3.3 Hash functions and digital signatures

Hash functions and digital signatures are crucial components of modern cryptography. They provide essential security features like data integrity, authentication, and non-repudiation. These tools are the backbone of many secure systems, from password storage to blockchain technology.

Understanding how hash functions and digital signatures work together is key to grasping modern cryptographic systems. They're used in everything from verifying software downloads to securing online transactions. Their applications are widespread and fundamental to digital security.

Cryptographic Hash Functions

Properties and Characteristics

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• Cryptographic hash functions are mathematical algorithms that take an input of arbitrary size and produce a fixed-size output, known as a hash value or message digest
• The main properties of cryptographic hash functions are pre-image resistance, second pre-image resistance, and collision resistance
  • Pre-image resistance means that given a hash value, it is computationally infeasible to find the original input message
  • Second pre-image resistance means that given an input message, it is computationally infeasible to find another message that produces the same hash value
  • Collision resistance means that it is computationally infeasible to find two different input messages that produce the same hash value
• Cryptographic hash functions are deterministic, meaning that the same input always produces the same output
• Hash functions are designed to be computationally efficient and fast to compute

Applications and Use Cases

• Cryptographic hash functions are used in various applications, such as data integrity verification, password storage, digital signatures, and blockchain technology
  • Data integrity verification uses hash functions to create a message digest or fingerprint of a file or message, which can be used to verify its integrity (SHA-256)
  • Password storage systems often store the hash value of passwords instead of the plain text to enhance security
  • Digital signature schemes combine hash functions with asymmetric cryptography to provide authentication and non-repudiation (ECDSA)
  • Blockchain technology heavily relies on hash functions to ensure the integrity and immutability of transactions and blocks in the chain (SHA-256 in Bitcoin)
• Examples of widely used cryptographic hash functions include SHA-256, SHA-3, and BLAKE2

Hash Function Algorithms

Design Principles and Constructions

• The Merkle-Damgård construction is a common design principle used in many hash function algorithms, such as SHA-1 and SHA-2 family
  • It involves iteratively processing fixed-size blocks of the input message and combining them with a compression function
  • The final block is padded to ensure a fixed-size output
• The sponge construction is an alternative design principle used in modern hash functions like SHA-3 and BLAKE2
  • It operates on a state that absorbs the input message and then squeezes out the output hash value
  • The sponge construction provides better resistance against certain types of attacks (length extension attacks)

Specific Hash Function Algorithms

• The SHA-2 family includes hash functions like SHA-256 and SHA-512, which use a series of bitwise operations and modular additions to process the input message
  • SHA-256 produces a 256-bit hash value and is widely used in various applications (Bitcoin, TLS)
  • SHA-512 produces a 512-bit hash value and provides higher security but is slower than SHA-256
• The SHA-3 family, also known as Keccak, uses a permutation-based sponge construction and includes hash functions like SHA3-256 and SHA3-512
  • SHA-3 was selected as the winner of the NIST hash function competition in 2012
  • It offers better performance and security compared to SHA-2 in certain scenarios (memory-constrained devices)
• Analyzing the security of hash functions involves studying their resistance to attacks such as length extension attacks, padding attacks, and collision attacks
  • Length extension attacks exploit the structure of the Merkle-Damgård construction to forge valid hash values for extended messages
  • Padding attacks manipulate the padding scheme used in hash functions to create collisions or preimages
• The birthday attack is a probabilistic attack that exploits the birthday paradox to find collisions in hash functions more efficiently than brute force
  • It relies on the fact that in a group of 23 people, there is a 50% chance that two people share the same birthday
  • The birthday attack is used to find collisions in hash functions with a complexity of approximately $\sqrt{2^n}$ for an n-bit hash value

Digital Signature Schemes

Principles and Components

• Digital signature schemes are cryptographic protocols that provide authentication, integrity, and non-repudiation of digital messages or documents
• The main components of a digital signature scheme are a key generation algorithm, a signing algorithm, and a verification algorithm
  • The key generation algorithm creates a pair of keys: a private key used for signing and a public key used for verification
  • The signing algorithm takes the input message and the private key as inputs and produces a digital signature
  • The verification algorithm takes the message, the public key, and the digital signature as inputs and verifies the authenticity and integrity of the message
• The security of digital signature schemes relies on the underlying mathematical problems, such as the discrete logarithm problem or the integer factorization problem
  • The discrete logarithm problem is the basis for elliptic curve digital signature algorithms (ECDSA)
  • The integer factorization problem is the foundation of the RSA digital signature scheme

Digital Signature Algorithms and Protocols

• The Digital Signature Algorithm (DSA) is a widely used digital signature scheme based on the discrete logarithm problem
  • DSA uses the SHA-1 or SHA-2 hash functions to generate a message digest, which is then signed using the private key
  • The signature consists of two components:

    r

    and

    s

    , which are computed based on the private key, the message digest, and a random nonce
• The Elliptic Curve Digital Signature Algorithm (ECDSA) is a variant of DSA that uses elliptic curve cryptography
  • ECDSA provides similar security to DSA with shorter key sizes, making it more efficient and suitable for resource-constrained environments (IoT devices, smart cards)
  • ECDSA is used in various protocols and systems, such as Bitcoin, Ethereum, and Transport Layer Security (TLS)
• The Rivest-Shamir-Adleman (RSA) digital signature scheme is based on the integer factorization problem
  • RSA uses the private key to sign the message digest and the public key to verify the signature
  • RSA signatures are typically larger than ECDSA signatures but offer better compatibility with existing systems and protocols (email, code signing)

Data Integrity and Authentication

Ensuring Data Integrity with Hash Functions

• Hash functions can be used to create a message digest or fingerprint of a file or message, which can be used to verify its integrity
  • The sender computes the hash value of the original message and sends it along with the message
  • The recipient recomputes the hash value of the received message and compares it with the provided hash value to check for any modifications
  • If the hash values match, it indicates that the message has not been altered during transmission
• Examples of using hash functions for data integrity include file integrity checks, software downloads, and data deduplication (SHA-256 in BitTorrent)

Combining Hash Functions and Digital Signatures

• Digital signatures can be combined with hash functions to provide both integrity and authentication of messages
• In a digital signature scheme, the message is first hashed using a cryptographic hash function, and then the hash value is signed using the private key of the signer
  • The signing process ensures that the message originated from the claimed sender and has not been modified
• The recipient can verify the signature by hashing the received message, using the public key to verify the signature, and comparing the computed hash value with the one obtained from the signature
  • If the signature verification succeeds, it confirms the authenticity and integrity of the message
• Digital certificates, such as X.509 certificates, use digital signatures to bind a public key to an identity, providing authentication and trust in public key infrastructure (PKI) systems (SSL/TLS certificates)
• Blockchain technology heavily relies on hash functions and digital signatures to ensure the integrity and immutability of transactions and blocks in the chain
  • Each block in the blockchain contains a hash of the previous block, creating a chain of linked blocks
  • Transactions in the blockchain are digitally signed by the senders, ensuring their authenticity and non-repudiation (Bitcoin transactions)