For an elliptic curve over a finite field with prime field size p, the trace of Frobenius is the integer t that determines the curve’s point count: #E = p + 1 − t. This turns an abstract field-counting problem into a parameter used to evaluate cryptographic curves. section 6.2 section 6.5
Use t in the following operational checks:
- Bound the possible trace with |t| < 2√p, as required by Hasse’s theorem. section 6.2 section 6.5
- Compute the cardinality #E from p and t, then inspect its factorization for a large prime factor and a small cofactor. section 6.2 section 6.5
- For an extension field p^k, use t as t1 in the recurrence for tk before computing #Ek = p^k + 1 − tk. section 13.1
Equation 6.2
\[\label{fgc2} \#E=p+1-t\]Equation 6.2 provides the primary translation from the field prime and trace to the elliptic-curve cardinality. section 6.2
Equation 6.3
\[\label{fgc3} |t| \leq 2 \sqrt{p}\]Equation 6.3 supplies the admissible range for t and constrains candidate point counts. section 6.2
Curve-search use
A curve search computes #E for each candidate, removes powers of 2 and selected small factors, and accepts curves whose cardinality has a suitable large prime factor. A negative odd t can produce a cardinality larger than p while retaining p’s bit size. section 6.2 section 6.5
