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December 4, 2025 19:31
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Cornacchia's algorithm for Gaussian Integers paper
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| #Complete walkthrough: https://leetarxiv.substack.com/p/computation-of-discrete-logarithms | |
| import sympy as sp | |
| p = sp.Integer("5213619424271520371687014113170182341777563603680354416779") | |
| # Step 1: find root of -2 mod p | |
| r = sp.sqrt_mod(-2, p, all_roots=False) # choose one root | |
| if r is None: | |
| raise ValueError("No solution exists.") | |
| # Step 2: Cornacchia | |
| a0, a1 = p, r | |
| while a1 > sp.sqrt(p): | |
| a0, a1 = a1, a0 % a1 | |
| T = a1 | |
| # Step 3: compute V | |
| rhs = (p - T*T) // 2 | |
| if rhs < 0 or not int(sp.sqrt(rhs))**2 == rhs: | |
| raise ValueError("No integer V exists; try the other root.") | |
| V = int(sp.sqrt(rhs)) | |
| print("T =", T) | |
| print("V =", V) | |
| print("Check:", (T*T + 2*V*V) % p) |
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