(a) E: Y² = X³ +23X + 13, p = 83, P = (24, 14),
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Elliptic Curves and Point Multiplication
#### Exercise 5.10
**Objective:** Use the double-and-add algorithm (refer to Table 5.3 in the textbook) to compute \( nP \) in \( E(\mathbb{F}_p) \) for each of the following curves and points, similarly to the method illustrated in Figure 5.4.
#### Example (a)
**Given:**
- elliptic curve equation \( E \):
\[
Y^2 = X^3 + 23X + 13
\]
- prime number \( p \):
\[
p = 83
\]
- point \( P \) on the curve \( E \):
\[
P = (24, 14)
\]
- scalar \( n \):
\[
n = 19
\]
**Instructions:**
1. Implement the double-and-add algorithm, which involves repeating the steps of doubling a point and adding points as specified in Table 5.3.
2. Compute the multiplication of the point \( P \) by the scalar \( n \), which is denoted by \( nP \).
The solution involves breaking down the scalar \( n \) into binary form and systematically applying the point doubling and point addition operations according to the binary representation of \( n \).
Please refer to the relevant sections in your textbook for detailed steps and algorithmic procedures. Figures and tables in the textbook, such as Table 5.3 and Figure 5.4, will guide you through the step-by-step point multiplication process on elliptic curves.
For further reference, consult the computational examples provided in your course materials.
Interactive graphing tools and calculator scripts can also be utilized for visualizing the elliptical curves and the results of point multiplications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa332eac-d846-4704-9340-0a50b86bfcea%2F5cb8661b-3a5a-4391-b46c-5840e99f0602%2Fv0d8trp_processed.png&w=3840&q=75)
Transcribed Image Text:### Elliptic Curves and Point Multiplication
#### Exercise 5.10
**Objective:** Use the double-and-add algorithm (refer to Table 5.3 in the textbook) to compute \( nP \) in \( E(\mathbb{F}_p) \) for each of the following curves and points, similarly to the method illustrated in Figure 5.4.
#### Example (a)
**Given:**
- elliptic curve equation \( E \):
\[
Y^2 = X^3 + 23X + 13
\]
- prime number \( p \):
\[
p = 83
\]
- point \( P \) on the curve \( E \):
\[
P = (24, 14)
\]
- scalar \( n \):
\[
n = 19
\]
**Instructions:**
1. Implement the double-and-add algorithm, which involves repeating the steps of doubling a point and adding points as specified in Table 5.3.
2. Compute the multiplication of the point \( P \) by the scalar \( n \), which is denoted by \( nP \).
The solution involves breaking down the scalar \( n \) into binary form and systematically applying the point doubling and point addition operations according to the binary representation of \( n \).
Please refer to the relevant sections in your textbook for detailed steps and algorithmic procedures. Figures and tables in the textbook, such as Table 5.3 and Figure 5.4, will guide you through the step-by-step point multiplication process on elliptic curves.
For further reference, consult the computational examples provided in your course materials.
Interactive graphing tools and calculator scripts can also be utilized for visualizing the elliptical curves and the results of point multiplications.
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