Isogenies occur throughout the theory of elliptic curves.Recently,the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols.Given two elliptic curves E1,E...Isogenies occur throughout the theory of elliptic curves.Recently,the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols.Given two elliptic curves E1,E2 defined over a finite field k with the same trace,there is a nonconstant isogeny b from E2 to E1 defined over k.This study gives out the index of Homk(E1,E2)b as a nonzero left ideal in Endk(E2)and figures out the correspondence between isogenies and kernel ideals.In addition,some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided.展开更多
Verifiable delay functions(VDFs)and delay encryptions(DEs)are two important primitives in decentralized systems,while existing constructions are mainly based on time-lock puzzles.A disparate framework has been establi...Verifiable delay functions(VDFs)and delay encryptions(DEs)are two important primitives in decentralized systems,while existing constructions are mainly based on time-lock puzzles.A disparate framework has been established by applying isogenies and pairings on elliptic curves.Following this line,we first employ Richelot isogenies and non-degenerate pairings from hyperelliptic curves for a new verifiable delay function,such that no auxiliary proof and interaction are needed for the verification.Then,we demonstrate that our scheme satisfies all security requirements,in particular,our VDF can resist several attacks,including the latest attacks for SIDH.Besides,resorting to the same techniques,a secure delay encryption from hyperelliptic curves is constructed by modifying Boneh and Frankiln's IBE scheme,which shares the identical setup with our VDF scheme.As far as we know,these schemes are the first cryptographic applications from high-genus isogenies apart from basic protocols,i.e.,hash functions and key exchange protocols.展开更多
An elliptic curve is a pair (E,O), where E is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2+a1xy+a3y=x2+a2x2+...An elliptic curve is a pair (E,O), where E is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2+a1xy+a3y=x2+a2x2+a4x+a6.Let Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1, 2, 3, 4, 6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E defined over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsWe say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism φ: E → E' such that φ(O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E'(Q)tors is in the form Z/mZ where m = 9, 10, 12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit model of all their isogenous curves with cyclic kernels consisting of rational points.展开更多
基金National Key Research and Development Project No.2018YFA0704705.
文摘Isogenies occur throughout the theory of elliptic curves.Recently,the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols.Given two elliptic curves E1,E2 defined over a finite field k with the same trace,there is a nonconstant isogeny b from E2 to E1 defined over k.This study gives out the index of Homk(E1,E2)b as a nonzero left ideal in Endk(E2)and figures out the correspondence between isogenies and kernel ideals.In addition,some results about the non-trivial minimal degree of isogenies between two elliptic curves are also provided.
基金supported by the National Natural Science Foundation of China(No.62272491)the Guangdong Major Project of Basic and Applied Basic Research(2019B030302008)the National R&D Key Program of China under Grant(2022YFB2701500).
文摘Verifiable delay functions(VDFs)and delay encryptions(DEs)are two important primitives in decentralized systems,while existing constructions are mainly based on time-lock puzzles.A disparate framework has been established by applying isogenies and pairings on elliptic curves.Following this line,we first employ Richelot isogenies and non-degenerate pairings from hyperelliptic curves for a new verifiable delay function,such that no auxiliary proof and interaction are needed for the verification.Then,we demonstrate that our scheme satisfies all security requirements,in particular,our VDF can resist several attacks,including the latest attacks for SIDH.Besides,resorting to the same techniques,a secure delay encryption from hyperelliptic curves is constructed by modifying Boneh and Frankiln's IBE scheme,which shares the identical setup with our VDF scheme.As far as we know,these schemes are the first cryptographic applications from high-genus isogenies apart from basic protocols,i.e.,hash functions and key exchange protocols.
基金This research was supported by the TMR programme of the European Community under contract ERBFMBICT960848.
文摘An elliptic curve is a pair (E,O), where E is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2+a1xy+a3y=x2+a2x2+a4x+a6.Let Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1, 2, 3, 4, 6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E defined over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsWe say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism φ: E → E' such that φ(O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E'(Q)tors is in the form Z/mZ where m = 9, 10, 12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit model of all their isogenous curves with cyclic kernels consisting of rational points.
