equation Consider the elliptic curve group based on the y² = x³ + ax + b mod p where a = 852, b = 29, and p = 1831. According to Hasse's theorem, what are the minimum and maximum number of elements this group might have? ≤ #E<
equation Consider the elliptic curve group based on the y² = x³ + ax + b mod p where a = 852, b = 29, and p = 1831. According to Hasse's theorem, what are the minimum and maximum number of elements this group might have? ≤ #E<
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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![**Problem: Elliptic Curve Group**
Consider the elliptic curve group based on the equation:
\[ y^2 \equiv x^3 + ax + b \pmod{p} \]
where \( a = 852 \), \( b = 29 \), and \( p = 1831 \).
According to Hasse's theorem, what are the minimum and maximum number of elements this group might have?
\[ \_\_\_\_\_ \leq \#E \leq \_\_\_\_\_ \]
**Explanation:**
The elliptic curve considered follows the equation \( y^2 \equiv x^3 + ax + b \pmod{p} \). Given values for \( a \), \( b \), and \( p \) provide specific conditions under modulo \( p \). Hasse's theorem helps in determining the range of the number of elements in the group. The question asks for these minimum and maximum bounds, indicated by the expression \(\_\_\_\_\_ \leq \#E \leq \_\_\_\_\_\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F648d38bc-f5b9-4baf-95bc-853621e34b39%2F9b48e18a-4db3-4f50-bdb8-aeea1e9d71fc%2Fln7dpdc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Elliptic Curve Group**
Consider the elliptic curve group based on the equation:
\[ y^2 \equiv x^3 + ax + b \pmod{p} \]
where \( a = 852 \), \( b = 29 \), and \( p = 1831 \).
According to Hasse's theorem, what are the minimum and maximum number of elements this group might have?
\[ \_\_\_\_\_ \leq \#E \leq \_\_\_\_\_ \]
**Explanation:**
The elliptic curve considered follows the equation \( y^2 \equiv x^3 + ax + b \pmod{p} \). Given values for \( a \), \( b \), and \( p \) provide specific conditions under modulo \( p \). Hasse's theorem helps in determining the range of the number of elements in the group. The question asks for these minimum and maximum bounds, indicated by the expression \(\_\_\_\_\_ \leq \#E \leq \_\_\_\_\_\).
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