Method and system for Cheon resistant static Diffie-Hellman security
A method for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the method including providing a system for message communication between a pair of correspondents, a message being exchanged in accordance with ECDH instructions executable on computer processors of the respective correspondents, the ECDH instructions using a curve selected from a plurality of curves, the selecting including choosing a range of curves; selecting, from the range of curves, curves matching a threshold efficiency; excluding, within the selected curves, curves which may include intentional vulnerabilities; and electing, from non-excluded selected curves, a curve with Cheon resistance, the electing comprising a curve from an additive group of order q, wherein q is prime, such that q−1=cr and q+1=ds, where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd≤48.
1. A non-transitory computer readable medium for storing instruction code for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the instruction code, when executed by a processor of a first computing device causing the first computing device to:
select a curve for message communication between the first computing device and a second computing device, the selecting comprising:
choosing a range of curves;
selecting, from the range of curves, curves matching a threshold efficiency;
excluding, within the selected curves, curves which includes intentional vulnerabilities; and
electing, from non-excluded selected curves, a curve with Cheon resistance, the electing comprising electing a curve from an additive group of order q, wherein q is prime, such that q−1=cr and q+1=ds, where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd≤48;
select a private key for the first computing device;
compute a public key for the first computing device from curve parameters of the curve with Cheon resistance and the private key for the first computing device;
transmit the curve parameters of the curve with Cheon resistance and the public key for the first computing device to the second computing device;
receive a public key for the second computing device;
compute a shared secret based on the public key for the second computing device and the private key for the first computing device; and
communicate with the second computing device using the shared secret.
2. The non-transitory computer readable medium of claim 1 , wherein the first computing device is caused to choose a range of curves having a length matching a threshold security level.
3. The non-transitory computer readable medium of claim 1 , wherein the first computing device is caused to select by selecting curves having a field size close to a power of two, and having efficient endomorphism.
4. The non-transitory computer readable medium of claim 1 , wherein the first computing device is caused to exclude by excluding curves which are non-compact or difficult to generate.
5. The non-transitory computer readable medium of claim 1 , wherein the curve has an affine equation in the form y 2 =x 3 +ix, where i=√{square root over (−1)}.
6. The non-transitory computer readable medium of claim 5 , wherein the curve has length of 454 bits.
7. The non-transitory computer readable medium of claim 6 , wherein a field size p=2 454 +(3×17×11287) 2 .
8. The non-transitory computer readable medium of claim 6 , wherein an order q=2 452 +(7×41117) 2 .
9. The non-transitory computer readable medium of claim 7 , wherein an order q=2 452 +(7×41117) 2 .
10. The non-transitory computer readable medium of claim 6 , wherein r=(q−1)/8; and s=(q+1)/6.
11. The non-transitory computer readable medium of claim 7 , wherein r=(q−1)/8; and s=(q+1)/6.
12. The non-transitory computer readable medium of claim 8 , wherein r=(q−1)/8; and s=(q+1)/6.
13. The non-transitory computer readable medium of claim 9 , wherein r=(q−1)/8; and s=(q+1)/6.
14. A non-transitory computer readable medium for storing instruction code for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the instruction code, when executed by a processor of a first computing device causing the first computing device to:
select a curve for message communication between the first computing device and a second computing device, the curve comprising:
an additive group of order q, wherein q is prime, such that q−1=cr and q+1=ds , where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd≤48;
an affine equation in the form y 2 =x 3 +ix, where i=√{square root over (−1)};
a length of 454 bits;
a field size p=2 454 +(3×17×11287) 2 ;
an order q=2 452 +(7×41117) 2 ;
r=(q−1)/8; and
s=(q+1)/6; and
select a private key for the first computing device;
compute a public key for the first computing device from curve parameters of the curve and the private key for the first computing device;
transmit the curve parameters of the curve and the public key for the first computing device to the second computing device;
receive a public key for the second computing device;
compute a shared secret based on the public key for the second computing device and the private key for the first computing device; and
communicate with the second computing device using the shared secret.