Modified elliptic curve signature algorithm for message recovery
A modified Chinese State Encryption Management Bureau's SM2 Elliptic Curve Signature Algorithm that offers partial message recovery and lowers the signature size for a given cryptographic strength. The modified SM2 Elliptic Curve Signature Algorithm includes a signature and verification algorithm that modifies a signature generation primitive to compute a key derived from the ephemeral signing key, and a multiple of the signer's public key.
1. A digital signature generation method for a transmitting-entity, comprising:
partitioning, by a processing circuit, a message to be transmitted into a first portion and a second portion;
generating an ephemeral key;
calculating, by the processing circuit, an encryption key based on the generated ephemeral key and a public key;
computing, by the processing circuit, a ciphertext by encrypting the first portion of the message using the calculated encryption key;
computing, by the processing circuit, a hash value by applying a first hash function to a concatenation of an identity hash value, the computed ciphertext, and the second portion of the message;
calculating, by the processing circuit, a signature key using a private key, the calculated encryption key, and the computed hash value; and
outputting an enciphered message that includes the calculated signature key, the computed ciphertext, and the second portion of the message, wherein the signature key, s, is calculated by the equation:
s =(1 +d A ) −1 ( k−r·d A )mod n,
wherein d A is the private key of the transmitting-entity, k is the ephemeral key, n is the order of the base point of the elliptic curve, and r is a first signature component calculated by the equation:
r=e+x Q mod n , and
wherein e is an integer value derived from the hash value h, and x Q is an integer conversion of the x-coordinate of the ephemeral key.
2. The digital signature generation method of claim 1 , wherein the step of calculating the encryption key, K e , is calculated by the equation:
K e =KDF( Q−x Q P A ),
wherein KDF is a key derivation function, P A is the public key, Q is the ephemeral key, and x Q is an integer conversion of the x-coordinate of Q.
3. The digital signature generation method of claim 1 , further comprising:
calculating the identity hash value by
forming a two-byte string of a distinguishing identifier;
forming a byte string of the distinguishing identifier, the public key, and a set of elliptic curve parameters; and
computing the identity hash value by applying a second hash function to the byte string.
4. The digital signature generation method of claim 1 , wherein the step of calculating the ciphertext, c, comprises calculating c by the equation:
c=E ( N,K e ),
wherein E is an encryption scheme, N is the first portion of the message to be transmitted, and k is the encryption key.
5. The digital signature generation method of claim 1 , wherein the step of calculating the hash value, h, comprises calculating h by the equation:
h =Hash( Z A ∥c∥V ),
wherein Z A is the identity hash value, c is the ciphertext, V is the second portion of the message to be transmitted, and Hash is the first hash function.
6. A digital signature verification method for a receiving-entity, comprising:
receiving an enciphered message that includes a signature key, a ciphertext, and a second portion of a message;
computing, by a processing circuit, a hash value by applying a first hash function to a concatenation of an identity hash value, the received ciphertext, and the received second portion of the message;
computing, by the processing circuit, an ephemeral key based on a public key, the received signature key, and the computed hash value;
computing, by the processing circuit, a decryption key based on the computed ephemeral key;
calculating, by the processing circuit, a first portion of the message by decrypting the received ciphertext using the computed decryption key; and
checking that the calculated first portion of the message has necessary redundancy to verify the message, wherein the ephemeral key, Q′, is calculated by the equation:
Q′=sG +( e+s ) P A , and
wherein s is the signature key, P A is the public key, e is the hash value expressed as an integer, and G is a base point of an elliptic curve.
7. The digital signature verification method of claim 6 , wherein the step of computing the decryption key, K e ′, comprises calculating the decryption key by the equation:
K e ′=KDF( Q ′),
wherein KDF is a key derivation function, and Q′ is the ephemeral key.
8. The digital signature verification method of claim 6 , wherein the step of computing the first portion of the message, N′, comprises calculating the first portion by the equation:
N′=E −1 ( c,K e ′),
wherein c is the received ciphertext, K e ′ is the computed decryption key, and E −1 is a decryption scheme.
9. A digital signature generation apparatus, comprising:
a processing circuit configured to:
partition a message to be transmitted into a first portion and a second portion;
generate an ephemeral key;
calculate an encryption key based on the generated ephemeral key and a public key;
compute a ciphertext by encrypting the first portion of the message using the calculated encryption key;
compute a hash value by applying a first hash function to a concatenation of an identity hash value, the computed ciphertext, and the second portion of the message;
calculate a signature key using a private key, the calculated encryption key, and the computed hash value; and
output an enciphered message that includes the calculated signature key, the computed ciphertext, and the second portion of the message, wherein the signature key, s, is calculated by the equation:
s =(1 +d A ) −1 ( k−r·d A )mod n,
wherein d A is the private key of the transmitting-entity, k is the ephemeral key, n is the order of the base point of the elliptic curve, and r is a first signature component calculated by the equation:
r=e+x Q mod n , and
wherein e is an integer value derived from the hash value h, and x Q is an integer conversion of the x-coordinate of the ephemeral key.
10. A digital signature verification apparatus, comprising:
a processing circuit configured to:
receive an enciphered message that includes a signature key, a ciphertext, and a second portion of a message;
compute a hash value by applying a first hash function to a concatenation of an identity hash value, the received ciphertext, and the received second portion of the message;
compute an ephemeral key based on a public key, the received signature key, and the computed hash value;
compute a decryption key based on the computed ephemeral key;
calculate a first portion of the message by decrypting the received ciphertext using the computed decryption key; and
check that the calculated first portion of the message has necessary redundancy to verify the message, wherein the ephemeral key, Q′, is calculated by the equation:
Q′=sG +( e+s ) P A , and
wherein s is the signature key, P A is the public key, e is the hash value expressed as an integer, and G is a base point of an elliptic curve.
11. A non-transitory computer-readable medium having stored thereon a program that when executed by a computer causes the computer to execute the steps of:
partitioning a message to be transmitted into a first portion and a second portion;
generating an ephemeral key;
calculating an encryption key based on the generated ephemeral key and a public key;
computing a ciphertext by encrypting the first portion of the message using the calculated encryption key;
computing a hash value by applying a first hash function to a concatenation of an identity hash value, the computed ciphertext, and the second portion of the message;
calculating a signature key using a private key, the calculated encryption key, and the computed hash value; and
outputting an enciphered message that includes the calculated signature key, the computed ciphertext, and the second portion of the message, wherein the signature key, s, is calculated by the equation:
s =(1 +d A ) −1 ( k−r·d A )mod n,
wherein d A is the private key of the transmitting-entity, k is the ephemeral key, n is the order of the base point of the elliptic curve, and r is a first signature component calculated by the equation:
r=e+x Q mod n , and
wherein e is an integer value derived from the hash value h, and x Q is an integer conversion of the x-coordinate of the ephemeral key.
12. A non-transitory computer-readable medium having stored thereon a program that when executed by a computer causes the computer to execute the steps of:
receiving an enciphered message that includes a signature key, a ciphertext, and a second portion of a message;
computing a hash value by applying a first hash function to a concatenation of an identity hash value, the received ciphertext, and the received second portion of the message;
computing an ephemeral key based on a public key, the received signature key, and the computed hash value;
computing a decryption key based on the computed ephemeral key;
calculating a first portion of the message by decrypting the received ciphertext using the computed decryption key; and
checking that the calculated first portion of the message has necessary redundancy to verify the message, wherein the ephemeral key, Q′, is calculated by the equation:
Q′=sG +( e+s ) P A , and
wherein s is the signature key, P A is the public key, e is the hash value expressed as an integer, and G is a base point of an elliptic curve.