848 research outputs found
Quantum resource estimates for computing elliptic curve discrete logarithms
We give precise quantum resource estimates for Shor's algorithm to compute
discrete logarithms on elliptic curves over prime fields. The estimates are
derived from a simulation of a Toffoli gate network for controlled elliptic
curve point addition, implemented within the framework of the quantum computing
software tool suite LIQ. We determine circuit implementations for
reversible modular arithmetic, including modular addition, multiplication and
inversion, as well as reversible elliptic curve point addition. We conclude
that elliptic curve discrete logarithms on an elliptic curve defined over an
-bit prime field can be computed on a quantum computer with at most qubits using a quantum circuit of at most Toffoli gates. We are able to classically simulate the
Toffoli networks corresponding to the controlled elliptic curve point addition
as the core piece of Shor's algorithm for the NIST standard curves P-192,
P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to
recent resource estimates for Shor's factoring algorithm. The results also
support estimates given earlier by Proos and Zalka and indicate that, for
current parameters at comparable classical security levels, the number of
qubits required to tackle elliptic curves is less than for attacking RSA,
suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added.
ASIACRYPT 201
A Survey Report On Elliptic Curve Cryptography
The paper presents an extensive and careful study of elliptic curve cryptography (ECC) and its applications. This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems. It also presents different forms of elliptic curve in various coordinate system , specifying which is most widely used and why. It also explains how isogenenies between elliptic curve provides the secure ECC. Exentended form of elliptic curve i.e hyperelliptic curve has been presented here with its pros and cons. Performance of ECC and HEC is also discussed based on scalar multiplication and DLP. Keywords: Elliptic curve cryptography (ECC), isogenies, hyperelliptic curve (HEC) , Discrete Logarithm Problem (DLP), Integer Factorization , Binary Field, Prime FieldDOI:http://dx.doi.org/10.11591/ijece.v1i2.8
Quantum Attacks on Modern Cryptography and Post-Quantum Cryptosystems
Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial encryption, these steps can be taken immediately to ensure that the impending development of large quantum computers does not compromise sensitive data
Revisiting Shor's quantum algorithm for computing general discrete logarithms
We heuristically demonstrate that Shor's algorithm for computing general
discrete logarithms, modified to allow the semi-classical Fourier transform to
be used with control qubit recycling, achieves a success probability of
approximately 60% to 82% in a single run. By slightly increasing the number of
group operations that are evaluated quantumly, and by performing a limited
search in the classical post-processing, we furthermore show how the algorithm
can be modified to achieve a success probability exceeding 99% in a single run.
We provide concrete heuristic estimates of the success probability of the
modified algorithm, as a function of the group order, the size of the search
space in the classical post-processing, and the additional number of group
operations evaluated quantumly. In analogy with our earlier works, we show how
the modified quantum algorithm may be simulated classically when the logarithm
and group order are both known. Furthermore, we show how slightly better
tradeoffs may be achieved, compared to our earlier works, if the group order is
known when computing the logarithm.Comment: The pre-print has been extended to show how slightly better tradeoffs
may be achieved, compared to our earlier works, if the group order is known.
A minor issue with an integration limit, that lead us to give a rough success
probability estimate of 60% to 70%, as opposed to 60% to 82%, has been
corrected. The heuristic and results reported in the original pre-print are
otherwise unaffecte
Aspects of hardware methodologies for the NTRU public-key cryptosystem
Cryptographic algorithms which take into account requirements for varying levels of security and reduced power consumption in embedded devices are now receiving additional attention. The NTRUEncrypt algorithm has been shown to provide certain advantages when designing low power and resource constrained systems, while still providing comparable security levels to higher complexity algorithms. The research presented in this thesis starts with an examination of the general NTRUEncrypt system, followed by a more practical examination with respect to the IEEE 1363.1 draft standard. In contrast to previous research, the focus is shifted away from specific optimizations but rather provides a study of many of the recommended practices and suggested optimizations with particular emphasis on polynomial arithmetic and parameter selection. Various methods are examined for storing, inverting and multiplying polynomials used in the system. Recommendations for algorithm and parameter selection are made regarding implementation in software and hardware with respect to the resources available. Although the underlying mathematical principles have not been significantly questioned, stable recommended practices are still being developed for the NTRUEncrypt system. As a further complication, recommended optimizations have come from various researchers and have been split between hardware and software implementations. In this thesis, a generic VHDL model is presented, based on the IEEE 1363.1 draft standard, which is designed for adaptation to software or hardware implementation while providing flexibility for changes in recommended practices
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
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