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 ↓


Symmetric and Asymmetric Cryptography


Cryptography is an important aspect of computer security, designed to protect information by converting it into a secure format, encrypting data so that it can only be read by someone who has the correct decryption key. There are two main types of cryptography algorithms - symmetric and asymmetric cryptography. Both have their own specific functions and use cases, methods and technicalities, which we will look at in detail.

Symmetric cryptography

Symmetric cryptography, also known as secret-key cryptography, involves the use of a single key for both encryption and decryption. This single key must be shared between the parties that wish to exchange encrypted data. It is the oldest and most common form of cryptography. Some popular symmetric encryption algorithms include Data Encryption Standard (DES), Advanced Encryption Standard (AES), and Blowfish.

How symmetric encryption works

In symmetric encryption, both the sender and the recipient share a key, which they use to encrypt and decrypt messages. The process can be summarized as follows:

Step 1: Choose a secret key. Step 2: Use the key to encrypt the message into ciphertext. Step 3: Send the ciphertext to the receiver. Step 4: The receiver uses the same secret key to decrypt the ciphertext back to the original message.

Example of symmetric encryption

Let's consider a simple symmetric encryption example:

  • Original message: "HELLO"
  • Secret key: "K"
  • Encrypted message: Transform each letter by moving the 'K' place in the alphabet.

For simplicity let's assume a basic Caesar cipher method. If 'K' refers to a shift of 3 positions, then:

H -> KE -> HL -> OL -> OO -> R

The ciphertext becomes: "KHOOR".

The receiver, who knows the key is 3, can convert this change back to "hello".

Advantages and limitations of symmetric cryptography

Benefit

  • Efficiency: Symmetric encryption is faster than asymmetric encryption and requires less computational power.
  • Simplicity: The use of a single key simplifies the process, which helps in faster implementation.

Boundaries

  • Key distribution: It is often difficult to securely share keys between parties. If the key is compromised, the encrypted data is no longer secure.
  • Scalability: For networks with many users, the number of keys required grows rapidly, complicating key management.

Asymmetric cryptography

Asymmetric cryptography, also known as public-key cryptography, uses a pair of keys: a public key and a private key. In this system, the public key can be openly shared, while the private key remains confidential to its owner. Data encrypted with a public key can only be decrypted by the corresponding private key and vice versa. Some common algorithms include RSA, ECC (Elliptic-Curve Cryptography), and DSA (Digital Signature Algorithm).

How asymmetric encryption works

The asymmetric encryption process works as follows:

Step 1: Generate a pair of keys – public and private. Step 2: Share the public key while keeping the private key secret. Step 3: Encrypt the message using the public key. Step 4: Send the encrypted message (ciphertext) to the receiver. Step 5: The receiver uses their private key to decrypt the message.

Example of asymmetric encryption

Here's an example of asymmetric encryption:

  • Sender's message: "HELLO"
  • The recipient shares their public key: PubKey_B
  • The sender encrypts the message using PubKey_B

The ciphertext will look different, and without the private key it cannot be easily converted to "HELLO":

Ciphertext Example: 5a3b8c...

Only the receiver can decrypt using his private key: PrivKey_B.

Advantages and limitations of asymmetric cryptography

Benefit

  • Advanced Security: No need to share private keys, security is enhanced by limiting the exposure of keys.
  • Scalability: In a network of users, each user holds a single key-pair, simplifying key management.

Boundaries

  • Slower speed: Asymmetric encryption requires more computational resources, making it slower than symmetric methods.
  • Complexity: Generation and management of key pairs is more complex than symmetric key management.

Combination of symmetric and asymmetric cryptography

In practice, many secure systems use a combination of symmetric and asymmetric cryptography to achieve both efficiency and security through a process called hybrid encryption. In such systems, asymmetric cryptography is used to securely exchange a symmetric key, which is then used for fast encryption and decryption of data. Here's an overview of how it works:

Step 1: Generate a symmetric key for data encryption. Step 2: Encrypt the symmetric key using the receiver's public key. Step 3: Send both the encrypted data and the encrypted key to the receiver. Step 4: Receiver decrypts the symmetric key with their private key. Step 5: Use the symmetric key to decrypt the data.

Visual example of hybrid encryption

SenderAESMessage Encrypted messagesReceiverReceiver uses the private keyMessage Decrypted

In the above example, the message is first encrypted with a symmetric key using AES. This symmetric key is then encrypted using the receiver's public key and sent along with the encrypted message. The receiver decrypts the symmetric key using his private key and can then decrypt the message.

Applications of cryptography

Cryptography is essential for securing data in a variety of industries and applications:

  • Banking: secure transactions, privacy of communications, identity verification.
  • Internet: Secure browsing using SSL/TLS, email security with PGP/GPG.
  • Healthcare: patient data protection, secure communications between health care providers.
  • Government: protection of classified information, secure communication channels.

Conclusion

Both symmetric and asymmetric cryptography play a vital role in securing data and maintaining privacy in the modern digital age. Understanding these cryptographic methods helps in choosing the right approach for particular security needs and applications. By combining the strengths of each technology, we are able to provide robust security solutions that meet both efficiency and security demands.


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Number Theory Applications in Cryptography
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Symmetric and Asymmetric Cryptography

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