Choosing Between $\mathbb{Z}_p$ and $\mathbb{Z}_p^*$ in Cryptographic Schemes

In pairing-based cryptographic protocols, the choice of algebraic structures for randomness and messages—specifically whether to use the full field $\mathbb{Z}_p$ (integers modulo a prime $p$, i.e., ${0, 1, \ldots, p-1}$) or its multiplicative subgroup $\mathbb{Z}_p^*$ (non-zero elements, i.e., ${1, \ldots, p-1}$)—has subtle but critical implications for security and correctness. Below, we analyze their tradeoffs.

Why $\mathbb{Z}_p^*$ is Safer

Using $\mathbb{Z}_p^*$ (non-zero exponents) is often preferred for $\textit{blinding factors}$ and $\textit{randomness}$ due to:

• Avoiding Degenerate Cases: A zero exponent (e.g., $r = 0$) can produce trivial group elements:

  $g^0 = 1 \in \mathbb{G}_1, \quad 0 \cdot P = \mathcal{O} \in \text{EC groups}$

  which may leak secrets or break protocol unlinkability. For instance, in PS-style commitments $\text{cm} = g^t \prod Y_i^{m_i}$, a $t = 0$ would expose $\prod Y_i^{m_i}$.

• Security Proof Compatibility: Many zero-knowledge proofs (e.g., Schnorr-type responses $z = \tilde{x} + c \cdot x$) implicitly assume $x \neq 0$ to avoid division-by-zero errors in reductions. This is particularly critical because such errors prevent the simulator from properly generating valid transcripts in security proofs, potentially invalidating the entire security argument.

• Invertibility Guarantees: Non-zero elements in $\mathbb{Z}_p^*$ are invertible, simplifying operations like computing $r^{-1} \bmod p$ in signature schemes.

Practical Implementation of $\mathbb{Z}_p^*$ Sampling

When implementing schemes requiring sampling from $\mathbb{Z}_p^*$, developers typically use one of two approaches:

• Rejection Sampling: Generate random elements from $\mathbb{Z}_p$ and retry if zero is obtained. This is probabilistically efficient given the negligible probability $(1/p)$ of sampling zero.

• Offset Method: Generate a random element $r$ from $\mathbb{Z}_p$ and use $r + 1 \bmod p$. This guarantees a non-zero result but may introduce slight biases that should be analyzed in security-critical applications.

Why Some Schemes Use $\mathbb{Z}_p$

Despite the risks, protocols like the Pointcheval-Sanders (PS) scheme often sample from $\mathbb{Z}_p$ because:

• Pairing Algebraic Requirements: Pairing equations (e.g., $e(g^a, h^b) = e(g, h)^{ab}$) require exponents to span the full field $\mathbb{Z}_p$ to preserve algebraic relationships. Polynomial evaluations, which are fundamental to PS credentials and many other pairing-based schemes, are defined over the entire field $\mathbb{Z}_p$. Restricting to $\mathbb{Z}_p^*$ would break these polynomial properties and their crucial role in constructing witnesses and proofs.

• Negligible Failure Probability: For large $p$, the probability of sampling $r = 0$ is $1/p$, which is considered cryptographically negligible. Schemes often accept this risk to simplify implementations.

• Message Flexibility: Messages (e.g., attributes $m_j$) may need to include 0 as a valid value. For example, a credential might encode $m_j = 0$ to represent “no value” for an optional field.

Practical Recommendations

• For Blinding Factors: Always use $\mathbb{Z}_p^*$ to eliminate edge cases and align with security assumptions in proofs. Implement proper sampling methods as discussed above.

• For Messages/Attributes: Use $\mathbb{Z}_p$ if 0 is a valid semantic value (e.g., default states). Consider adding range proofs or other validation when zero values might impact security.

• In Pairing-Based Schemes: Follow the scheme’s specification—PS uses $\mathbb{Z}_p$ for exponents to maintain pairing correctness, but ensure other safeguards (e.g., range proofs) mitigate risks.

Summary

While $\mathbb{Z}_p^*$ is theoretically safer for randomness, practical schemes like PS often use $\mathbb{Z}_p$ for compatibility with pairing algebra. Developers must weigh algebraic requirements against edge-case risks when choosing structures, and implement appropriate sampling and validation mechanisms based on their specific security requirements.