DocumentCode :
943353
Title :
An efficient solution of the congruence x^2 + ky^2 = m\\pmod{n}
Author :
Pollard, John M. ; Schnorr, Claus P.
Volume :
33
Issue :
5
fYear :
1987
fDate :
9/1/1987 12:00:00 AM
Firstpage :
702
Lastpage :
709
Abstract :
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} .
Keywords :
Cryptography; Feedback communication; Number theory; Polynomials; Digital signatures; Equations; Helium; Polynomials; Public key; Public key cryptography;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.1987.1057350
Filename :
1057350
Link To Document :
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