Missing Signer Checks in RSA Signatures

• Missing Signer Checks are flaws in signature verification where not all bits, especially high-order bits, are validated.
• The Diebold Signature Variant exemplifies how partial verification, using a fixed exponent and restricted bit checks, enables adversarial signature forgery.
• Mitigation strategies involve full bitwise checks, adopting robust standardized padding schemes, and using safe public exponents to secure RSA signatures.

A missing signer check (MSC) refers to a class of implementation flaws in cryptographic signature verification procedures where not all critical components of the recovered message, particularly those embedded in the high-order bits, are examined and verified. In the context of RSA digital signatures, missing signer checks can permit adversarial forgery of signature values even when commonly used cryptographic transformations and hash functions are present. The Diebold Signature Variant recently examined in deployed voting systems exemplifies the security risk introduced by inadequate verification of the high-order bits in signature padding (Gardner et al., 2 Mar 2024).

1. Standard RSA Signatures and the Role of Padding

The conventional RSA signature scheme relies on a transformation $T$ that encodes the message $m$ into a padded value $M$, typically using hash functions and structured padding. The signer computes the signature as $\sigma \equiv M^d \pmod{n}$, where $d$ is the private RSA exponent, and $n$ is the public RSA modulus. Verification proceeds by computing $M' = T(m')$ from the provided message and $M'' = \sigma^e \pmod{n}$, with $e$ the public exponent; the signature is valid if $M' = M''$. The precise design and verification of padding within $T$ is fundamental to precluding structural forgeries. Absent comprehensive padding integrity checks, the mathematical properties of RSA are susceptible to exploitation.

2. The Diebold Signature Variant: Implementation Characteristics

Diebold voting machine implementations use a "textbook" RSA signature scheme with distinct departures from best practice:

• The public exponent is fixed at $e = 3$ to optimize computational efficiency.
• The conversion function $T$ applies SHA-1 hashing and encodes redundancy, but the verifier checks only the $b$ least significant bits (LSBs) of the recovered message—often corresponding to the digest—while deliberately ignoring the high-order bits.

Specifically, the verification step becomes: $[M']_{0}^{b-1} = [M'']_{0}^{b-1}$, i.e., it solely compares the LSBs of the canonical encoded message and the recovered signature value. Secure RSA schemes require comparison against the entirety of $M$, not only a subset. Neglecting the high-order bits constitutes a missing signer check, as the bulk of the signature’s structure remains unvalidated.

3. Mathematical Description and Exploitation of MSC

Exploitation arises from incomplete bitwise comparison. An adversary seeks $\sigma$ such that:

$\sigma^3 \equiv M + 2^b z \pmod{n}$

for some $z$ with $2^b z < n$. Since only the bottom $b$ bits are verified, forging a valid signature is reduced to engineering the high-order bits to arbitrary values without detection.

Attack Construction

• Odd $M$: Let $M$ be odd (in $\mathbb{Z}_{2^b}^*$). Compute integer $r$ with $3r \equiv 1 \pmod{2^{b-1}}$. Construct $σ \equiv M^r \cdot 2^b$ mod $2^b$, ensuring $\sigma^3 \equiv M \pmod{2^b}$.
• Even $M$: For even $M$, select $c$ so $(2^b c)^3 > n$ and apply a modified construction with $r$ as above for $M + n$, yielding $\sigma = 2^b c + ((M+n)^r \cdot 2^b)$.

These constructions yield forgeable signatures for arbitrary $M$. The critical condition is:

$\sigma^3 \equiv M \pmod{2^b}$

which is satisfied through the arithmetic manipulations above. Expanding the cube, the attack ensures the high bits encode undetected data while verification proceeds on the checked lower bits alone. The attack is mathematically elementary, and the time required is negligible.

4. Security Implications and Systemic Risks

The absence of complete signer checks undermines the foundational integrity of RSA-based signature systems. Adversaries may forge signatures for any $m$, bypassing cryptographic security through manipulation of unchecked high-order bits. For systems requiring exceptional guarantees, such as electronic voting machines, this vulnerability is critical; forgeries are undetectable by the deployment’s verification logic.

A plausible implication is that any deployment paralleling the Diebold variant—using partial bit checks and low public exponent ($e=3$)—is exposed to trivial forgeries. The security model collapses even with robust cryptographic hash functions present, as the unchecked bits may encode arbitrary structure.

5. Mitigation Strategies and Implementation Recommendations

Eliminating missing signer checks requires a multidimensional approach:

• Full Bitwise Verification: Ensure every bit of the recovered message, including all redundancy and hash portions, is compared during verification. No segment may be omitted.
• Adoption of Standardized Padding: Employ robust, standardized padding like PKCS#1 v1.5 or RSA-PSS, which introduce randomness and additional structure making mathematical forging infeasible.
• Selection of Safe Public Exponents: Prefer larger exponents (e.g., $e=65537$) to mitigate vulnerabilities intrinsic to small values such as $e=3$, especially in padding edge cases.
• Structured Redundancy and Hash Integration: Transformation functions must securely mix cryptographic hashes and redundancy bits, such that tampering is detectable in all portions of the encoded message.
• Rigorous Code Review: Systematic review, including constant-time operations and exhaustive check of all verification logic, is necessary to preclude both side-channel leakage and logic errors related to incomplete checks.

6. Context, Common Misconceptions, and Conclusions

A common misconception is that the presence of a secure hash or structured redundancy alone suffices to guarantee signature security. The Diebold case makes clear that unless all bits of the padded message $M$ are compared in the verification process, the system remains trivially vulnerable. The mathematical simplicity of the attack underscores the ease with which neglecting high-order bits can be leveraged in practice. Implementations must avoid reliance on partial-bit checks regardless of performance gains attributed to low exponents or reduced computation.

In conclusion, missing signer checks constitute a fundamental security flaw in signature verification systems, especially when only lower-order bits are scrutinized. Complete verification, careful selection of cryptographic parameters, adoption of standardized padding schemes, and thorough implementation review are requisite to maintaining signature integrity. The Diebold Signature Variant vividly demonstrates the consequences of such lapses in systems deployed for critical infrastructure.