Cryptography

4.

Asymmetric Key Cryptography

4.1.

Public Key Cryptography Principles

4.1.1.

Key Pair Concept

4.1.1.1.

Public Key Properties

4.1.1.2.

Private Key Properties

4.1.1.3.

Key Relationship

4.1.2.

One-Way Functions

4.1.2.1.

Computational Assumptions

4.1.3.

Trapdoor Functions

4.1.3.1.

Trapdoor Information

4.1.3.2.

Easy vs. Hard Directions

4.1.4.

Computational Hardness Assumptions

4.1.4.1.

Integer Factorization Problem

4.1.4.2.

Discrete Logarithm Problem

4.1.4.3.

Elliptic Curve Discrete Logarithm Problem

4.1.5.

Public Key Advantages

4.1.5.1.

Key Distribution Solution

4.1.5.2.

Digital Signatures

4.1.5.3.

Non-repudiation

4.1.6.

Public Key Disadvantages

4.1.6.1.

Computational Overhead

4.1.6.2.

Key Size Requirements

4.1.6.3.

Implementation Complexity

4.1.7.

Hybrid Cryptosystems

4.1.7.1.

Key Encapsulation Mechanism

4.1.7.2.

Data Encapsulation Mechanism

4.1.7.3.

Performance Benefits

4.2.

Key Agreement Protocols

4.2.1.

Diffie-Hellman Key Exchange

4.2.1.1.

Protocol Description

4.2.1.2.

Mathematical Foundation

4.2.1.3.

Security Analysis

4.2.1.4.

Man-in-the-Middle Vulnerability

4.2.1.5.

Discrete Logarithm Assumption

4.2.2.

Ephemeral Diffie-Hellman

4.2.2.1.

Forward Secrecy

4.2.2.2.

Session Key Generation

4.2.2.3.

Implementation Considerations

4.2.3.

Elliptic Curve Diffie-Hellman

4.2.3.1.

Elliptic Curve Operations

4.2.3.2.

Key Size Advantages

4.2.3.3.

Curve Selection Criteria

4.2.4.

Station-to-Station Protocol

4.2.4.1.

Authentication Integration

4.2.4.2.

Digital Signature Usage

4.2.5.

4.2.5.1.

Implicit Authentication

4.2.5.2.

Security Properties

4.3.

Public Key Encryption

4.3.1.

4.3.1.1.

Mathematical Foundation

4.3.1.1.1.

Modular Exponentiation

4.3.1.1.2.

Euler's Theorem Application

4.3.1.2.

Key Generation Process

4.3.1.2.1.

Prime Selection

4.3.1.2.2.

Modulus Computation

4.3.1.2.3.

Exponent Selection

4.3.1.2.4.

Private Key Computation

4.3.1.3.

Encryption Process

4.3.1.4.

Decryption Process

4.3.1.5.

Security Analysis

4.3.1.5.1.

Factoring Problem

4.3.1.5.2.

Chosen Ciphertext Attacks

4.3.1.6.

Padding Schemes

4.3.1.6.1.

4.3.1.6.2.

4.3.1.6.3.

4.3.1.7.

Implementation Attacks

4.3.1.7.1.

4.3.1.7.2.

4.3.1.8.

Key Size Recommendations

4.3.2.

ElGamal Encryption

4.3.2.1.

Mathematical Foundation

4.3.2.2.

4.3.2.3.

Encryption Process

4.3.2.4.

Decryption Process

4.3.2.5.

Security Properties

4.3.2.6.

Ciphertext Expansion

4.3.3.

Rabin Cryptosystem

4.3.3.1.

Quadratic Residues

4.3.3.2.

Decryption Ambiguity

4.3.3.3.

Security Equivalence to Factoring

4.3.4.

Paillier Cryptosystem

4.3.4.1.

Homomorphic Properties

4.4.

Elliptic Curve Cryptography

4.4.1.

Elliptic Curve Mathematics

4.4.1.1.

Curve Equations

4.4.1.2.

Point Addition Rules

4.4.1.3.

4.4.1.4.

Scalar Multiplication

4.4.1.5.

4.4.2.

Elliptic Curve Parameters

4.4.2.1.

Curve Coefficients

4.4.2.2.

Base Point Selection

4.4.2.3.

Order Computation

4.4.2.4.

4.4.3.

Standard Curves

4.4.3.1.

4.4.3.2.

Brainpool Curves

4.4.3.3.

4.4.3.4.

4.4.4.

4.4.4.1.

Smaller Key Sizes

4.4.4.2.

Computational Efficiency

4.4.4.3.

Bandwidth Savings

4.4.5.

ECC Implementation Considerations

4.4.5.1.

Point Representation

4.4.5.2.

Coordinate Systems

4.4.5.3.

Side-Channel Resistance

4.4.6.

Elliptic Curve Discrete Logarithm Problem

4.4.6.1.

Problem Definition

4.4.6.2.

4.4.6.3.

Security Parameters

4.5.

Digital Signature Schemes

4.5.1.

Digital Signature Principles

4.5.1.1.

Authentication Property

4.5.1.2.

Integrity Property

4.5.1.3.

Non-repudiation Property

4.5.2.

Signature Generation and Verification

4.5.2.1.

4.5.2.2.

Signing Algorithm

4.5.2.3.

Verification Algorithm

4.5.3.

4.5.3.1.

Basic RSA Signature

4.5.3.2.

Hash-and-Sign Paradigm

4.5.3.3.

Padding Schemes

4.5.3.3.1.

4.5.3.3.2.

4.5.3.4.

Security Considerations

4.5.4.

Digital Signature Algorithm

4.5.4.1.

Mathematical Foundation

4.5.4.2.

Parameter Generation

4.5.4.3.

4.5.4.4.

Signature Generation

4.5.4.5.

Signature Verification

4.5.4.6.

Security Analysis

4.5.5.

Elliptic Curve Digital Signature Algorithm

4.5.5.1.

ECDSA Parameters

4.5.5.2.

Signature Generation

4.5.5.3.

Signature Verification

4.5.5.4.

Deterministic ECDSA

4.5.5.5.

4.5.6.

Schnorr Signatures

4.5.6.1.

Protocol Description

4.5.6.2.

Security Properties

4.5.6.3.

Batch Verification

4.5.7.

Blind Signatures

4.5.7.1.

Blinding Process

4.5.7.2.

Unblinding Process

4.5.8.

Ring Signatures

4.5.8.1.

Anonymity Properties

4.5.8.2.

Construction Methods

4.5.9.

Threshold Signatures

4.5.9.1.

Secret Sharing Integration

4.5.9.2.

Distributed Signing

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5. Cryptographic Hash Functions