Digital signature

As explained in Hacking Cryptography

A cryptographic value associated with data that enables verification of its integrity and origin by parties possessing the required verification information.

A digital signature is a cryptographic proof that a message came from the holder of a private key and was not altered. Anyone with the corresponding public key can verify it, but verification does not give them the ability to create new valid signatures. This asymmetric design avoids the shared-secret problem of MACs, where every verifier could also forge messages. section 9.0 section 9.1

Signing and verification with RSA

For a schoolbook RSA keypair, the private key contains d and the public key contains (e, n), with ed ≡ 1 mod φ(n). Signing computes s = m^d mod n; verification computes s′ = s^e mod n and accepts when s′ equals m. This simple construction is unsafe by itself: an attacker can choose a random s, calculate m = s^e mod n, and obtain a valid signature-message pair. Real schemes therefore enforce formatting and padding rules, and some add randomness. section 9.4 section 9.7

Digital signatures support several security and trust workflows:

• Authenticate messages without sharing the signing private key with verifiers. section 9.2
• Establish nonrepudiation when a valid signature verifies under a public key and the private key has remained secure. section 9.2
• Authorize software before execution by checking signatures against trusted manufacturer keys and a hardware-protected root of trust. section 9.2.2
• Authorize ownership transfers and transaction details in blockchains and chip-based payment systems. section 9.2.3

Implementation discipline matters. A signature scheme is broken if attackers recover its private key or forge signatures. ECDSA requires a unique nonce for each signing operation; reusing one nonce can reveal the nonce and then the private key. Forgery attacks also differ: universal forgery covers arbitrary messages, selective forgery covers constrained messages, and existential forgery produces at least one valid pair. section 9.7

Figure 9.2 Digital signatures provide an asymmetric alternative to proving message authenticity.

Use the asymmetric signing diagram to trace private-key signature generation and public-key verification. section 9.1

How Hacking Cryptography covers it

How deep it goes

Derives schoolbook RSA signing and verification, demonstrates existential forgery, and flags padding and nonce-reuse failures.

Examples

Uses schoolbook RSA, signed software, Bitcoin ownership transfers, and chip-based payment transactions as concrete examples.

What it assumes

Assumes familiarity with cryptographic integrity, authenticity, hashing, MACs, and modular arithmetic while contrasting symmetric and asymmetric keys.

What you can do after

Evaluate/understand asymmetric signing, recognize basic forgery risks, and identify catastrophic ECDSA nonce reuse.

How it connects

Connects digital signatures to message authentication codes, public-key cryptography, nonrepudiation, digital certificates, blockchains, and code integrity.

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Also explained in

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  Software Security for DevelopersA digital signature is a cryptographic value that authenticates a message and binds it to the signer so that the signature can be independently verified.
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  Real-World CryptographyA digital signature is a cryptographic value computed with a private signing key and verified with the corresponding public key to authenticate data or its signer.
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  Secure APIsA token signature is cryptographic data that allows a recipient to verify that a token was issued by a trusted party and has not been altered.
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  API Design PatternsA digital signature is a cryptographic value generated with a private key and verifiable with the corresponding public key to authenticate the signer and detect changes to signed data.
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  Elliptic Curve Cryptography for DevelopersA digital signature is a cryptographic value produced with a private key that authenticates data and can be checked with the corresponding public key.
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  Grokking Web Application SecurityA cryptographic signature is data generated with a private key that enables verification of a message's authenticity and integrity using the corresponding public key.
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  Learn PowerShell Scripting in a Month of Lunches, Second Edition: Write and organize scripts and toolsA digital signature is cryptographic metadata attached to a file that authenticates its signer and detects changes to the signed content.

Compare how 8 books explain digital signature →

Related concepts

• private keyproduced by · A private key is the secret component of an asymmetric cryptographic key pair used by its holder for operations such as authentication, decryption, or digital signing.
• ECDSAimplemented by · ECDSA is a public-key digital signature algorithm that uses elliptic-curve arithmetic to generate and verify signatures.
• Schnorr signatureincludes · A Schnorr signature is a public-key digital signature scheme based on a proof of knowledge of a discrete logarithm.
• CRYSTALS-Dilithiumincludes · CRYSTALS-Dilithium is a lattice-based digital signature scheme based on the learning with errors problem.
• hash-based signature schemeincludes · A hash-based signature scheme is a digital signature scheme whose security is based on cryptographic hash-function properties rather than algebraic number-theoretic assumptions.
• public keyuses · A public key is the openly distributable component of an asymmetric cryptographic key pair, used for encryption, authentication, or signature verification.

This explainer was generated by Manning's knowledge graph from the text of Hacking Cryptography. The section links point to the parts of the book it draws on.