A field prime is the prime modulus that defines a prime field for elliptic-curve computation. For an ordinary curve, coordinates and the curve equation are evaluated modulo p: y² = x³ + a₄x + a₆ mod p. The choice of p therefore fixes the base field in which curve points and coefficients live. section 6.2
Equation 6.1
\[\label{fgc1} E: y^2=x^3+a_4 x + a_6 \text{ mod }p\]The curve equation makes the operational role of p explicit: every equality is taken modulo the field prime. section 6.2
The field prime is closely constrained by curve size. If #E is the number of curve points, then #E = p + 1 − t, where t is the trace of Frobenius and |t| ≤ 2√p. A negative odd trace can make the cardinality prime and slightly larger than the field while retaining p’s bit size. section 6.2
When selecting parameters, p participates in several linked checks:
- Choose a large prime field and curve coefficients; candidate curves are then tested by their cardinality and cofactor. section 6.2
- For extension-field curves, use p to determine a compatible irreducible trinomial; together they determine the elliptic-curve order. section 13.3
- Compare p with the large subgroup prime r through ρ = log₂(p)/log₂(r), a size ratio relevant to pairing-friendly parameters. section 14.1
- Fix p as public system data before key generation, alongside curve, extension-field, cardinality, cofactor, and base-point parameters. section 18.3
A tiny field with p = 43
A toy pairing curve uses field prime p = 43, curve cardinality 55, and embedding degree 2. Here p also determines which irreducible trinomial works for the extension field; the base curve’s large prime factor is 11, making the relationships small enough to inspect directly. section 13.3
In pairing-based multi-signatures, the field prime is the lowest parameter level and is larger than the torsion value by ρ. It defines irreducible-polynomial coefficients, while G₁ and G₂ must share curve coefficients for pairings to work. Keeping these values in one public parameter structure makes every curve calculation use the same field definition. section 18.3
