A Relation between Group Order of Elliptic Curve and Extension Degree of Definition Field
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1 A Relation between Group Order of Elliptic Curve and Extension Degree of Definition Field Taichi Sumo, Yuki Mori (Okayama University) Yasuyuki Nogami (Graduate School of Okayama University) Tomoko Matsushima (Polytechnic University) Satoshi Uehara (University of Kitakyushu)
2 Research Background Recent innovative cryptographic applications are based on Pairing-based cryptography Elliptic curve cryptography ID-based cryptography Group signature authentication Time release cryptography EC discrete logarithm problem Elliptic curve addition Finite field Prime field and Extension field Arithmetic operations Addition/Subtraction Multiplication/Division 2/18
3 Research Background ID-based cryptography We can use ID-based information as public key. User name E-main address Phone number etc. Group signature authentication Anonymous authentication Attribute-based authentication Time release cryptography It keeps the data encrypted until the day for release comes. 3/18
4 Research Background Pairing-based cryptography is based on elliptic curve cryptography. Pairing-based cryptography Elliptic curve cryptography ID-based cryptography Group signature authentication Time release cryptography EC discrete logarithm problem Elliptic curve addition Finite field Extension field Arithmetic operations Addition Multiplication 4/18
5 *Finite and discrete mathematics Mathematical notations E : y 2 x 3 ax b, a, b, x, y F p n F p F p n y P rational point 0 x E F EF p p n # E F, # EF p p n 5/18
6 Elliptic curve cryptography (1/2) Elliptic curve cryptography E : y 2 x 3 ax b, y P Q P 0 a, b, x, y F Arithmetic operations p n x R P Q [2] P P P rational point 6/18
7 Elliptic curve cryptography (2/2) Elliptic curve cryptography the order r is larger than 160 bits [ r] P Infinity point O P Rational point P P [2] P [ 2] P P [3] P Cyclic group ECDLP Solve the scalar [i] Q [ i] P 7/18
8 Research Background Pairing-based cryptography uses a special class of elliptic curve. Pairing-based cryptography Elliptic curve cryptography ID-based cryptography Group signature authentication Time release cryptography EC discrete logarithm problem Elliptic curve addition Finite field Extension field Arithmetic operations Addition Multiplication 8/18
9 Pairing-based cryptography (1/3) Pairing-based cryptography over elliptic curve cryptography r rational points Requirement: torsion group structure Q [2] Q [ r] P P [2] P [ r] Q O Pairing uses two cyclic groups among r + 1 9/18
10 Pairing-based cryptography (2/3) Pairing-friendly curves It is defined over extension field The defining equation is n-th vector space F p n E : y 2 x 3 ax b, a, b, x, y F p n Some conditions should be satisfied Torsion group structure The number of rational points # E Fp n needs to be divisible by r 2. 10/18
11 Pairing-based cryptography (3/3) How to prepare pairing-friendly curves It is difficult except for some special curves Barreto-Naehrig (BN) curve : n = 12 Brezing-Weng (BW) curve : n = 8 Setting parameters : p, a, b #rational points r dimension n E : y 2 # E Fp n x 3 ax b a, b, x, y F r p n 11/18
12 Target of this research There are r points in each cyclic group n n : the extension degree (dimension) ( r 1) There are some previous works. BN and BW curves O n ( r 1) There are few reports n ( r 1) n r Torsion structure The target of this work 12/18
13 Algebraic closure F Prime field and n-th Extension field p F p n Fp F p n : Field elements 13/18
14 In the same Elliptic curve closure EF E Fp n Over Prime field p and ex. field E Fp n E Fp : Rational points 14/18
15 Our contribution (theoretic proof was given) If F and n r r # E p E Fp n Torsion structure appears E Fp : Rational points 15/18
16 Our contribution (theoretic proof was given) If F and n r r # E p E Fp n Torsion structure appears E Fp : Rational points 16/18
17 Example 17/18
18 Conclusion This work has focused on Torsion structure appears n r Further consideration Consider pairing-based cryptographic applications. Thank you for your attentions. 18/18
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