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Batch Verification with Applications to Cryptography and Checking
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Weierstraß Elliptic Curves and Side-Channel Attacks
PKC '02 Proceedings of the 5th International Workshop on Practice and Theory in Public Key Cryptosystems: Public Key Cryptography
Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems
CHES '99 Proceedings of the First International Workshop on Cryptographic Hardware and Embedded Systems
Preventing SPA/DPA in ECC Systems Using the Jacobi Form
CHES '01 Proceedings of the Third International Workshop on Cryptographic Hardware and Embedded Systems
Randomized Addition-Subtraction Chains as a Countermeasure against Power Attacks
CHES '01 Proceedings of the Third International Workshop on Cryptographic Hardware and Embedded Systems
Hessian Elliptic Curves and Side-Channel Attacks
CHES '01 Proceedings of the Third International Workshop on Cryptographic Hardware and Embedded Systems
The Montgomery Powering Ladder
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
Low-Cost Solutions for Preventing Simple Side-Channel Analysis: Side-Channel Atomicity
IEEE Transactions on Computers
Advances in Elliptic Curve Cryptography (London Mathematical Society Lecture Note Series)
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The Complexity of Certain Multi-Exponentiation Techniques in Cryptography
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Efficient scalar multiplication by isogeny decompositions
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Curve25519: new diffie-hellman speed records
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SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
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Research in Cryptology
CHES '08 Proceeding sof the 10th international workshop on Cryptographic Hardware and Embedded Systems
Pairing Computation on Twisted Edwards Form Elliptic Curves
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Practical-Sized Instances of Multivariate PKCs: Rainbow, TTS, and lIC-Derivatives
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Another Approach to Pairing Computation in Edwards Coordinates
INDOCRYPT '08 Proceedings of the 9th International Conference on Cryptology in India: Progress in Cryptology
Twisted Edwards Curves Revisited
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Generalized Scalar Multiplication Secure against SPA, DPA, and RPA
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication
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EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Double-Base Number System for Multi-scalar Multiplications
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Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Novel Precomputation Schemes for Elliptic Curve Cryptosystems
ACNS '09 Proceedings of the 7th International Conference on Applied Cryptography and Network Security
Jacobi Quartic Curves Revisited
ACISP '09 Proceedings of the 14th Australasian Conference on Information Security and Privacy
Elliptic Curve Scalar Multiplication Combining Yao's Algorithm and Double Bases
CHES '09 Proceedings of the 11th International Workshop on Cryptographic Hardware and Embedded Systems
On the number of distinct elliptic curves in some families
Designs, Codes and Cryptography
Fast Reconfigurable Elliptic Curve Cryptography Acceleration for GF(2m) on 32 bit Processors
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New formulae for efficient elliptic curve arithmetic
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
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AFRICACRYPT'08 Proceedings of the Cryptology in Africa 1st international conference on Progress in cryptology
Faster group operations on elliptic curves
AISC '09 Proceedings of the Seventh Australasian Conference on Information Security - Volume 98
Coordinate blinding over large prime fields
CHES'10 Proceedings of the 12th international conference on Cryptographic hardware and embedded systems
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
Differential addition in generalized Edwards coordinates
IWSEC'10 Proceedings of the 5th international conference on Advances in information and computer security
Toric forms of elliptic curves and their arithmetic
Journal of Symbolic Computation
Arithmetic of split kummer surfaces: montgomery endomorphism of Edwards products
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
High-speed high-security signatures
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
To infinity and beyond: combined attack on ECC using points of low order
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
International Journal of Applied Cryptography
Efficient arithmetic on hessian curves
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Refinement of miller's algorithm over edwards curves
CT-RSA'10 Proceedings of the 2010 international conference on Topics in Cryptology
International Journal of Applied Cryptography
The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines
Finite Fields and Their Applications
Complete atomic blocks for elliptic curves in jacobian coordinates over prime fields
LATINCRYPT'12 Proceedings of the 2nd international conference on Cryptology and Information Security in Latin America
CHES'12 Proceedings of the 14th international conference on Cryptographic Hardware and Embedded Systems
An 8-bit AVR-Based Elliptic Curve Cryptographic RISC Processor for the Internet of Things
MICROW '12 Proceedings of the 2012 45th Annual IEEE/ACM International Symposium on Microarchitecture Workshops
Secure JTAG Implementation Using Schnorr Protocol
Journal of Electronic Testing: Theory and Applications
Twisted edwards-form elliptic curve cryptography for 8-bit AVR-based sensor nodes
Proceedings of the first ACM workshop on Asia public-key cryptography
Elligator: elliptic-curve points indistinguishable from uniform random strings
Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security
SN-SEC: a secure wireless sensor platform with hardware cryptographic primitives
Personal and Ubiquitous Computing
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Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field. This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only 3M + 4S, i.e., 3 field multiplications and 4 field squarings. If curve parameters are chosen to be small then the algorithm for mixed addition uses only 9M + 1S and the algorithm for non-mixed addition uses only 10M + 1S. Arbitrary Edwards curves can be handled at the cost of just one extra multiplication by a curve parameter. For comparison, the fastest algorithms known for the popular "a4=-3 Jacobian" form use 3M + 5S for doubling; use 7M + 4S for mixed addition; use 11M + 5S for non-mixed addition; and use 10M + 4S for non-mixed addition when one input has been added before. The explicit formulas for non-mixed addition on an Edwards curve can be used for doublings at no extra cost, simplifying protection against side-channel attacks. Even better, many elliptic curves (approximately 1/4 of all isomorphism classes of elliptic curves over a non-binary finite field) are birationally equivalent--over the original field--to Edwards curves where this addition algorithm works for all pairs of curve points, including inverses, the neutral element, etc. This paper contains an extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.