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 method for Cheon-resistance for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the method comprising:
receiving, at a second computing device and from a first computing device, a public key for the first computing device along with curve parameters of a curve with Cheon resistance, the curve with Cheon resistance comprising a curve chosen from a range of curves matching a threshold efficiency and excluding curves which may include intentional vulnerabilities, and being elected 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;
selecting a private key for the second computing device;
computing a public key for the second computing device based on the curve parameters and the private key for the second computing device;
transmitting the public key for the second computing device to the first computing device;
computing a shared secret based on the public key for the first computing device and the private key for the second computing device; and
communicating with the first computing device using the shared secret.
2. The method of claim 1 , wherein the curve with Cheon resistance has a length matching a threshold security level.
3. The method of claim 1 , wherein the curve with Cheon resistance has a field size close to a power of two, and having efficient endomorphism.
4. The method of claim 1 , wherein the curve with Cheon resistance excludes curves which are non-compact or difficult to generate.
5. The method 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 method of claim 5 , wherein the curve has length of 454 bits.
7. The method of claim 6 , wherein the curve has a field size p=2 454 +(3×17×11287) 2 ; order q=2 452 +(7×41117) 2 ; r=(q−1)/8; and s=(q+1)/6.
8. A computing device for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the computing device comprising a hardware processor for executing program instructions configured to:
receive, at the computing device and from a first computing device, a public key for the first computing device along with curve parameters of a curve with Cheon resistance, the curve with Cheon resistance comprising a curve chosen from a range of curves matching a threshold efficiency and excluding curves which may include intentional vulnerabilities, and being elected 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 computing device;
compute a public key for the computing device based on the curve parameters and the private key for the computing device;
transmit the public key for the computing device to the first computing device;
compute a shared secret based on the public key for the first computing device and the private key for the computing device; and
communicating with the first computing device using the shared secret.
9. The computing device of claim 8 , wherein the curve with Cheon resistance has a length matching a threshold security level.
10. The computing device of claim 8 , wherein the curve with Cheon resistance has a field size close to a power of two, and has efficient endomorphism.
11. The computing device of claim 8 , wherein the curve with Cheon resistance excludes curves that are non-compact or difficult to generate.
12. The computing device of claim 8 , wherein the curve has an affine equation in the form y 2 =x 3 +ix, where i=√{square root over (−1)}.
13. The computing device of claim 12 , wherein the curve has length of 454 bits.
14. The computing device of claim 13 , wherein the curve has a field size p=2 454 +(3×17×11287) 2 ; order q=2 452 +(7×41117) 2 ; r=(q−1)/8; and s=(q+1)/6.
15. 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 second computing device causing the second computing device to:
receive, at a second computing device and from a first computing device, a public key for the first computing device along with curve parameters of a curve with Cheon resistance, the curve with Cheon resistance comprising a curve chosen from a range of curves matching a threshold efficiency and excluding curves which may include intentional vulnerabilities, and being elected 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 second computing device;
compute a public key for the second computing device based on the curve parameters and the private key for the second computing device;
transmit the public key for the second computing device to the first computing device;
compute a shared secret based on the public key for the first computing device and the private key for the second computing device; and
communicate with the first computing device using the shared secret.
16. The non-transitory computer readable medium of claim 15 , wherein the curve with Cheon resistance has a length matching a threshold security level.
17. The non-transitory computer readable medium of claim 15 , wherein the curve with Cheon resistance has a field size close to a power of two, and having efficient endomorphism.
18. The non-transitory computer readable medium of claim 15 , wherein the curve with Cheon resistance excludes curves which are non-compact or difficult to generate.
19. The non-transitory computer readable medium of claim 15 , wherein the curve has an affine equation in the form y 2 =x 3 +ix, where i=√{square root over (−1)}.
20. The non-transitory computer readable medium of claim 19 , wherein the curve has length of 454 bits.
21. The non-transitory computer readable medium of claim 20 , wherein the curve has a field size p=2 454 +(3×17×11287) 2 .