Method and system for distributed digital signature computation

A system comprising:

• a memory storing instructions;
• one or more communication interfaces;
• and
• one or more hardware processors coupled to the memory via the one or more communication interfaces, wherein the one or more hardware processors are configured by the instructions to:
• share, via one or more hardware processors, a plurality of input parameters and a message digest (M) with the first party and the second party, wherein the plurality of input parameters comprises an elliptic curve (E) defined as an elliptic curve equation over a finite field, a group base point in the elliptic curve (G), a fixed prime number indicating an order of points in the elliptic curve (q), a Paillier Encryption parameters including a Paillier modulus (N), a Paillier private key (λ(N)), a Paillier public key (g), a first random number (a), a second random number (b), a third random number (k1) and a fourth random number (k2);
• generate a public key, via one or more hardware processors, by the first party and the second party based on a public key generation technique, wherein the public key is generated by the first party and the second party using the plurality of input parameters, the first random number and the second random number;
• generate a session key, via one or more hardware processors, by the first party and the second party based on a session key generation technique, wherein the session key is generated by the first party and the second party using the plurality of input parameters, the third random number (k1) and the fourth random number (k2);
• and
• generate a digital signature (S) by the first party and the second party, via one or more hardware processors, based on a digital signature generation technique, wherein the digital signature generation technique comprises:
• compute a first part of the digital signature (X1) and a second part of the digital signature (X2) at the first party, wherein the first part of the digital signature and the second part of the digital signature are computed based on the first random number, the third random number, the session key (r), and the message digest (M);
• compute a first ciphertext (C1), a first zero-knowledge proof (ZKP_C1) of the first ciphertext, a second ciphertext (C2), and a second zero-knowledge proof (ZKP_C2) of the second ciphertext, at the first party, wherein the first ciphertext is generated based on a Paillier Encryption technique using the first part of the digital signature (X1) and the Paillier public key and the second ciphertext is generated based on the Paillier Encryption technique using the second part of the digital signature (X2) and the Paillier public key;
• share the first ciphertext (C1), the first zero-knowledge proof (ZKP_C1), the second ciphertext (C2) and the second zero-knowledge proof (ZKP_C1) with the second party for a zero-knowledge proof verification;
• generate a third part of the digital signature (X3) and a fourth part of the digital signature (X4) upon successful validation of (a) the first zero-knowledge proof (ZKP_C1), and (b) the second zero-knowledge proof (ZKP_C1), wherein the third part of the digital signature and the fourth part of the digital signature are generated using the second random number, the fourth random number, the session key (r), the message digest (M) and the private key share of the second party (b);
• compute a third ciphertext (C3) of the third signature and a fourth ciphertext (C4) of the fourth signature at the second party, wherein the third ciphertext and the fourth ciphertext (C4) are generated using the Paillier Encryption technique;
• generate an encrypted form of a full signature (Z) at the second party and sharing the encrypted form of the full signature (Z) with the first party, wherein the encrypted form of the full signature (Z) is generated by a summation of the third ciphertext (C3) and the fourth ciphertext (C4) using the Paillier encryption parameter;
• and
• obtain the digital signature (S) at the first party, wherein the digital signature is obtained by decrypting the encrypted form of full signature using the Paillier private key based on the Paillier Encryption technique and a ECDSA signature verification technique.

One or more non-transitory machine-readable information storage mediums comprising one or more instructions which when executed, cause one or more hardware processors to:

• share a plurality of input parameters and a message digest (M) with the first party and the second party, wherein the plurality of input parameters comprises an elliptic curve (E) defined as an elliptic curve equation over a finite field, a group base point in the elliptic curve (G), a fixed prime number indicating an order of points in the elliptic curve (q), a Paillier Encryption parameters including a Paillier modulus (N), a Paillier private key (λ(N)), a Paillier public key (g), a first random number (a), a second random number (b), a third random number (k1) and a fourth random number (k2);
• generate a public key by the first party and the second party based on a public key generation technique, wherein the public key is generated by the first party and the second party using the plurality of input parameters, the first random number and the second random number;
• generate a session key by the first party and the second party based on a session key generation technique, wherein the session key is generated by the first party and the second party using the plurality of input parameters, the third random number (k1) and the fourth random number (k2);
• and
• generate a digital signature (S) by the first party and the second party based on a digital signature generation technique, wherein the digital signature generation technique comprises:
• compute a first part of the digital signature (X1) and a second part of the digital signature (X2) at the first party, wherein the first part of the digital signature and the second part of the digital signature are computed based on the first random number, the third random number, the session key (r), and the message digest (M);
• compute a first ciphertext (C1), a first zero-knowledge proof (ZKP_C1) of the first ciphertext, a second ciphertext (C2), and a second zero-knowledge proof (ZKP_C2) of the second ciphertext, at the first party, wherein the first ciphertext is generated based on a Paillier Encryption technique using the first part of the digital signature (X1) and the Paillier public key and the second ciphertext is generated based on the Paillier Encryption technique using the second part of the digital signature (X2) and the Paillier public key;
• share the first ciphertext (C1), the first zero-knowledge proof (ZKP_C1), the second ciphertext (C2) and the second zero-knowledge proof (ZKP_C1) with the second party for a zero-knowledge proof verification;
• generate a third part of the digital signature (X3) and a fourth part of the digital signature (X4) upon successful validation of (a) the first zero-knowledge proof (ZKP_C1), and (b) the second zero-knowledge proof (ZKP_C1), wherein the third part of the digital signature and the fourth part of the digital signature are generated using the second random number, the fourth random number, the session key (r), the message digest (M) and the private key share of the second party (b);
• compute a third ciphertext (C3) of the third signature and a fourth ciphertext (C4) of the fourth signature at the second party, wherein the third ciphertext and the fourth ciphertext (C4) are generated using the Paillier Encryption technique;
• generate an encrypted form of a full signature (Z) at the second party and sharing the encrypted form of the full signature (Z) with the first party, wherein the encrypted form of the full signature (Z) is generated by a summation of the third ciphertext (C3) and the fourth ciphertext (C4) using the Paillier encryption parameter;
• and
• obtain the digital signature (S) at the first party, wherein the digital signature is obtained by decrypting the encrypted form of full signature using the Paillier private key based on the Paillier Encryption technique and a ECDSA signature verification technique.