ANTS / 5 / 20 Katsuyuki Takashima Mitsubishi Electric
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1 Efficiently Computable Distortion Maps for Supersingular Curves ANTS / 5 / 20 Katsuyuki Takashima Mitsubishi Electric 1
2 Our results Galbraith-Pujolas-Ritzenthaler-Smith [GPRS] gave unsolved problems on distortion maps for special supersingular curves. We solve them based on explicit construction of a basis of - vector space consisting of eigenvectors of the Frobenius endomorphism ( -eigenvector basis ) a -basis of - vector space We explicitly determine the discrete logarithms of the Weil pairing to one base where We obtain an efficiently constructible (semi-)symplectic -eigenvector basis. 2
3 Agenda Target supersingular curves Distortion maps Computational problems on distortion maps Results and unsolved problem given in [GPRS] Our approach Our results on Our results on Conclusions 3
4 Target supersingular curves : proj., nonsingular, geom. irred. curve. Def. Def. :supersingular :supersingular isogeneous to a product of supersingular elliptic curves prime, prime s.t. -power Frobenius endomorphism action of a primitive -th root of unity induced by on on -power Frobenius endomorphism Action of an extra-special 2-group of order 32 [vdgvdv]. 4
5 Distortion maps : prime s.t. s.t. : nondegenerate bilinear pairing from to Definition [GPRS] For a pair is called a distortion map. Theorem 1 [GPRS] Let be a target supersingular curve. endo. of endo. defined over - vector space In particular, for every pair there exists a distortion map 5
6 Computational problems on distortion maps Theorem 1 doesn t assure the existence of an efficiently computable distortion map. Computational problem 1 For every pair a distortion map can we efficiently compute? Cf. [GR] for the case of supersingular elliptic curves. Computational problem 2 Is there a basis of s.t. are efficiently computable? Basis in problem 2 an answer (efficient algorithm) to problem 1. 6
7 Results and unsolved problem given in [GPRS] [GPRS] gave bases of For For -vector space for target curves. is a and are -basis. -bases. Unsolved problem given in [GPRS] Are the above and -bases of? We show that it holds for 1-st curve when and 2-nd curve when by using a direct approach different from theirs. positive answer to problem 2 (and 1) for target curves. 7
8 Our approach We construct a with a nonzero -eigenvector basis of and explicit generating operators s.t. For example, for are given by Gauss sums for the 1-st curve. We show that are invertible and are also efficiently computable. A key fact: : projection to Since where are eigenvalues of where : matrix units w.r.t. we know that (and ) are -bases of 8
9 Our results on. where We show that is a -basis of when for (it holds if ) -eigenvector basis of 1. Generate a nonzero 2. for : Gauss sum operator multiplicative character of of order additive character of 9
10 Our results on. where is a -eigenvector basis of where and is a Jacobi sum. is a basis of From and for we see that is a basis of
11 Fundamental properties of the Weil pairing. where and : the dual of e.g. [Mil, p.132] In particular, we use the following two cases. For example, we calculate 11
12 Weil pairing on. Using the fundamental properties of we obtain where and when for any nonzero for (Corollary 2) If we normalize to for we obtain an efficiently constructible (semi-)symplectic basis w.r.t. the Weil pairing. 12
13 . (full) embedding degree for is 12, i.e., order of is 12. Action of an extra-special 2-group of order 32. For any where is a root of the quadratic eq. The dihedral subgroup of order 8. 13
14 Our results on. where We show that and are -bases of when We consider the following 1. Generate a nonzero 2. where and 14
15 Our results on. when (Lemma 5) A basis of consisting of eigenvectors of for when for is a -eigenvector basis of 15
16 Our results on. and are since -bases of is the dihedral group. By the fundamental properties of : (semi-)symplectic basis w.r.t. the Weil pairing 16
17 Conclusions We proved several facts on distortion maps given in [GPRS]. Our explicit results seem useful to use - dim. vector space in cryptography. Can we obtain a similar or general result for a broader class of curves? Cf. [GR] Is there another application of our results? 17
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Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.