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Exploration 3. Suppose (a/p) = 1 and consider the following procedure. Choose x € Up at random and check if x² = a. If not, then compute z := (x + y) (p-¹)/2 in the ring Zplyly-a. Write z in the form u + vy where u, v € Zp. If u = 0, we can use z to compute √a in Up. How? And, more importantly, how often is u = 0 (given a random choice for x)?

Exploration 3. Suppose (a/p) = 1 and consider the following procedure. Choose x € Up at random and check if x² = a. If not, then compute z := (x + y) (p-¹)/2 in the ring Zplyly-a. Write z in the form u + vy where u, v € Zp. If u = 0, we can use z to compute √a in Up. How? And, more importantly, how often is u = 0 (given a random choice for x)?

Advanced Engineering Mathematics
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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Exploration 3. Suppose (a/p) = 1 and consider the following procedure. Choose x € Up at random and check if x² = a. If not, then compute 2 z := Z₂[y]„² (x + y)(p−¹)/² in the ring Zp[y]y-a. Write z in the form u + vy where u, v € Zp. If u = 0, we can use z to compute √a in Up. How? And, more importantly, how often is u = 0 (given a random choice for x)? 45
Transcribed Image Text:Exploration 3. Suppose (a/p) = 1 and consider the following procedure. Choose x € Up at random and check if x² = a. If not, then compute 2 z := Z₂[y]„² (x + y)(p−¹)/² in the ring Zp[y]y-a. Write z in the form u + vy where u, v € Zp. If u = 0, we can use z to compute √a in Up. How? And, more importantly, how often is u = 0 (given a random choice for x)? 45
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