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Digital signature forgery

Digital signature forgery is the process by which an adversary generates a valid digital signature for a message without possessing the signer's private key, thereby compromising the cryptographic assurance of authenticity, integrity, and non-repudiation provided by digital signature schemes.^[1] In cryptographic systems, digital signatures rely on asymmetric key pairs where the signer uses a private key to produce a signature verifiable by anyone using the corresponding public key, typically involving hash functions to bind the signature to the message content.^[2] Forgery attacks are categorized by their goals and the adversary's capabilities: existential forgery involves producing a valid signature for at least one message (often meaningless), selective forgery targets a specific pre-chosen message, and universal forgery enables signing arbitrary messages by deriving an equivalent signing mechanism.^[1] The security of digital signature schemes is evaluated against various attack models, ranging from key-only attacks (where the adversary has only the public key) to more powerful message attacks, including known-message (access to signatures on fixed messages), chosen-message (signatures on adversary-selected messages), and adaptive chosen-message attacks (where message choices depend on prior signatures obtained via a signing oracle).^[1] A scheme is considered secure if existential forgery under adaptive chosen-message attack is computationally infeasible, assuming underlying hard problems like integer factorization or discrete logarithms remain unsolved.^[2] Foundational work, such as the Goldwasser-Micali-Rivest (GMR) scheme from 1988, demonstrated provable security against adaptive attacks using claw-free permutations, influencing modern standards like those in NIST's Digital Signature Standard (DSS).^[2] Contemporary concerns include vulnerabilities in outdated algorithms (e.g., RSA without proper padding) and the need for post-quantum resistant signatures to counter future quantum computing threats that could enable efficient forgery via Shor's algorithm. In response, NIST standardized post-quantum digital signatures in 2024, including ML-DSA (FIPS 204) and SLH-DSA/FN-DSA (FIPS 205).^[3]

Fundamentals

Digital Signatures

A digital signature is a cryptographic value generated from a message and a private key, which can be verified using the corresponding public key to confirm the message's authenticity, integrity, and the signer's non-repudiation of authorship.^[4] This mechanism ensures that the signed data has not been altered since signing and that it originates from the claimed sender, serving as a digital analogue to a handwritten signature.^[5] The core components of a digital signature scheme include a private key used exclusively by the signer for generation, a public key available for verification by any recipient, and a cryptographic hash function—such as SHA-256—to condense the message into a fixed-size digest before signing.^[6] In the basic process, the signer computes the signature \sigma = \text{Sign}(sk, \text{Hash}(m)), where sk is the private key, m is the message, and \text{Hash} produces the message digest; the verifier then checks \text{Verify}(pk, \text{Hash}(m), \sigma) using the public key pk, accepting the signature if it matches.^[7] Digital signatures originated in the late 1970s and early 1980s as part of the development of public-key cryptography, with the RSA algorithm published in 1978 providing one of the first practical constructions for generating and verifying signatures based on the difficulty of integer factorization. Common schemes in use today include RSA, which relies on modular exponentiation; the Digital Signature Algorithm (DSA), standardized by NIST in 1991 and based on the discrete logarithm problem; and the Elliptic Curve Digital Signature Algorithm (ECDSA), an elliptic curve variant of DSA offering smaller key sizes for equivalent security.^[8] Unforgeability under chosen-message attacks is a fundamental security property required of these schemes.^[8]

Unforgeability in Security Models

Unforgeability is a fundamental security property of digital signature schemes, ensuring that it is computationally infeasible for an adversary to generate a valid signature on a new message that has not been previously signed by the legitimate signer. Formally, a signature scheme is unforgeable if no efficient probabilistic polynomial-time (PPT) adversary, given the public verification key and access to signatures on messages of its choice, can produce a valid message-signature pair (m, σ) where m is distinct from all previously queried messages and σ verifies correctly under the public key.^[9] This property is typically formalized using a game-based security model, where a challenger generates the key pair for the signature scheme, provides the verification key to the adversary, and responds to the adversary's queries for signatures on adaptively chosen messages. The adversary interacts with the challenger in phases, potentially querying signatures multiple times, and at the end outputs a candidate forgery (m*, σ*). The adversary succeeds if (m*, σ*) is valid according to the verification algorithm and m* was not among the queried messages. The scheme is secure if the probability of the adversary winning this game is negligible in the security parameter.^[9] Security models for unforgeability vary in strength based on the adversary's capabilities, ranging from weak to strong notions. The weakest level is existential unforgeability under key-only attack (EUF-KOA), where the adversary has access only to the public key and must forge a signature on any message without seeing any signatures. Stronger models include existential unforgeability under known-message attack (EUF-KMA), where the adversary observes a fixed set of message-signature pairs, and existential unforgeability under chosen-message attack (EUF-CMA), the standard strong notion where the adversary can adaptively choose messages to sign via queries to the challenger. EUF-CMA captures the most realistic threat scenario and is the minimal requirement for practical digital signature schemes.^[9] The adversary's advantage in breaking unforgeability is quantified as the probability of success in the security game minus the trivial success probability (typically 0), taken over the randomness of the keys, queries, and adversary's coins; a scheme is unforgeable if this advantage is negligible for all PPT adversaries. Unforgeability directly implies message integrity for signed data, as any alteration would invalidate the signature, but it does not provide confidentiality, as signatures and messages are publicly verifiable.^[9]

Adversary Capabilities

Key-Only Attack

In the key-only attack model for digital signatures, the adversary possesses only the signer's public key and description of the verification algorithm, with no access to any valid signatures or messages. The adversary aims to produce a valid message-signature pair (m, σ) that verifies correctly under the public key, typically targeting an existential forgery where m can be any message, even a randomly generated one. This model represents the minimal adversarial resources, making it the weakest security scenario considered in formal definitions of signature unforgeability.^[2] A digital signature scheme is said to satisfy key-only existential unforgeability if no probabilistic polynomial-time adversary can succeed in forging such a pair with non-negligible probability. Security in this model relies fundamentally on the computational hardness of the underlying mathematical problem assumed to be intractable, such as the difficulty of integer factorization for RSA-based schemes or the existence of claw-free permutation pairs in more general constructions. Without these hardness assumptions, even basic trapdoor-based signatures become trivially forgeable, as an adversary can generate a random candidate signature σ and derive a corresponding m via the public verification process to ensure validity.^[2] For instance, the plain RSA signature scheme—where signing computes σ = m^d mod N and verification checks m = σ^e mod N—is existentially forgeable under a key-only attack. An adversary can select a random σ in {0,1}^n, compute m = σ^e mod N, and obtain a valid pair (m, σ), though m lacks semantic meaning. This vulnerability highlights why modern RSA signatures incorporate padding and hashing to elevate security beyond key-only resistance.^[10]

Known-Message Attack

In a known-message attack on a digital signature scheme, the adversary gains access to a fixed set of valid message-signature pairs (m_i, \sigma_i) that were selected and signed by the legitimate signer prior to the attack, without any input from the adversary into the message choices.^[2] The adversary observes these pairs alongside the signer's public key but cannot interact further with the signing process. The primary objective of the adversary is to forge a new valid signature pair (m, \sigma) such that m is distinct from all observed m_i and \sigma verifies correctly using the public key. This model assumes the public key and basic signature verification format are public, allowing the adversary to test potential forgeries.^[2] Known-message attacks are particularly insightful for evaluating schemes susceptible to message dependencies, such as algebraic signature constructions where linear relations among multiple observed signatures enable the computation of a forgery for an unseen message via linear algebra techniques.^[11] For instance, in certain algebraic schemes over groups of known order, an adversary can combine at least n+1 signatures—where n relates to the scheme's algebraic dimension—to derive a linear dependency yielding a valid signature on a fresh message.^[11] The security property addressing this threat is known-message unforgeability, which requires that no efficient adversary can produce such a forgery with non-negligible probability; this notion occupies an intermediate position in attack strength, exceeding key-only scenarios but falling short of chosen-message models where the adversary selects messages. In practice, contemporary digital signature schemes are predominantly assessed for security under the more stringent chosen-message paradigm, as it aligns better with potential real-world threats, rendering standalone known-message evaluations uncommon.^[12]

Chosen-Message Attack

In a chosen-message attack, an adversary gains access to a signing oracle that produces valid signatures on messages of the adversary's choosing, allowing the attacker to adaptively query the oracle with selected messages m_i and receive corresponding signatures \sigma_i, before attempting to forge a new signature \sigma for a fresh message m that was not previously queried.^[9] This model assumes the adversary operates within probabilistic polynomial time and aims to demonstrate the unforgeability of a signature scheme by showing that no such forgery occurs with more than negligible probability.^[9] The attack can be adaptive, where each query may depend on previous oracle responses, or non-adaptive, where all messages are chosen in advance without relying on intermediate signatures; adaptive variants represent the more powerful and realistic threat, as they model dynamic adversary behavior.^[9] Security against chosen-message attacks is formalized as existential unforgeability under chosen-message attacks (EUF-CMA), the standard notion requiring that the adversary's advantage in producing a valid forgery—defined as the probability of success minus the negligible base probability—is negligible for any polynomial-time attacker, assuming underlying computational hardness like the difficulty of factoring or discrete logarithms.^[9] Signature schemes such as RSA with the Probabilistic Signature Scheme (PSS) achieve EUF-CMA security in the random oracle model, where the signing process incorporates randomized padding to prevent predictable patterns exploitable by the adversary.^[13] Similarly, Edwards-curve Digital Signature Algorithm (EdDSA) variants, like Ed25519, are proven EUF-CMA secure under the elliptic curve discrete logarithm assumption, providing efficient protection against such attacks through deterministic nonce generation and hash-based message encoding.^[14] The advantage is quantified relative to the scheme's parameters, such as key size, ensuring that forgery probability remains bounded by $2^{-t} for security level t bits against computationally bounded adversaries.^[13] This attack model captures practical scenarios like insider threats, where an attacker has temporary access to a legitimate signing device, or compromised oracles in distributed systems, emphasizing the need for schemes resilient to interactive querying without revealing the private key.^[9]

Forgery Types

Total Break

A total break in a digital signature scheme occurs when an adversary recovers the signer's private key, enabling the generation of valid signatures for any arbitrary message without further restrictions. This represents the most complete form of compromise, as the private key is the core secret used in the signing process to produce verifiable outputs.^[1] Such a break can be achieved under various adversary capabilities, including key-only, known-message, or chosen-message attacks, though it most commonly arises from vulnerabilities in the key generation algorithm itself, such as weaknesses in randomness or factorization. In practice, the adversary exploits mathematical properties of the scheme to invert the public key and derive the private components.^[1] From a security perspective, a total break renders the entire digital signature scheme insecure, as the adversary gains equivalent signing authority to the legitimate signer, undermining the scheme's core guarantees of authenticity and non-repudiation. Security models for digital signatures explicitly aim to prevent this outcome, deeming any scheme susceptible to a total break as fundamentally broken.^[1] Historically, early digital signature schemes like textbook RSA were vulnerable to total breaks through factoring attacks on the modulus, allowing recovery of the private exponent and thus unlimited forgery.^[15] These vulnerabilities highlighted the need for probabilistic signing and padding to elevate security beyond basic trapdoor functions.^[1] Detection of a total break typically becomes evident only after private key exposure, which facilitates forgeries on both past messages (by recomputing signatures) and future ones, often leading to widespread compromise in deployed systems.

Universal Forgery

Universal forgery refers to an attack in which an adversary obtains the ability to generate valid signatures for any arbitrary message of their choice, effectively producing an algorithm equivalent to the signer's private signing procedure without recovering the private key itself. This capability allows the adversary to completely impersonate the signer across all possible messages, simulating legitimate signing operations. The concept was formalized in early cryptographic literature as one of the strongest forms of forgery short of a total break.^[1] In terms of security hierarchies, universal forgery represents a failure of universal unforgeability, which provides stronger protection than resistance to selective or existential forgery but is encompassed by preventing a total break, where the private key is explicitly recovered. Such an attack implies a profound breakdown in the scheme's unforgeability guarantees, enabling indefinite impersonation without key extraction.^[1] Universal forgery typically arises from exploiting inherent structural vulnerabilities in the signature scheme, such as deterministic signature generation that is predictable or invertible based on public information and observed signatures. For instance, if the signing process lacks sufficient randomization or relies on easily reversible computations, an adversary may derive a universal signing method from a limited set of valid signature examples.^[1] In contemporary digital signature schemes, universal forgery is exceedingly rare, as modern constructions are designed to achieve existential unforgeability under adaptive chosen-message attacks (EUF-CMA), a security notion that precludes even weaker forms of forgery and thus inherently resists universal attacks; achieving such a forgery generally demands interaction with a chosen-message oracle under a computationally bounded adversary model.

Selective Forgery

Selective forgery refers to an attack in which an adversary preselects a specific message and subsequently generates a valid signature for that exact message without access to the signer's private key or a prior signature on it from the legitimate signer. This concept was introduced in the foundational paper by Goldwasser, Micali, and Rivest, defining it as the ability to "forge a signature for a particular message chosen a priori by the enemy." Unlike broader forgery types, the adversary commits to the target message before initiating the attack, limiting the scope but enabling precise targeting.^[9] The adversary typically operates within a chosen-message attack model, where it can query a signing oracle for valid signatures on arbitrary messages of its choice (excluding the preselected target) after receiving the public key. This access allows the adversary to gather information about the signature scheme's behavior while attempting to forge the signature for the fixed message. Security against selective forgery, termed selective unforgeability, requires that no efficient adversary succeeds with more than negligible probability. This notion provides protection against targeted forgeries but is weaker than existential unforgeability under chosen-message attacks (EUF-CMA), the standard modern security model, which resists forgery of any new message by a more powerful adaptive adversary and thus subsumes resistance to selective forgery. Selective unforgeability is also weaker than universal unforgeability, where the adversary could forge signatures for any message without precommitment. In practice, selective forgery poses significant risks in targeted scenarios, such as falsifying a specific legal contract, financial transaction, or software update where the adversary has prior knowledge of the desired message content. For instance, an attacker might aim to forge a digital signature on a particular policy document to deceive verifiers into accepting it as authentic. The formal experiment for selective unforgeability proceeds as follows: the adversary first outputs the target message m; the challenger generates the key pair and provides the public key; the adversary then queries the signing oracle on messages other than m and finally outputs a purported signature σ for m; success occurs if σ verifies correctly under the public key without prior oracle response for m. Schemes provably secure under stronger models like EUF-CMA, such as certain lattice-based constructions, ensure robustness against such precommitted attacks.

Existential Forgery

Existential forgery, also referred to as existential unforgeability (EUF), is achieved when an adversary generates a valid message-signature pair (m, \sigma) such that m has not been previously signed by the legitimate signer, irrespective of whether m holds any practical significance. This represents the weakest form of forgery that still violates the core security property of a digital signature scheme, as the adversary need only produce one such pair after interacting with the system.^[12] The prevailing security model for assessing resistance to existential forgery is existential unforgeability under chosen-message attacks (EUF-CMA), where a probabilistic polynomial-time adversary receives a public verification key, adaptively queries a signing oracle for signatures on chosen messages, and must ultimately output a forgery (m^*, \sigma^*) that verifies correctly under the public key, with m^* distinct from all queried messages. A scheme is considered EUF-CMA-secure if no such adversary succeeds with more than negligible probability in the security parameter.^[12] This model captures the adversary's ability to learn from legitimate signatures while requiring the forgery to demonstrate new knowledge beyond mere verification.^[16] Such forgeries are concerning because they compromise non-repudiation, a fundamental goal of digital signatures, allowing the adversary to attribute a forged signature to the signer who cannot plausibly deny its origin despite the message's potential meaninglessness.^[17] Common vulnerabilities enabling existential forgery include the use of weak hash functions susceptible to collisions, as demonstrated in attacks on MD5 where colliding messages produced identical hashes, permitting forged signatures on arbitrary content in systems like X.509 certificates.^[18] Similarly, padding oracle attacks on RSA-PKCS#1 v1.5 signatures, such as Bleichenbacher's 2006 method exploiting verification feedback, enable adversaries to construct valid existential forgeries for low public exponents without the private key.^[19] Existential unforgeability under EUF-CMA serves as the foundational security notion for digital signatures, forming the basis for more stringent variants like strong existential unforgeability that address additional forgery scenarios.^[16] Modern schemes, including the Boneh-Lynn-Shacham (BLS) signatures based on bilinear pairings, achieve EUF-CMA security in the random oracle model, assuming the computational Diffie-Hellman problem's hardness in asymmetric bilinear groups.^[20]

Strong Existential Forgery

Strong existential forgery, also referred to as a failure of strong existential unforgeability (sEUF or SUF), occurs when an adversary generates a valid signature pair (m, σ') where m is a message previously signed by the legitimate signer with a different signature σ (i.e., σ' ≠ σ), in the context of an adaptive chosen-message attack.^[21] This type of forgery targets the ability to produce any new valid signature, including modifications to existing ones, making it a stricter notion than basic existential unforgeability under chosen-message attacks (EUF-CMA), which only prohibits new messages but allows alternative signatures on old ones.^[16] The sEUF-CMA security model formalizes this through a game where the adversary receives the public key, adaptively queries the signing oracle for up to q messages to obtain signatures, and then outputs a pair (m, σ); the adversary succeeds if (m, σ) verifies correctly but does not match any previously queried pair exactly (neither the message nor the signature).^[22] This model addresses vulnerabilities like signature malleability, where an attacker can alter components of a signature—such as re-randomizing or inverting values—while preserving validity for the same message, thereby breaking applications that rely on signature uniqueness.^[17] Achieving sEUF-CMA is essential for protecting against modification attacks in advanced cryptographic constructions, including chosen-ciphertext secure encryption systems and group or aggregate signature schemes, where altered signatures could enable unauthorized actions or privacy breaches.^[21] Without this stronger security, even EUF-CMA-secure schemes may fail in multi-party settings, as a single malleable signature could compromise the integrity of combined proofs or verifications.^[23] Vulnerabilities to strong existential forgery often arise in probabilistic schemes lacking sufficient randomization to prevent malleability; for instance, ECDSA allows an attacker to forge a new signature on a signed message by negating the s-component (replacing s with the curve order minus s), yielding a distinct but valid pair since verification equations hold symmetrically.^[24] Similarly, deterministic signatures without randomization, such as early full-domain hash RSA variants, can expose systems to related forgeries by enabling predictable modifications or collisions that mimic strong attacks.^[17] Most modern digital signature schemes developed after 2000, including RSA-PSS in the random oracle model, are explicitly designed to meet sEUF-CMA security, incorporating randomized padding and masking to ensure that valid signatures are unique and non-malleable under the RSA assumption.^[25] These constructions provide the robust protection required for contemporary applications like secure messaging and blockchain transactions.^[23]

Examples and Implications

Existential Forgery Example

A classic example of existential forgery arises in the textbook RSA signature scheme, where the signature on a message m is computed as \sigma = m^d \mod n, with d the private exponent and n = pq the modulus from primes p and q. This scheme exhibits a multiplicative homomorphic property: if \sigma_1 = m_1^d \mod n is a valid signature on message m_1, and \sigma_2 = m_2^d \mod n is a valid signature on message m_2, then \sigma_1 \cdot \sigma_2 \mod n = (m_1 \cdot m_2)^d \mod n, providing a valid signature \sigma = \sigma_1 \cdot \sigma_2 \mod n on the new message m = m_1 \cdot m_2 \mod n. Under a known-message attack, an adversary obtains two valid signed messages (\sigma_1, m_1) and (\sigma_2, m_2) from the public key. The adversary then computes \sigma = \sigma_1 \cdot \sigma_2 \mod n and m = m_1 \cdot m_2 \mod n. If m has not been previously signed (ensuring it is a novel message), this yields a valid but typically meaningless message-signature pair (\sigma, m), constituting an existential forgery. This attack succeeds because the scheme lacks padding or hashing to disrupt the multiplicativity, and it requires only publicly available signatures without needing the private key. To mitigate such existential forgeries, modern RSA implementations employ probabilistic padding schemes like RSA-PSS, which incorporate random salts and hashing to eliminate the multiplicative property and provide provable security against existential forgery under the RSA assumption.^[13]

Real-World Vulnerabilities

In 2012, the Flame malware exploited a chosen-prefix collision in the MD5 hash function to forge valid Microsoft code-signing certificates, enabling the distribution of malicious payloads disguised as legitimate Windows updates.^[26] Attackers constructed two different X.509 certificates—one legitimate from the Microsoft Terminal Services chain and one forged—sharing the same MD5-based signature, which bypassed validation on Windows systems including Vista and later versions.^[27] This attack, building on earlier MD5 collision techniques from the late 2000s, represented an existential forgery by producing a valid signature for an unauthorized certificate without access to the private key. In 2010, hackers from the fail0verflow group compromised the Sony PlayStation 3's security by exploiting nonce reuse in the ECDSA signature scheme used for firmware validation.^[28] Sony's implementation generated signatures with the same ephemeral nonce value across multiple firmware updates, allowing attackers to apply a lattice-based reduction attack to recover the elliptic curve private key from as few as two signatures.^[29] This total break enabled the creation of arbitrary signed firmware, leading to widespread piracy and custom software installation on millions of consoles.^[30] In 2006, Daniel Bleichenbacher demonstrated flaws in implementations of the PKCS#1 v1.5 signature verification scheme that could be exploited for forgery attacks, particularly with low public exponents such as 3. These flaws allowed verifiers to accept signatures that did not strictly conform to the padding format, enabling an attacker to construct and validate signatures for arbitrary messages without the private key.^[31] This vulnerability affected various cryptographic libraries, compromising the integrity of signed data and leading to patches for stricter padding validation in standards like those from RSA Laboratories. The 2017 SHAttered attack by researchers from Google and Centrum Wiskunde & Informatica produced the first practical collision for the full SHA-1 hash function, with two distinct PDF files yielding identical 160-bit digests after approximately 2^63 SHA-1 operations.^[32] In digital signatures relying on SHA-1, such as those in X.509 certificates, this enables selective forgery by signing a benign document and later replacing it with a malicious one sharing the same hash.^[32] For version control systems like Git, which use SHA-1 for commit integrity, the attack allows creating two repositories with identical commit hashes but differing contents, potentially inserting backdoors into trusted code histories; Git responded by introducing collision detection in version 2.13.^[32] These incidents underscore the critical need for randomized nonces in schemes like ECDSA to prevent key recovery via lattice attacks, as seen in the PS3 breach.^[29] Secure hash functions beyond MD5 and SHA-1, such as SHA-256, are essential to resist collision-based forgeries in certificate chains and signatures.^[27] Ongoing cryptanalysis and timely migration to robust standards remain vital to mitigate such practical vulnerabilities.^[32]

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