Method and system for delegating calculation of a bilinear pairing value to a calculation server

An obtaining method enabling an entity to obtaining a bilinear pairing value e(A,B) between two values A and B from a value calculated by a calculation server, said obtaining method comprising:

• selecting two public elements P1 and P2 and two elements S1 and S2 that said entity keeps secret, two of the elements P1, P2, S1, and S2 being selected from among the values A and B respectively;
• generating four elements:
• R1=vS1
• R2=uS2
• T1=uP1+S1, and
• T2=vP2+S2
• where u and v designate random numbers;
• transmitting said elements R1, R2, T1, and T2 to said calculation server;
• receiving two values a1 and a2 calculated by said calculation and satisfying:
• (a1)y=e(T1,T2)[e(R1,P2)e(P1,R2)]−1, and
• (a2)z=e(D1,D2)
• y and z designating two predetermined integers, each of said integers being equal to 1 or to a predetermined integer c, and D1 and D2 designating two public elements selected from among the values A and B or the elements R1 and R2;
• and
• obtaining said pairing value e(A,B) from the value a1 or from the value a2.

A transmission method performed by a calculation server, said transmission method comprising:

• receiving from an entity four elements R1, R2, T1, and T2;
• calculating two values a1 and a2 satisfying:
• (a1)y=e(T1,T2)[e(R1,P2)e(P1,R2)]−1, and
• (a2)z=e(D1,D2)
• wherein e(A,B) designates a bilinear pairing between two values A and B, P1 and P2 designate two public elements, y and z designate two predetermined integers, each of said integers being equal to 1 or to a predetermined integer c, and D1 and D2 designate two public elements selected from among two values A and B or from among the elements R1 and R2;
• and
• transmitting the values a1 to a2 to said entity to enable the entity to obtain the bilinear pairing value e(A,B) between the two values A and B.

An entity configured to obtain a bilinear pairing value e(A,B) between two values A and B from a value calculated by a calculation server, said entity comprising:

• a selector configured to select two public elements P1 and P2 and two elements S1 and S2 that said entity keeps secret, two of the elements P1, P2, S1, and S2 being selected from among the values A and B respectively;
• a generator configured to generate four elements:
• R1=vS1
• R2=uS2
• T1=uP1+S1, and
• T2=vP2+S2
• where u and v designate random numbers;
• a transmitter configured to transmit said elements R1, R2, T1, and T2 to said calculation server;
• a receiver configured to receive two values a1 and a2 calculated by said calculation server and satisfying:
• (a1)y=e(T1,T2)[e(R1,P2)e(P1,R2)]−1, and
• (a2)z=e(D1,D2)
• y and z designating two predetermined integers, each of said integers being equal to 1 or to a predetermined integer c, and D1 and D2 designating two public elements selected from among the values A and B or the elements R1 and R2;
• and
• an obtainer configured to obtain said pairing value e(A,B) from the value a1 or from the value a2.

A calculation server comprising:

• a receiver configured to receive from an entity four elements R1, R2, T1, and T2;
• a calculator configured to calculate two values a1 and a2 satisfying:
• (a1)y=e(T1,T2)[e(R1,P2)e(P1,R2)]−1, and
• (a2)z=e(D1,D2)
• e designating a bilinear pairing, P1 and P2 designating two public elements, y and z designating two predetermined integers, each of said integers being equal to 1 or to a predetermined integer c, and D1 and D2 designating two public elements selected from among the values A and B or from among the elements R1 and R2;
• and
• a transmitter configured to transmit the values a1 to a2 to said entity.