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Pairing Friendly Elliptic Curves of Prime Order with Embedding degree 12 Paulo Barreto and Michael Naehrig Presented by Mike Scott

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BN Curves An elliptic curve E: y 2 =x 3 +B mod p, where #E=p+1-t, and defined by p(x) = 36x 4 +36x 3 +24x 2 +6x+1 #E(x)=36x 4 +36x 3 +18x 2 +6x+1 t(x)= 6x 2 +1

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BN Curves … has an embedding degree of 12 … has a CM discriminant of 3 … facilitates pairings at the 128-bit level of security … is good for all pairing applications (including short signature) … supports a sextic twist, so the P and Q parameters of the pairing can be over F p 2 and F p respectively

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BN Curves … supports pairing compression … is efficient for both the Tate and Ate pairings (half length loop) … curves are plentiful and are easily found. … I could go on… … The End

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Elliptic Curve Public Key Cryptography Why ? ● ECC offers greater security for a given key size. ● The smaller key size also makes possible much more compact.

Elliptic Curve Public Key Cryptography Why ? ● ECC offers greater security for a given key size. ● The smaller key size also makes possible much more compact.

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