The present article explains how to generalize the hash function SwiftEC (in an elementary quasi-unified way) to any elliptic curve E over any finite field \(\mathbb {F}_{\!q}\)  of characteristic \(> 3\) . The new result apparently brings the theory of hash functions onto elliptic curves to its logical conclusion. To be more precise, this article provides compact formulas that define a hash function \(\{0,1\}^* \rightarrow E(\mathbb {F}_{\!q})\) (deterministic and indifferentible from a random oracle) with the same working principle as SwiftEC. In particular, both of them equally compute only one square root in \(\mathbb {F}_{\!q}\) (in addition to two cheap Legendre symbols). However, the new hash function is valid with much more liberal conditions than SwiftEC, namely when \(3 \mid q-1\) . Since in the opposite case \(3 \mid q-2\) there are already indifferentiable constant-time hash functions to E with the cost of one root in \(\mathbb {F}_{\!q}\) , this case is not processed in the article. If desired, its approach nonetheless allows to easily do that mutatis mutandis.