2 research outputs found
Fast Multiple Point Multiplication on Elliptic Curves over Prime and Binary Fields using the Double-Base Number System
Multiple-point multiplication on elliptic curves is
the highest computational complex operation in the elliptic curve
cyptographic based digital signature schemes. We describe three
algorithms for multiple-point multiplication on elliptic curves
over prime and binary fields, based on the representations
of two scalars, as sums of mixed powers of 2 and 3. Our
approaches include sliding window mechanism and some precomputed
values of points on the curve. A proof for formulae to
calculate the number of double-based elements, doublings and
triplings below 2^n is listed. Affine coordinates and Jacobian
coordinates are considered in both prime fields and binary fields.
We have achieved upto 24% of improvements in new algorithms
for multiple-point multiplication
ΠΠΎΠ΄ΠΈΡΡΠΊΠΎΠ²Π°Π½ΠΈΠΉ ΠΌΠ΅ΡΠΎΠ΄ Π±Π°Π³Π°ΡΠΎΠΊΡΠ°ΡΠ½ΠΎΠ³ΠΎ ΡΠΊΠ°Π»ΡΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Π½Ρ ΡΠΎΡΠΎΠΊ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ Ρ ΡΠΊΡΠ½ΡΠ΅Π½Π½ΠΈΡ ΠΏΠΎΠ»ΡΡ
ΠΠ°Π½ΠΈΠΉ Π΄ΠΈΠΏΠ»ΠΎΠΌΠ½ΠΈΠΉ ΠΏΡΠΎΠ΅ΠΊΡ ΠΏΡΠΈΡΠ²ΡΡΠ΅Π½ΠΈΠΉ ΡΠΎΠ·ΡΠΎΠ±Π»Π΅Π½Π½Ρ ΠΌΠΎΠ΄ΠΈΡΡΠΊΠ°ΡΡΡ ΠΌΠ΅ΡΠΎΠ΄Ρ Π±Π°Π³Π°ΡΠΎΠΊΡΠ°ΡΠ½ΠΎΠ³ΠΎ ΡΠΊΠ°Π»ΡΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΡΠΎΡΠΎΠΊ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ Ρ ΡΠΊΡΠ½ΡΠ΅Π½Π½ΠΈΡ
ΠΏΠΎΠ»ΡΡ
. ΠΠ°Π½Π° ΡΠΎΠ·ΡΠΎΠ±ΠΊΠ° ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡ ΡΠΎΠ±ΠΎΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠ½ΠΈΠΉ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡ Π΄Π»Ρ Π²ΠΈΠΊΠΎΠ½Π°Π½Π½Ρ ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ Π½Π°Π΄ ΡΠΎΡΠΊΠ°ΠΌΠΈ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ, Π·ΠΎΠΊΡΠ΅ΠΌΠ° ΠΎΠΏΠ΅ΡΠ°ΡΡΡ Π΅Π»Π΅ΠΊΡΡΠΎΠ½Π½ΠΎ-ΡΠΈΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΡΠ΄ΠΏΠΈΡΡ Π²ΠΊΠ»ΡΡΠ½ΠΎ Π· ΠΏΡΠΎΠ³ΡΠ°ΠΌΠ½ΠΎΡ ΡΠ΅Π°Π»ΡΠ·Π°ΡΡΡΡ ΡΡΠ½ΡΡΡΠΈΡ
ΡΠ° ΡΠΎΠ·ΡΠΎΠ±Π»Π΅Π½ΠΎΠ³ΠΎ ΠΌΠ΅ΡΠΎΠ΄Ρ Π±Π°Π³Π°ΡΠΎΠΊΡΠ°ΡΠ½ΠΎΠ³ΠΎ ΡΠΊΠ°Π»ΡΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΡΠΎΡΠΎΠΊ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ Π½Π° ΡΠΈΡΠ»ΠΎ. Π€ΡΠ½ΠΊΡΡΠΎΠ½Π°Π»ΡΠ½ΡΡΡΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠ½ΠΎΠ³ΠΎ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΡ Π·Π°Π±Π΅Π·ΠΏΠ΅ΡΡΡ Π²ΠΈΠΊΠΎΠ½Π°Π½Π½Ρ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΠ½ΠΈΡ
ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ Π½Π°Π΄ ΡΠΎΡΠΊΠ°ΠΌΠΈ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ ΡΠΊ ΡΠΎ Π΄ΠΎΠ΄Π°Π²Π°Π½Π½Ρ, Π²ΡΠ΄Π½ΡΠΌΠ°Π½Π½Ρ, ΠΌΠ½ΠΎΠΆΠ΅Π½Π½Ρ ΡΠΎΡΠΊΠΈ Π½Π° ΡΠΈΡΠ»ΠΎ, Π·Π½Π°Ρ
ΠΎΠ΄ΠΆΠ΅Π½Π½Ρ ΠΎΠ±Π΅ΡΠ½Π΅Π½ΠΎΡ ΡΠΎΡΠΊΠΈ, Π° ΡΠ°ΠΊΠΎΠΆ ΠΎΠ±ΡΠΈΡΠ»Π΅Π½Π½Ρ Π΅Π»Π΅ΠΊΡΡΠΎΠ½Π½ΠΎ-ΡΠΈΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΡΠ΄ΠΏΠΈΡΡ ΡΠ° ΠΉΠΎΠ³ΠΎ ΠΏΠ΅ΡΠ΅Π²ΡΡΠΊΠ°.
Π£ Π΄Π°Π½ΠΎΠΌΡ Π΄ΠΈΠΏΠ»ΠΎΠΌΠ½ΠΎΠΌΡ ΠΏΡΠΎΠ΅ΠΊΡΡ ΡΠΎΠ·ΡΠΎΠ±Π»Π΅Π½ΠΎ: Π°ΡΡ
ΡΡΠ΅ΠΊΡΡΡΡ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠ½ΠΎΠ³ΠΎ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΡ, ΠΌΠΎΠ΄ΡΠ»Ρ ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ Ρ ΡΠΊΡΠ½ΡΠ΅Π½Π½ΠΈΡ
ΠΏΠΎΠ»ΡΡ
, ΠΌΠΎΠ΄ΡΠ»Ρ ΠΎΠΏΠ΅ΡΠ°ΡΡΠΉ ΡΠ· ΡΠΎΡΠΊΠ°ΠΌΠΈ Π΅Π»ΡΠΏΡΠΈΡΠ½ΠΎΡ ΠΊΡΠΈΠ²ΠΎΡ, ΠΌΠΎΠ΄ΡΠ»Ρ Π΅Π»Π΅ΠΊΡΡΠΎΠ½Π½ΠΎ-ΡΠΈΡΡΠΎΠ²ΠΎΠ³ΠΎ ΠΏΡΠ΄ΠΏΠΈΡΡ.This diploma project is devoted to the development of the modification of the method of multiple scalar multiplication of the points of an elliptic curve in finite fields. This development is a software package for performing operations on points of an elliptic curve, in particular electronic digital signature operations, including the program implementation of the existing and developed method of multiple scalar multiplication of points of an elliptic curve to a number.
The functionality of the software complex provides execution of arithmetic operations over the points of the elliptic curve, such as adding, subtracting, multiplying a point by number, finding the inverse of a point, and calculating the digital signature and checking it.
This project consists of: software architecture, operations module in finite fields, module of operations with points of an elliptic curve, module of electronic-digital signature