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In EUROCRYPT 2009 Galbraith, Lin and Scott constructed an efficiently computable endomorphism for faster elliptic curve cryptosystem for a large family of elliptic curves defined over finite fields of large characteristic.In this paper we extend the construction to any genus 2 hyperelliptic curve defined over a finite field of even characteristic.We propose an efficient algorithm to generate a random genus 2 hyperelliptic curve and its quadratic twist equipped with a fast endomorphism on the Jacobian.The analysis of the scalar multiplication amount is also given.