scholarly journals Application of Montgomery’s Trick to Scalar Multiplication for Elliptic and Hyperelliptic Curves Using a Fixed Base Point

2004 ◽  
pp. 41-54 ◽  
Author(s):  
Pradeep Kumar Mishra ◽  
Palash Sarkar
Keyword(s):  
Base Point ◽  
Hyperelliptic Curves ◽  
Scalar Multiplication ◽  
Fixed Base ◽  
Elliptic And Hyperelliptic Curves

scholarly journals Hyper-and-elliptic-curve cryptography

2014 ◽  
Vol 17 (A) ◽  
pp. 181-202 ◽  
Author(s):  
Daniel J. Bernstein ◽  
Tanja Lange
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Base Point ◽  
Scalar Multiplication ◽  
Key Generation ◽  
Diffie Hellman ◽  
Fixed Base ◽  
Shared Secret ◽  
Genus 2 ◽  
Point Scalar Multiplication

AbstractThis paper introduces ‘hyper-and-elliptic-curve cryptography’, in which a single high-security group supports fast genus-2-hyperelliptic-curve formulas for variable-base-point single-scalar multiplication (for example, Diffie–Hellman shared-secret computation) and at the same time supports fast elliptic-curve formulas for fixed-base-point scalar multiplication (for example, key generation) and multi-scalar multiplication (for example, signature verification).

scholarly journals Concrete quantum cryptanalysis of binary elliptic curves

2020 ◽  
pp. 451-472
Author(s):  
Gustavo Banegas ◽  
Daniel J. Bernstein ◽  
Iggy Van Hoof ◽  
Tanja Lange
Keyword(s):  
Elliptic Curves ◽  
Quantum Computer ◽  
Discrete Logarithm ◽  
Base Point ◽  
Scalar Multiplication ◽  
Logarithmic Factor ◽  
Fixed Base ◽  
Logical Qubits ◽  
Toffoli Gates ◽  
Point Scalar Multiplication

This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2(n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2(n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography.

scholarly journals A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group

2009 ◽  
Vol 146 (1) ◽  
pp. 109-118 ◽  
Author(s):  
YUSUKE KUNO
Keyword(s):  
Mapping Class Group ◽  
Homology Group ◽  
Class Group ◽  
Base Point ◽  
Mapping Class ◽  
Combinatorial Formula ◽  
Hyperelliptic Involution ◽  
Fixed Base ◽  
Genus 2 ◽  
Closed Oriented Surface

AbstractWe present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

scholarly journals The Spectrum Dip of Deck Mounted Systems

2010 ◽  
Vol 17 (1) ◽  
pp. 55-69
Author(s):  
R.J. Scavuzzo ◽  
G.D. Hill ◽  
P.W. Saxe
Keyword(s):  
Vertical Motion ◽  
Base Point ◽  
Degree Of Freedom ◽  
Mass System ◽  
Detailed Model ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Fixed Base ◽  
Grid Points ◽  
Modal Mass

In this paper, a detailed model of a ship deck and attached dynamic systems was developed and subjected to dynamic studies using two different shock inputs: a triangular shaped velocity pulse and the vertical motion of the innerbottom of the standard Floating Shock Platform (FSP). Two studies were conducted, one considering four single degree-of-freedom systems attached at various deck locations and another considering a three-mass system attached at one location. The two shock inputs were used only for the multi-mass system study. The triangular pulse was used for the four single degree-of-freedom systems study. For the single degree-of-freedom systems study, shock spectra were first calculated at the four mounting locations assuming the oscillators were not present. Then the oscillator systems were added to these grid points to determine the change in the shock spectra. First, the oscillators were added one at a time, and then all the oscillators were added to the deck. The multi-mass system was analyzed using both shock inputs. First, the fixed-base modal masses and frequencies were determined. Then, the system as a whole was attached to the deck and the spectrum values at the base point were determined and compared to those for the free deck case. In the last step each mode of the multi-mass system, represented by a single degree-of-freedom system with the modal mass and appropriate spring stiffness, was considered individually to determine the spectrum responses. Results of the free deck, the entire system and individual modal responses are compared.

scholarly journals Efficient Fixed-base exponentiation and scalar multiplication based on a multiplicative splitting exponent recoding

2018 ◽  
Vol 9 (2) ◽  
pp. 115-136
Author(s):  
Jean-Marc Robert ◽  
Christophe Negre ◽  
Thomas Plantard
Keyword(s):  
Scalar Multiplication ◽  
Fixed Base
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