Digital Signature Scheme Overview

Updated 8 January 2026

Digital Signature Scheme is a cryptographic framework that uses key generation, signing, and verification algorithms to produce secure, non-repudiable proofs of data authenticity.
It incorporates classical methods like RSA and ECDSA as well as modern post-quantum techniques such as lattice-based and code-based approaches to meet diverse security needs.
The scheme's security is guaranteed by properties like EUF-CMA, forward security, and designated verifiability, making it essential for secure protocols and access control applications.

A digital signature scheme is a set of algorithms enabling a party to produce a cryptographically verifiable proof of authorization, authenticity, or integrity for a message or data item. Such schemes are used to secure protocols, authenticate access rights, guarantee non-repudiation, and build privacy-preserving primitives across classical and quantum communication networks. The essential construction consists of a key generation process yielding a private signing key and a public verification key, a signing algorithm that derives a message-dependent signature, and a verification algorithm accepting valid signatures. The security of a digital signature scheme is measured by its resilience—typically existential unforgeability under chosen-message attack (EUF-CMA)—and may incorporate additional guarantees such as forward security, blindness, or designated verifiability, depending on the protocol variant and intended use case.

1. Formal Definitions and Core Algorithms

A digital signature scheme Σ is formally specified as a triple of probabilistic polynomial-time algorithms (KeyGen, Sign, Verify) (Nia et al., 2014):

KeyGen(1^λ) → (pk, sk): On input the security parameter λ, output a public key pk and a secret key sk.
Sign(sk, m) → σ: On message m, outputs signature σ using sk.
Verify(pk, m, σ) → {accept, reject}: Verifies (m, σ) under pk.

Correctness requires that for all λ, all keypairs, and all messages, if σ ← Sign(sk, m) then Verify(pk, m, σ) = accept.

Security goals span several attack models:

Existential Unforgeability Under Adaptive Chosen-Message Attack (EUF-CMA): No efficient adversary, given signing oracle access, can forge (m*,σ) such that Verify(pk,m,σ)=accept and m is fresh (Nia et al., 2014).
Additional properties: forward security (time-evolving keys), blindness (signer learns nothing about the message being signed), proxy delegation, batch signing, and designated verifiability.

Underlying computational hardness assumptions include the difficulty of the RSA problem, the discrete logarithm problem (DLP), the syndrome decoding problem (SDP for code-based systems), lattice problems such as SIS/LWE, and, in post-quantum variants, algebraic problems resistant to quantum algorithms.

2. Representative Classical Signature Schemes

Classical schemes most frequently leverage number-theoretic primitives:

RSA-based Signature: KeyGen involves generating an RSA modulus N, public exponent e, private exponent d. Signing evaluates σ = H(m)^d mod N, verification accepts if σ^e ≡ H(m) mod N. Security derives from the assumed intractability of RSA inversion (Nia et al., 2014).
ECDSA/Schnorr Signatures: Use the group structure of elliptic curves; signatures are produced via group exponentiation, verification relies on public-key group operations. ECDSA achieves security with much shorter keys at equivalent computational cost and is favored in resource-constrained environments (Imem, 2015).

Digital signature schemes are also available in variants tailored for efficiency or additional security:

Scheme	Key Size	Signature Size	Throughput/Overhead
RSA (classic)	2048-bit	2048-bit	1 exp. per sign/verify
Batch RSA	2048-bit	k·2048-bit + overhead	k+1 exp., parallelizable
Proxy-Signature	RSA/ECC	sig + warrant	delegation overhead
ECDSA	~256-bit	~512-bit	fast sign, moderate verify

Optimal scheme selection depends on the application's requirements—mass verification favors batch schemes; long-lived keys subject to exposure favor forward-security; privacy-sensitive tasks use blind or designated-verifier schemes (Nia et al., 2014, Renan, 20 Jul 2025).

3. Modern Post-Quantum and Exotic Schemes

Recent advances target quantum-resistant and nonstandard security needs, exploiting algebraic or structured problems outside classical group theory:

Code-Based Signatures: LEDAsig employs the syndrome decoding problem for QC-LDGM codes, achieving sub-ms signing, multi-kilobyte signatures, and quantum-resistance (Baldi et al., 2018). RYDE adopts rank-metric codes with zero-knowledge proofs (MPC-in-the-head paradigm), supporting compact signatures and minimal public keys for post-quantum security (Bidoux et al., 2023).
Lattice-Based/Module-SIS/LWE Signatures: Digital signatures built on lattice problems, like module-SIS and module-LWE, deliver strong theoretical security and flexible key/signature sizes suited for post-quantum protocols (Al-Jabbari et al., 2024, Dey et al., 2024). Undeniable signatures require interactive zero-knowledge confirmation/disavowal, so only designated verifiers can validate signatures (Dey et al., 2024).
Multivariate Quadratic (MQ) Signatures: Schemes such as VDOO insert diagonal layers into oil-vinegar constructions to optimize Gaussian elimination complexity, yielding very short signatures at NIST PQC levels with substantial speed advantages (Ganguly et al., 2023).
Algebraic Matrix/Automorphism Schemes: Protocols leveraging boolean automorphisms (BASS), non-square matrices, or scrap automorphisms in polynomial rings for post-quantum security via hardness of inverting or reconstructing automorphic mappings; these tend to have larger key/signature sizes (Grigoriev et al., 2023, Chen et al., 2023).

Scheme	Security Basis	Signature Size	Key Size	Notable Features
LEDAsig	SDP (code)	3–112 KB	0.3–15 MB	Fast sign, large public key
RYDE	RSD (rank)	~6–9 KB	~0.1 KB	Zero-knowledge, PQ security
BASS	Boolean Aut.	~4 KB	~12 KB	#P-hard verification
VDOO	MQ (UOV/Rainbow)	96–316 B	67–343 KB	Minimal signature, fast verification
CSI-SDVS	Isogeny, MT-GAIP	~272 B	~1 KB	Designated verifier, PQ compactness

Quantum digital signature schemes exploit principles of quantum mechanics for information-theoretic security and transferability not achievable classically, using entanglement (EPR channels), quantum key distribution (QKD), and universal hashing (Nadeem et al., 2015, Arrazola et al., 2015, Xiong et al., 2024).

4. Security Concepts and Provable Guarantees

Signature scheme security is generally defined and proved in the random oracle model or standard model. Major security notions include:

Unforgeability (EUF-CMA): The scheme is robust against an adversary with signing oracle access (Nia et al., 2014). Demonstrated by reductions to computational hardness problems—RSA inversion, syndrome decoding, lattice SVP/SIS, etc. (Lee et al., 2015, Baldi et al., 2018, Al-Jabbari et al., 2024).
Non-Repudiation: Signer cannot deny a valid signature; achieved by unambiguous verification steps and key-dependent signatures (Lee et al., 2015, Nadeem et al., 2015).
Designated Verification and Invisibility: Only specific verifiers can validate, or adversaries cannot distinguish real signatures from simulated transcripts (CSI-SDVS, undeniable signatures) (Renan, 20 Jul 2025, Dey et al., 2024).
Transferability: Necessary in multiparty or quantum systems, allowing a valid signature to be sequentially handed off and verified across participants (Arrazola et al., 2015).
Freshness/Expiry: Tokens/signatures include explicit expiry, enforced by verification logic (Lee et al., 2015).

Security proofs often rely on reductions: forgers breaking the scheme would yield efficient solvers to intractable problems, contradicting established computational or information-theoretic bounds (Lee et al., 2015, Ganguly et al., 2023).

5. Efficiency, Implementation, and Performance

Performance and practicality are dictated by arithmetic complexity, key/signature sizes, and protocol operational cost:

RSA/Classic: Modular exponentiation is cubic in key size; signature and key sizes scale linearly with parameter length (Lee et al., 2015, Nia et al., 2014).
ECC/ECDSA: Group operations on short keys, rapid keygen/signing, but slower verification relative to RSA (Imem, 2015).
Post-Quantum Schemes: Code-based (LEDAsig, RYDE) and lattice/module-based systems require fast polynomial or syndrome arithmetic, often with kilobyte-scale keys and signatures (Baldi et al., 2018, Bidoux et al., 2023, Al-Jabbari et al., 2024).
Quantum/Transferable Schemes: Key consumption and signature calculation scale with number of receivers and security parameters, but efficient arbitrated quantum digital signature (AQDS) achieves two orders of magnitude lower key usage than prior art (Xiong et al., 2024).

Empirical results confirm sub-ms signing and verification (LEDAsig), 12 ms EC-based signing on 8-bit MCUs (SEMECS) (Yavuz et al., 2019), and scalable verification for mass/batch scenarios (Nia et al., 2014).

6. Specializations and Comparative Dimensions

Schemes are adapted for unique application domains and advanced security requirements:

Access Control: SignEPC issues RSA-based tokens encoding user-specific access rights and expiry, allowing for local verification and scalable network operation without access queries (Lee et al., 2015).
Blind/Proxy/Fwd-Secure: Blind schemes protect privacy even from the signer, proxy signatures delegate authority securely, forward-secure signatures withstand key exposure (Nia et al., 2014).
Designated Verifier/Undeniable: CSI-SDVS implements strong designated verification in the isogeny framework, while module-lattice undeniable signatures enforce interactive confirmation/disavowal (Renan, 20 Jul 2025, Dey et al., 2024).
Quantum/Multiparty: Transferable information-theoretic QS schemes are formally characterized by nested verification levels and rigorous analysis of forging, repudiation, and non-transferability events (Arrazola et al., 2015).

7. Practical and Future Considerations

Choice of digital signature scheme should respect the trade-offs between security level, key and signature size, performance on constrained hardware, scalability, and post-quantum resilience. Guidelines suggest batch schemes for high-throughput logging, forward security for long-lived systems, blind or designated-verifier for privacy, and code/lattice/MQ-based for post-quantum deployments (Nia et al., 2014, Renan, 20 Jul 2025, Ganguly et al., 2023, Baldi et al., 2018).

Notably, emerging quantum and algebraic primitives—module-lattice, isogeny, neural network-based, Boolean and automorphism systems—continue to expand the available design space, emphasizing parameter flexibility, compactness, and security against known quantum attack paradigms (Dey et al., 2024, Renan, 20 Jul 2025, Grigoriev et al., 2023, Kumar et al., 28 Jul 2025). Signature schemes are a locus of ongoing progress, with robustness against cryptanalytic advances and efficient deployment in distributed, IoT, and quantum-secure infrastructures remaining central research challenges.