A cryptographic method comprising determining two corresponding points (x, y) and (X, Y, Z) belonging to an elliptic curve defined over a finite field of characteristic p > 3, where p is a prime number satisfying the property p = 3 mod 4, said determination step performing a simplified SWU encoding function from a secret element u belonging to the finite field Fp, the cryptographic method further comprising the use of the two points in a cryptographic method between a first device and a second device, the simplified SWU encoding function being performed in an electronic component of the first device determining the points (x, y) in affine coordinates, by the following steps: a) α = -u 2 mod p b) X 2 = -b ((1 + α + α 2)l(a(α + α 2))) mod p c) X 3 = α X 2 mod p d) h 2 = (X 2) 3 + a X 2 + b mod p f) U = u 3 h 2 mod p g) A = (h 2) p-1-(p + 1) / 4 mod p h) if A 2 h 2 = 1 mod p then (x, y) = (X 2, Ah 2 mod p) i) else (x, y) = (X 3, AU mod p) the method being characterized in that the simplified SWU encoding function being performed in an electronic component of the second device determines the points (X, Y, Z) in Jacobian coordinates by one of the following two sets of steps: a') α = -u 2 mod p b') V = a(α + a 2) mod p c') Z = V 2 mod p d') X 2 = -bZV(1 + α + α 2) mod p e') X 3 = α X 2 mod p f') h 2 = (X 2) 3 + aX 2 Z 4 + bZ 6 mod p h') U = u 3 h 2 mod p i') A = (h 2) p-1-(p+1) / 4 mod p j') if A 2 h 2 = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) k') else (X, Y, Z) = (X 3, A U mod p, Z) or a) α = -u 2 mod p b) Z = a(α+α 2) mod p c) X 2 = -bZ(1 + α + a 2) mod p d) X 3 = α X 2 mod p e) h 2 = (X 2) 3 + aX 2 Z 4 + bZ 6 mod p g) U = u 3 h 2 mod p h) A = Z.(h 2 Z 2) p-1-(p+1) / 4 mod p i) if A 2 h 2 = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) j) else (X, Y, Z) = (X 3, A U mod p, Z). or the simplified SWU encoding function being performed in an electronic component of the second device determines the points (X, Y, Z) in homogeneous projective coordinates by the following steps: a) α = -u 2 mod p b) Z = a(α + α 2) mod p c) X 2 = -b(1 + α + α 2) mod p d) X 3 = α X 2 mod p e) h 2 = (X 2) 3 + aX 2 Z 2 + bZ 3 mod p f) U = u 3 h 2 mod p g) A = (h 2 Z) p-1-(p+1) / 4 mod p h) if A 2 h 2 Z = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) i) else (X, Y, Z) = (X 3, A U mod p, Z). a) α = -u 2 mod p b) X 2 = -b ((1 + α + α 2)l(a(α + α 2))) mod p c) X 3 = α X 2 mod p d) h 2 = (X 2) 3 + a X 2 + b mod p f) U = u 3 h 2 mod p g) A = (h 2) p-1-(p + 1) / 4 mod p h) if A 2 h 2 = 1 mod p then (x, y) = (X 2, Ah 2 mod p) i) else (x, y) = (X 3, AU mod p) a') α = -u 2 mod p b') V = a(α + a 2) mod p c') Z = V 2 mod p d') X 2 = -bZV(1 + α + α 2) mod p e') X 3 = α X 2 mod p f') h 2 = (X 2) 3 + aX 2 Z 4 + bZ 6 mod p h') U = u 3 h 2 mod p i') A = (h 2) p-1-(p+1) / 4 mod p j') if A 2 h 2 = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) k') else (X, Y, Z) = (X 3, A U mod p, Z) a) α = -u 2 mod p b) Z = a(α+α 2) mod p c) X 2 = -bZ(1 + α + a 2) mod p d) X 3 = α X 2 mod p e) h 2 = (X 2) 3 + aX 2 Z 4 + bZ 6 mod p g) U = u 3 h 2 mod p h) A = Z.(h 2 Z 2) p-1-(p+1) / 4 mod p i) if A 2 h 2 = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) j) else (X, Y, Z) = (X 3, A U mod p, Z). a) α = -u 2 mod p b) Z = a(α + α 2) mod p c) X 2 = -b(1 + α + α 2) mod p d) X 3 = α X 2 mod p e) h 2 = (X 2) 3 + aX 2 Z 2 + bZ 3 mod p f) U = u 3 h 2 mod p g) A = (h 2 Z) p-1-(p+1) / 4 mod p h) if A 2 h 2 Z = 1 mod p then (X, Y, Z) = (X 2, Ah 2 mod p, Z) i) else (X, Y, Z) = (X 3, A U mod p, Z).