An identity-based identification scheme based on discrete logarithms modulo a composite number
EUROCRYPT '90 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves
Mathematics of Computation
A new elliptic curve based analogue of RSA
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
Verifiable Partial Sharing of Integer Fractions
SAC '98 Proceedings of the Selected Areas in Cryptography
Identity-Based Encryption from the Weil Pairing
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
New Public-Key Schemes Based on Elliptic Curves over the Ring Zn
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
Security and Performance of Server-Aided RSA Computation Protocols
CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Old and New Deterministic Factoring Algorithms
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
A One Round Protocol for Tripartite Diffie-Hellman
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
Short Signatures from the Weil Pairing
Journal of Cryptology
The Weil Pairing, and Its Efficient Calculation
Journal of Cryptology
A public key cryptosystem based on elliptic curves over Z/nZ equivalent to factoring
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Elliptic curve cryptosystems using curves of smooth order over the ring Zn
IEEE Transactions on Information Theory
The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems
IEEE Transactions on Information Theory
CT-RSA '09 Proceedings of the The Cryptographers' Track at the RSA Conference 2009 on Topics in Cryptology
Hidden pairings and trapdoor DDH groups
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Comparison-based encryption for fine-grained access control in clouds
Proceedings of the second ACM conference on Data and Application Security and Privacy
Efficient pairing computation on ordinary elliptic curves of embedding degree 1 and 2
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
Additively homomorphic encryption with a double decryption mechanism, revisited
ISC'12 Proceedings of the 15th international conference on Information Security
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The Weil and Tate pairings are defined for elliptic curves over fields, including finite fields. These definitions extend naturally to elliptic curves over ℤ/Nℤ, for any positive integer N, or more generally to elliptic curves over any finite commutative ring, and even the reduced Tate pairing makes sense in this more general setting. This paper discusses a number of issues which arise if one tries to develop pairing-based cryptosystems on elliptic curves over such rings. We argue that, although it may be possible to develop some cryptosystems in this setting, there are obstacles in adapting many of the main ideas in pairing-based cryptography to elliptic curves over rings. Our main results are: (i) an oracle that computes reduced Tate pairings over such rings (or even just over ℤ/Nℤ) can be used to factorise integers; and (ii) an oracle that determines whether or not the reduced Tate pairing of two points is trivial can be used to solve the quadratic residuosity problem.