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FACTOR4 AND 6 COMPRESSION OF CYCLOTOMIC Subgroups Of . . .
, 2009
"... Bilinear pairings derived from supersingular elliptic curves of embedding degrees 4 and 6 over finite fields F2 m and F3m, respectively, have been used to implement pairingbased cryptographic protocols. The pairing values lie in certain primeorder subgroups of the cyclotomic subgroups of orders ..."
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Bilinear pairings derived from supersingular elliptic curves of embedding degrees 4 and 6 over finite fields F2 m and F3m, respectively, have been used to implement pairingbased cryptographic protocols. The pairing values lie in certain primeorder subgroups of the cyclotomic subgroups of orders 22m + 1 and 32m − 3m + 1, respectively, of the multiplicative groups F ∗ 24m and F ∗ 36m. It was previously known how to compress the pairing values over characteristic two fields by a factor of 2, and the pairing values over characteristic three fields by a factor of 6. In this paper, we show how the pairing values over characteristic two fields can be compressed by a factor of 4. Moreover, we present and compare several algorithms for performing exponentiation in the primeorder subgroups using the compressed representations. In particular, in the case where the base is fixed, we expect to gain at least a 54 % speed up over the fastest previously known exponentiation algorithm that uses factor6 compressed representations.
SQUARING IN CYCLOTOMIC SUBGROUPS
"... Abstract. We propose new squaring formulae for cyclotomic subgroups of certain finite fields. Our formulae use a compressed representation of elements having the property that decompression can be performed at a very low cost. The squaring formulae lead to new exponentiation algorithms in cyclotomic ..."
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Abstract. We propose new squaring formulae for cyclotomic subgroups of certain finite fields. Our formulae use a compressed representation of elements having the property that decompression can be performed at a very low cost. The squaring formulae lead to new exponentiation algorithms in cyclotomic subgroups which outperform the fastest previouslyknown exponentiation algorithms when the exponent has low Hamming weight. Our algorithms can be adapted to accelerate the final exponentiation step of pairing computations. 1.
DOUBLEEXPONENTIATION IN FACTOR4 GROUPS AND ITS APPLICATIONS
"... Abstract. In previous work we showed how to compress certain primeorder subgroups of the cyclotomic subgroups of orders 22m + 1 of the multiplicative groups of F ∗ 24m by a factor of 4. We also showed that singleexponentiation can be efficiently performed using compressed representations. In this ..."
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Abstract. In previous work we showed how to compress certain primeorder subgroups of the cyclotomic subgroups of orders 22m + 1 of the multiplicative groups of F ∗ 24m by a factor of 4. We also showed that singleexponentiation can be efficiently performed using compressed representations. In this paper we show that doubleexponentiation can be efficiently performed using factor4 compressed representation of elements. In addition to giving a considerable speed up to the previously known fastest singleexponentiation algorithm for general bases, doubleexponentiation can be used to adapt our compression technique to ElGamal type signature schemes. 1.