An algorithm for modular exponentiation
Information Processing Letters
Elliptic Curves: Number Theory and Cryptography
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A Tree-Based Approach for Computing Double-Base Chains
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Optimizing double-base elliptic-curve single-scalar multiplication
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
Faster addition and doubling on elliptic curves
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
Extending scalar multiplication using double bases
ASIACRYPT'06 Proceedings of the 12th international conference on Theory and Application of Cryptology and Information Security
Scalar multiplication on koblitz curves using double bases
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
Efficient and secure elliptic curve point multiplication using double-base chains
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
FPGA implementation of point multiplication on koblitz curves using kleinian integers
CHES'06 Proceedings of the 8th international conference on Cryptographic Hardware and Embedded Systems
Short memory scalar multiplication on koblitz curves
CHES'05 Proceedings of the 7th international conference on Cryptographic hardware and embedded systems
Handbook of Elliptic and Hyperelliptic Curve Cryptography, Second Edition
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Bounds and trade-offs for Double-Base Number Systems
Information Processing Letters
Sublinear scalar multiplication on hyperelliptic koblitz curves
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
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The Joint Sparse Form is currently the standard representation system to perform multi-scalar multiplications of the form [n ]P + m [Q ]. We introduce the concept of Joint Double-Base Chain, a generalization of the Double-Base Number System to represent simultaneously n and m . This concept is relevant because of the high redundancy of Double-Base systems, which ensures that we can find a chain of reasonable length that uses exactly the same terms to compute both n and m . Furthermore, we discuss an algorithm to produce such a Joint Double-Base Chain. Because of its simplicity, this algorithm is straightforward to implement, efficient, and also quite easy to analyze. Namely, in our main result we show that the average number of terms in the expansion is less than 0.3945log2 n . With respect to the Joint Sparse Form, this induces a reduction by more than 20% of the number of additions. As a consequence, the total number of multiplications required for a scalar multiplications is minimal for our method, across all the methods using two precomputations, P + Q and P *** Q . This is the case even with coordinate systems offering very cheap doublings, in contrast with recent results on scalar multiplications. Several variants are discussed, including methods using more precomputed points and a generalization relevant for Koblitz curves. Our second contribution is a new way to evaluate $\widehat\phi$, the dual endomorphism of the Frobenius. Namely, we propose formulae to compute $\pm{\widehat\phi}(P)$ with at most 2 multiplications and 2 squarings in the base field $\mathbb{F}_{2^d}$. This represents a speed-up of about 50% with respect to the fastest known techniques. This has very concrete consequences on scalar and multi-scalar multiplications on Koblitz curves.