641 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
Design of reverse converters for the multi-moduli residue number systems with moduli of forms 2a, 2b - 1, 2c + 1
Residue number system (RNS) is a non-weighted integer number representation system that is capable of supporting parallel, carry-free and high speed arithmetic. This system is error-resilient and facilitates error detection, error correction and fault tolerance in digital systems. It finds applications in Digital Signal Processing (DSP) intensive computations like digital filtering, convolution, correlation, Discrete Fourier Transform, Fast Fourier Transform, etc. The basis for an RNS system is a moduli set consisting of relatively prime integers. Proper selection of this moduli set plays a significant role in RNS design because the speed of internal RNS arithmetic circuits as well as the speed and complexity of the residue to binary converter (R/B or Reverse Converter) have a large dependency on the form and number of the selected moduli. Moduli of forms 2a, 2b- 1, 2c + 1 (a, b and c are natural numbers) have the most use in RNS moduli sets as these moduli can be efficiently implemented using usual binary hardware that lead to simple design. Another important consideration for the reverse converter design is the selection of an appropriate conversion algorithm from Chinese Remainder Theorem (CRT), Mixed Radix Conversion (MRC) and the new Chinese Remainder Theorems (New CRT I and New CRT II). This research is focused on designing reverse converters for the multi-moduli RNS sets especially four and five moduli sets with moduli of forms 2a, 2b- 1, 2c + 1 . The residue to binary converters are designed by applying the above conversion algorithms in different possible ways and facilitating the use of modulo (2k) and modulo (2k β 1) adders that lead to simple design of adder based architectures and VLSI efficient implementations (k is a natural number). The area and delay of the proposed converters is analyzed and an efficient reverse converter is suggested from each of the various four and five moduli set converters for a given dynamic range
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