Digital Signatures Explained: Public Key Cryptography Basics

ECDSA is the signature algorithm used in Bitcoin. To sign a message m, the signer: (1) chooses a random ephemeral key e, (2) computes R = eG and takes R mod Q as the x-coordinate, (3) computes S = e⁻¹(m + kR) mod Q. The signature is (R, S). To verify: (1) compute u₁ = mS⁻¹ mod Q and u₂ = RS⁻¹ mod Q, (2) compute point V = u₁G + u₂Q, (3) verify that the x-coordinate of V equals R. The ephemeral key e must never be reused, as this would allow attackers to recover the private key.

ECDSA (Elliptic Curve DSA) applies elliptic curve cryptography to achieve equivalent security with dramatically smaller key sizes. For 80-bit security, DSA requires 1024-bit keys while ECDSA needs only 160-bit keys, achieving up to 15x speed improvement over RSA. ECDSA uses elliptic curve points instead of integers: setup involves field F_p, curve E, base point G with order q, and secret scalar s. Signing computes random k, point U=kG, S1=x(U) mod q, and S2=(h(m)+sS1)k^(-1) mod q. Verification checks V1G+V2V equals point with x-coordinate S1. Security relies on the elliptic curve discrete logarithm problem, which lacks efficient solving algorithms, making ECDSA a robust, efficient choice for modern digital signatures.

Three major digital signature algorithms are: (1) RSA - deterministic, encrypts hash with private key, widely used, (2) ElGamal - probabilistic, uses discrete logarithm problem, generates different signatures for same message, (3) DSA - probabilistic, similar to ElGamal but uses different mathematical approach. All three algorithms involve computing the message hash, encrypting it with the private key, and verifying with the public key. RSA signatures are deterministic (same message produces same signature), while ElGamal and DSA are probabilistic (same message can produce different signatures). DSA is generally faster than RSA for signing operations due to smaller signature sizes. The choice depends on security requirements, performance needs, and specific application scenarios.

RSA signatures involve: (1) Setup - choosing primes p, q, computing n=p*q, finding e,d such that e*d≡1 mod φ(n), (2) Signing - hash message to y, compute σ=y^d mod n, (3) Verification - compute y'=σ^e mod n and verify. Security relies on factoring difficulty. Current recommendations use 2048-4096 bit moduli. ECDSA uses elliptic curves defined by y²=x³+ax+b over finite fields, forming abelian groups. ECDSA uses domain parameters (field, curve, generator, order, hash function). Signing involves: hash message, choose random k, compute R=k*G, r=x-coordinate mod n, s=k⁻¹(e+dr) mod n. ECDSA security is based on the discrete logarithm problem, which requires exhaustive search, providing stronger security per bit than RSA. ECDSA allows smaller key sizes (256-bit keys provide similar security to 2048-bit RSA), resulting in shorter signatures and faster operations.

Digital signatures provide authentication and integrity verification for messages. In Mathematica 12, ECDSA (Elliptic Curve Digital Signature Algorithm) is the primary supported signature method, using SHA-256 hashing. The signature process involves hashing the message and applying a mathematical operation with the private key, producing two values (R and S). RSA signatures are in development for version 12.1. Key pairs consist of a private key (kept secret) and public key (shared openly). The public key can be published on websites, sent to individuals, or shared via cloud services.

Digital signature is a cryptographic method used to verify the authenticity and integrity of digital messages and documents. Process: (1) Message creation - sender creates the document, (2) Hashing - message is converted to a fixed-size hash value, (3) Signing - sender encrypts the hash with their private key, (4) Sending - message and signature are transmitted, (5) Verification - recipient decrypts signature with sender's public key and compares hash values. Hash functions convert input data of any size into a fixed-size string of characters (hash value or digest). SHA-512 produces a 512-bit (64-byte) hash value, commonly used for data integrity checks and digital signatures. PGP (Pretty Good Privacy) is an encryption program developed by Phil Zimmermann in 1991 that provides cryptographic privacy and authentication for data communication. Real world protocols are rules that help devices and systems communicate over a network securely. HTTP is the foundation of web communication, FTP handles file transfers, and SMTP handles email transmission between servers.

Digital signatures are cryptographic mechanisms that provide authentication and non-repudiation for electronic communications, enabling verification that a message was signed by a specific party without revealing the message content. The two most widely used signature algorithms are RSA-based signatures, which rely on the computational difficulty of factoring large composite numbers, and Elliptic Curve Digital Signature Algorithm (ECDSA), which leverages the discrete logarithm problem on elliptic curves for more efficient operations with smaller key sizes. Public Key Infrastructure (PKI) addresses the fundamental trust challenge by using Certificate Authorities to bind public keys to identities through digital certificates, enabling secure communication over untrusted networks. Real-world applications include TLS/SSL for secure web browsing, GPG for software verification, and Bitcoin for decentralized electronic cash using ECDSA signatures and proof-of-work consensus. However, digital signatures face significant real-world vulnerabilities, as demonstrated by attacks like the Flame malware (which exploited MD5 hash collisions to forge Microsoft signatures) and compromised Certificate Authorities (such as DigiNotar and Comodo), highlighting the critical importance of robust cryptographic practices and PKI management.

This section introduces the authentication problem in TLS/SSL protocols. While previous lectures covered privacy and integrity, authentication ensures you're communicating with the intended party. The man-in-the-middle attack exploits this by intercepting key exchange and establishing separate encrypted channels with both parties. Electronic signatures solve this using asymmetric encryption: messages are encrypted with the sender's private key, allowing anyone with the public key to verify the signature. Combined with hash functions (like SHA-256), this creates a mechanism where the server signs a message hash with its private key, and clients verify using the server's public key.

Digital signatures use hash functions and asymmetric encryption (public/private key pairs) to ensure data integrity and authentication, while SSL/TLS protocols secure web communications through a handshake process that verifies server certificates issued by trusted Certificate Authorities (CAs) and establishes encrypted sessions using session keys.

Digital signature proves data authenticity using public key cryptography, where the sender encrypts a message digest with their private key, and the recipient decrypts it with the sender's public key from an electronic certificate. SSL/TLS secures web communications through session key cryptography, where the server sends its certificate, the client verifies it, creates an encrypted session key, and both parties use it for data encryption. The URL 'https' and lock icon indicate secure connections.

Digital signatures form the cryptographic foundation of blockchain security, enabling secure transaction authorization. Only the owner can produce a valid signature, while anyone can verify its authenticity. This system uses a pair of cryptographic keys: a public key (like an account address) and a private key (like a password). Private keys are generated from seed phrases (typically 24 random dictionary words) through cryptographic hashing. Public keys are mathematically derived from private keys using one-way algorithms. To sign a transaction, the sender uses their private key to create a signature from the transaction message. Verification requires checking the message, signature, and public key. In cryptocurrency systems like Bitcoin, ownership is defined by the ability to produce valid signatures. Possessing a private key means you can generate signatures that prove you own funds at a given address, making possession equivalent to ownership.

Digital signatures solve the trust problem in shared ledgers by allowing senders to cryptographically prove they approved transactions. Each participant generates a public-private key pair; the private key signs transactions while the public key verifies them. Signatures are unique to each message, preventing forgery. To prevent double-spending, transactions include unique identifiers requiring new signatures. This eliminates reliance on participant honesty while ensuring only authorized parties can initiate transactions.

A blockchain is a sequence of transactions partitioned into blocks with persistence (transactions cannot be removed) and consensus (everyone agrees on the transaction set). Digital signatures enable secure currency implementation through three steps: generating a public-private key pair (32 bytes each), signing messages with the secret key to produce a 64-byte signature, and verifying signatures with the public key. In cryptocurrency like Bitcoin, users generate key pairs where the public key is hashed to create a 32-byte address (34 bytes when encoded), allowing others to send funds while only the secret key holder can authorize spending through signed transactions.

Digital signatures form the cryptographic foundation of blockchain technology, functioning similarly to historical wax seals used for document authentication. In this analogy, the public key serves as the visible seal that others can examine, while the private key acts as the hidden stamp used to create authentic signatures. The signing process takes a secret private key and a message to produce a unique signature, which can then be verified against the corresponding public key. This system prevents forgery by making it computationally infeasible to create a valid signature without access to the private key. However, digital signatures alone cannot prevent certain types of fraud, such as double-spending in financial transactions, which requires additional mechanisms like blockchain.

Digital signatures enable secure authorization in blockchain systems. Unlike physical signatures that can be copied, digital signatures are generated by combining a secret signing key with the data being signed. The signature depends on both the data and the private key, making it impossible for anyone without the private key to generate valid signatures. Signatures are used everywhere on blockchains: authorizing transactions, governance votes, and consensus protocols. Each miner must verify all signatures independently, which represents redundant computation.