A course in number theory and cryptography
A course in number theory and cryptography
A course in computational algebraic number theory
A course in computational algebraic number theory
Algorithmic number theory
The Art of Computer Programming Volumes 1-3 Boxed Set
The Art of Computer Programming Volumes 1-3 Boxed Set
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
An improved quantum Fourier transform algorithm and applications
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Algorithms for quantum computation: discrete logarithms and factoring
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Circuit for Shor's algorithm using 2n+3 qubits
Quantum Information & Computation
One-more matching conjugate problem and security of braid-based signatures
ASIACCS '07 Proceedings of the 2nd ACM symposium on Information, computer and communications security
On the Design and Optimization of a Quantum Polynomial-Time Attack on Elliptic Curve Cryptography
Theory of Quantum Computation, Communication, and Cryptography
Transitive signatures from braid groups
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
Quantum addition circuits and unbounded fan-out
Quantum Information & Computation
Optimized quantum implementation of elliptic curve arithmetic over binary fields
Quantum Information & Computation
An O(m2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2m)a
Quantum Information & Computation
New constructions of public-key encryption schemes from conjugacy search problems
Inscrypt'10 Proceedings of the 6th international conference on Information security and cryptology
ETRICS'06 Proceedings of the 2006 international conference on Emerging Trends in Information and Communication Security
CSP-DHIES: a new public-key encryption scheme from matrix conjugation
Security and Communication Networks
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We show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around 1000 qubits while factoring the security-wise equivalent 1024 bit RSA modulus would require about 2000 qubits. In this paper we only consider elliptic curves over GF(p) and not yet the equally important ones over GF(2n) or other finite fields. The main technical difficulty is to implement Euclid's gcd algorithm to compute multiplicative inverses modulo p. As the runtime of Euclid's algorithm depends on the input, one difficulty encountered is the "quantum halting problem".