By Atanasov D.V.

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**Example text**

The aﬃne representation Karabina Karabina 4 6 I1 + 9M1 I1 + 24M1 3M1 I1 + 9M1 6 24 log rM log 3 rM1 1 2 w+1 2w+1 1 bit 1 trit the trace representation this work this work 4 6 2I1 + 16M1 2I1 + 44M1 I1 + 3M1 4I1 + SqRt + 15M1 6 24 log rM log3 rM1 1 2 w+1 2w+1 2 bits 1 bit and 1 trit Factor-4 and 6 (De)Compression for Values of Pairings Using Trace Maps 31 the square root, to solve degree-3 equation, to transform the solution and to 2 +1 calculate γ. We explain solving degree-3 equation in detail.

19–34, 2013. c Springer-Verlag Berlin Heidelberg 2013 20 T. Yonemura et al. problem in a prime-order group. To compress the public-key size is to represent the prime-order group with fewer bits than the size of the embedding ﬁeld. For instance, the recommended size of the ﬁnite ﬁeld is 2048 bits, and the corresponding size of the prime-order group is 224 bits [2], because the discrete logarithm problem in the ﬁnite ﬁeld is easier than in the general group, namely, the elliptic curve. The index calculus is a relatively eﬃcient algorithm to solve the discrete logarithm problem in ﬁnite ﬁelds.

Let p = 3 and m be odd. An element of groups G± , #G± = pm ± 3pm + 1, is identiﬁed by an element of Fpm without distinction among conjugates. The compression map is as follows: T r6/1 : F(pm )6 → Fpm m m 2 g → g + g p + g (p ) m 3 + g (p ) m 4 + g (p ) m 5 + g (p ) . Since #G+ #G− = Φ6 (pm ), The groups G± are subgroups of T6 (Fpm ). Such subgroups are related to supersingular elliptic curves of embedding degree 6. 4 Construction of Decompression for Trace Maps We propose the decompressible trace representation with additional information.

### 5th International Conference on Geometry and Applications by Atanasov D.V.

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