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An efficient solution of the congruence x^2 + ky^2 = m\pmod{n}

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2 Author(s)

The equation of the title arose in the proposed signature scheme of Ong-Schnorr-Shamir. The large integers n, k and m are given and we are asked to find any solution x, y . It was believed that this task was of similar difficulty to that of factoring the modulus n; we show that, on the contrary, a solution can easily be found if k and m are relatively prime to n . Under the assumption of the generalized Riemann hypothesis, a solution can be found by a probabilistic algorithm in O(\log n)^{2}|\log \log |k||) arithmetical steps on O(\log n) -bit integers. The algorithm can be extended to solve the equation X^{2} + KY^{2} = M \pmod {n} for quadratic integers K, M \in {\bf Z}[\sqrt {d}] and to solve in integers the equation x^{3} + ky_{3} + k^{2}z^{3} - 3kxyz = m \pmod {n} .

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Information Theory, IEEE Transactions on  (Volume:33 ,  Issue: 5 )