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**2002/088**

**Title:**

Constructing Elliptic Curves with Prescribed Embedding Degrees

**Authors:**
Paulo S. L. M. Barreto, Ben Lynn, Michael Scott

**Abstract:**

Pairing-based cryptosystems depend on the existence of groups where

the Decision Diffie-Hellman problem is easy to solve, but the

Computational Diffie-Hellman problem is hard. Such is the case of

elliptic curve groups whose embedding degree is large enough to

maintain a good security level, but small enough for arithmetic

operations to be feasible. However, the embedding degree is usually

enormous, and the scarce previously known suitable elliptic groups

had embedding degree k \leqslant 6. In this note, we examine

criteria for curves with larger k that generalize prior work by

Miyaji et al. based on the properties of cyclotomic

polynomials, and propose efficient representations for the

underlying algebraic structures.

**ePrint:**
https://eprint.iacr.org/2002/088

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