2021

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Quantities of points on some Edwards curves
(Igor Sikorsky Kyiv Polytechnic Institute, 2021) Rybak, Oleksandr
The Edwards curves of the form x2 + y2 = 1 + dx2y2 are investigated in this article. An exact formula for the quantity of points on x2 + y2 = 1 + dx2y2 over a field Fp is obtained for odd prime numbers p. The special attention is paid to the curves with exactly p+1 points over the field Fp. These curves are called supersingular. They are not recommended for usage in cryptography, because their structure is relatively simple. The supersingularity of the curve is proved for any prime p = 4m+3. Also, some other values of d, for which x2 + y2 is equivalent to 1 + dx2y2 (mod p) is supersingular, are found.