Cryptographic method on a binary Edwards curve defined by d ( x + x 2 + y + y 2)=( x + x 2)( y + y 2) in the affine plane by a binary extension field F 2 m t = F 2 X / H X in which F 2 X is the polynomial ring with coefficients in F 2 and H is an irreducible polynomial of F 2 X with degree m, the parameter d of the curve being an element of F 2 m t, the set of points of the curve being provided with a law conferring an additive group structure thereon, E d F 2 m, said method using at least one cryptographic primitive in which an integer k is transformed into kG in which G is a point in E d F 2 m, with affine coordinates ( G x, G y) called the generating point, said method implementing arithmetic operations by means of a differential addition on said curve, said differential addition using successive iterations on a Montgomery scale on said curve, each iteration using the element 1 w G in which w = G x + G y is the differential coordinate of the generating point, said method being characterised in that: - for an implementation of said method using a 32-bit or 64-bit hardware architecture, the binary extension field is one of the following fields: F 2 257 t = F 2 X / X 257 + X 65 + 1; F 2 431 t = F 2 X / X 431 + X 303 + X 239 + X 111 + 1; F 2 487 t = F 2 X / X 487 + X 295 + X 167 + X 39 + 1; F 2 569 t = F 2 X / X 569 + X 441 + X 313 + X 121 + 1; and - for an implementation using a 32-bit hardware architecture, the binary extension field can also be chosen from among: F 2 223 t = F 2 X / X 223 + X 159 + 1; F 2 479 t = F 2 X / X 479 + X 255 + 1 F 2 521 t = F 2 X / X 521 + X 489 + 1; - the parameter d of the binary Edwards curve on each of these binary extension fields being given by: - for F 2 223 [ t]: d = t 64 + t 36 + t 5 +1; - for F 2 257 [ t]: d = t 65 + t 31 + t 14 +1; - for F 2 313 [ t]: d = t 38 + t 33 + t 28 +1; - for F 2 431 [ t]: d = t 83 + t 66 + t 17 +1; - for F 2 479 [ t]: d = t 73 + t 29 + t 3 +1; - for F 2 487 [ t]: d = t 69 + t 33 + t 15 +1; - for F 2 521 [ t]: d = t 66 + t 29 + t 28 +1; - for F 2 569 [ t]: d = t 56 + t 45 + t 41 +1 and that the inverse of the differential coordinate of the generating point, 1 w G satisfies: - for F 2 223 t: 1 w G = t 32 + 1; - for F 2 257 t: 1 w G = t 192; - for F 2 313 t: 1 w G = t 64 + 1; - for F 2 431 t: 1 w G = t 64 + 1; - for F 2 479 t: 1 w G = t 64 + 1; - for F 2 487 t: 1 w G = t 64 + 1; - for F 2 521 t: 1 w G = t 32 + 1; - for F 2 569 t: 1 w G = t 64 + 1. - for an implementation of said method using a 32-bit or 64-bit hardware architecture, the binary extension field is one of the following fields: F 2 257 t = F 2 X / X 257 + X 65 + 1; F 2 431 t = F 2 X / X 431 + X 303 + X 239 + X 111 + 1; F 2 487 t = F 2 X / X 487 + X 295 + X 167 + X 39 + 1; F 2 569 t = F 2 X / X 569 + X 441 + X 313 + X 121 + 1; and - for an implementation using a 32-bit hardware architecture, the binary extension field can also be chosen from among: F 2 223 t = F 2 X / X 223 + X 159 + 1; F 2 479 t = F 2 X / X 479 + X 255 + 1 F 2 521 t = F 2 X / X 521 + X 489 + 1; - the parameter d of the binary Edwards curve on each of these binary extension fields being given by: - for F 2 223 [ t]: d = t 64 + t 36 + t 5 +1; - for F 2 257 [ t]: d = t 65 + t 31 + t 14 +1; - for F 2 313 [ t]: d = t 38 + t 33 + t 28 +1; - for F 2 431 [ t]: d = t 83 + t 66 + t 17 +1; - for F 2 479 [ t]: d = t 73 + t 29 + t 3 +1; - for F 2 487 [ t]: d = t 69 + t 33 + t 15 +1; - for F 2 521 [ t]: d = t 66 + t 29 + t 28 +1; - for F 2 569 [ t]: d = t 56 + t 45 + t 41 +1 - for F 2 223 [ t]: d = t 64 + t 36 + t 5 +1; - for F 2 257 [ t]: d = t 65 + t 31 + t 14 +1; - for F 2 313 [ t]: d = t 38 + t 33 + t 28 +1; - for F 2 431 [ t]: d = t 83 + t 66 + t 17 +1; - for F 2 479 [ t]: d = t 73 + t 29 + t 3 +1; - for F 2 487 [ t]: d = t 69 + t 33 + t 15 +1; - for F 2 521 [ t]: d = t 66 + t 29 + t 28 +1; - for F 2 569 [ t]: d = t 56 + t 45 + t 41 +1 - for F 2 223 t: 1 w G = t 32 + 1; - for F 2 257 t: 1 w G = t 192; - for F 2 313 t: 1 w G = t 64 + 1; - for F 2 431 t: 1 w G = t 64 + 1; - for F 2 479 t: 1 w G = t 64 + 1; - for F 2 487 t: 1 w G = t 64 + 1; - for F 2 521 t: 1 w G = t 32 + 1; - for F 2 569 t: 1 w G = t 64 + 1.