Abstract: A method for establishing a shared key, includes: determining, by a first device, a braid group Bn having an index n as a public key; selecting, by the first device, a plurality of elements from the braid group Bn to generate a subgroup P, and selecting an element x from the subgroup P as a private key; receiving, by the first device, {y?1?1y,y?1?2y, . . . , y?1?n?1y} sent from a second device; sending, by the first device, {x?1?1x,x?1?2x, . . . , x?1?n?1x} calculated according to the first private key and each element of the public key, to the second device, to allow the second device to replace all occurrences of ?k in the second private key y with x??kx to obtain fy(x?1?1x,x?1?2x, . . . , x?1?n?1x)=x?1yx and to obtain the shared key x?1y?1xy by calculation; and replacing all ?k in the first private key of the first device with y?1?ky to obtain fx(y?1?1yy?1?2y, . . . , y?1?n?1y)=y?1xy, and calculating to obtain the shared key x?1y?1xy.

Abstract: A method for establishing a shared key, includes: determining, by a first device, a braid group Bn having an index n as a public key; selecting, by the first device, a plurality of elements from the braid group Bn to generate a subgroup P, and selecting an element x from the subgroup P as a private key; receiving, by the first device, {y?1?1y,y?1?2y, . . . , y?1?n?1y} sent from a second device; sending, by the first device, {x?1?1x,x?1?2x, . . . , x?1?n?1x} calculated according to the first private key and each element of the public key, to the second device, to allow the second device to replace all occurrences of ?k in the second private key y with x??kx to obtain fy(x?1?1x,x?1?2x, . . . , x?1?n?1x)=x?1yx and to obtain the shared key x?1y?1xy by calculation; and replacing all ?k in the first private key of the first device with y?1?ky to obtain fx(y?1?1yy?1?2y, . . . , y?1?n?1y)=y?1xy, and calculating to obtain the shared key x?1y?1xy.