption method is the Caesar cipher. This cipher is named after Julius Caesar, who, according to legend, used it to communicate with his generals.

In a Caesar cipher, each letter in the plain text is 'shifted' a certain number down in the alphabet. Here's how it works:

Plain Text : HELLO Shift : 3 Encrypted : KHOOR

How does it work? Consider the alphabet:

ABCDEFGHIJKLMNOPQRSTUVWXYZ

If we move three places to the right:

DEFGHIJKLMNOPQRSTUVWXYZABC

Thus, 'H' becomes 'K', 'E' becomes 'H', 'L' becomes 'O', and so on.

Types of cryptography

Cryptographic methods may be broadly classified into three types:

1. Symmetric key cryptography

In symmetric key cryptography, the same key is used for both encryption and decryption. This is similar to a locked box that both parties can open because they have the same key. It is efficient and fast, but the challenge lies in sharing the key securely. Some common symmetric algorithms include:

  • Data Encryption Standard (DES)
  • Advanced Encryption Standard (AES)

Visual example:

plain text Cipher The same key

2. Asymmetric key cryptography

Asymmetric algorithms use a pair of keys - a public key and a private key. The public key is shared openly and used for encryption, while the private key is kept secret for decryption. It has significant advantages over symmetric cryptography in terms of key distribution but is more computationally intensive.

  • Rivest–Shamir–Adleman (RSA)
  • Elliptic Curve Cryptography (ECC)

Visual example:

plain text Cipher public key plain text private key

3. Hash functions

Hash functions do not involve keys. Instead, they take an input (or 'message') and return a fixed-size string of bytes. The output, usually a 'digest', is unique for each unique input. It is nearly impossible to derive the original input from its hash output.

Hash functions are used to ensure data integrity and password storage, among other applications. Popular hash functions include:

  • MD5 (now considered insecure for many applications)
  • SHA-256

Visual example:

Message Hash Output Hash functions

Mathematical foundations of cryptography

Mathematics is the basis of cryptography, with number theory and algebra playing important roles. Some of the important concepts are as follows:

1. Modular arithmetic

Cryptographic algorithms often use modular arithmetic. This is like clockwork arithmetic where the numbers rotate after reaching a certain point - the 'modulus'. For example, in modulo 12 arithmetic, 13 equals 1:

13 mod 12 = 1

Modular arithmetic is essential in many cryptographic systems, notably RSA, which uses properties of prime numbers in its operations.

2. Prime numbers

Prime numbers, numbers that are divisible only by 1 and themselves, are fundamental in cryptography. They are used to generate keys in algorithms such as RSA. The difficulty of factoring large numbers into primes underlies the security of these systems.

3. Euler's totient function

Euler's totient function, denoted φ(n), is important in cryptography for computing public and private keys. It counts the number of integers up to a given integer n that are coprime with n.

4. Discrete logarithm

The discrete logarithm problem is another mathematical concept used in cryptography. It is similar to the regular logarithm, but within the set of integers. This problem underlies the security of some cryptographic systems such as Diffie-Hellman key exchange.

Applications of cryptography

Cryptography plays a vital role in securing information systems. Its major applications include:

1. Secure communication

Encryption allows private conversations over public channels. Classic uses include secure web browsing (HTTPS), which relies on cryptographic protocols to protect data transmitted over the Internet.

2. Authentication

Cryptography ensures that data comes from a verified source. Techniques such as digital signatures allow users to verify the authenticity of a message, software, or digital document.

3. Digital signature

These are the cryptographic equivalent of handwritten signatures, but they are much more secure. Digital signatures certify the integrity and origin of the message, ensuring it has not been altered.

4. Cryptographic hash functions in password security

Hash functions store passwords securely. When a user inputs a password, it is hashed and compared to the stored hash. This means that even if the hash database is tampered with, the passwords are not directly exposed.

Conclusion

Cryptography is a cornerstone of modern security practices, enabling secure transactions and communications across digital platforms. Its foundations in discrete mathematics, particularly in number theory and algebra, highlight the complex relationships between mathematical theory and practical applications. By protecting information, cryptography remains an essential field in maintaining privacy and trust in an increasingly connected world.


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Cryptography

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