Elliptic curve random number generation
An elliptic curve random number generator avoids escrow keys by choosing a point Q on the elliptic curve as verifiably random. An arbitrary string is chosen and a hash of that string computed. The hash is then converted to a field element of the desired field, the field element regarded as the x-coordinate of a point Q on the elliptic curve and the x-coordinate is tested for validity on the desired elliptic curve. If valid, the x-coordinate is decompressed to the point Q, wherein the choice of which is the two points is also derived from the hash value. Intentional use of escrow keys can provide for back up functionality. The relationship between P and Q is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to reconstruct the random number with the escrow key.
1. A computer-implemented method of establishing an escrow key for a security domain within a network, comprising:
establishing a pair of points (P, Q) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=eQ, and e is a relationship between the pair of points (P, Q);
storing the relationship e as an escrow key;
generating from the ECRNG a random number for use in cryptographic operations within the security domain;
using the random number in a cryptographic operation; and
logging an output of the ECRNG to reconstruct the random number with the escrow key.
2. The computer-implemented method of claim 1 , wherein the pair of points (P, Q) is a pair of elliptic curve points.
3. The computer-implemented method of claim 1 , wherein one point of the pair of points (P, Q) is obtained from an output of a one way function.
4. The computer-implemented method of claim 3 , wherein the one way function is a hash function.
5. The computer-implemented method of claim 3 , wherein the other point of the pair of points (P, Q) is utilized as an input to the one way function.
6. The computer-implemented method of claim 5 , wherein the one point of the pair of points (P, Q) is Q, and the other point of the pair of points (P, Q) is P.
7. A device, comprising:
a memory; and
at least one processor communicatively coupled with the memory and configured to:
establish a pair of points (P, Q) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=eQ, and e is a relationship between the pair of points (P, Q);
store the relationship e as an escrow key;
generate from the ECRNG a random number for use in cryptographic operations within a security domain;
using the random number in a cryptographic operation; and
logging an output of the ECRNG to reconstruct the random number with the escrow key.
8. The device of claim 7 , wherein the pair of points (P, Q) is a pair of elliptic curve points.
9. The device of claim 7 , wherein one point of the pair of points (P, Q) is obtained from an output of a one way function.
10. The device of claim 9 , wherein the one way function is a hash function.
11. The device of claim 9 , wherein the other point of the pair of points (P, Q) is utilized as an input to the one way function.
12. The device of claim 11 , wherein the one point of the pair of points (P, Q) is Q, and the other point of the pair of points (P, Q) is P.
13. A non-transitory computer readable medium storing instructions which, when executed, cause a computing device to perform operations comprising:
establishing a pair of points (P, Q) as respective inputs to an elliptic curve random number generator (ECRNG), wherein P=eQ, and e is a relationship between the pair of points (P, Q);
storing the relationship e as an escrow key;
generating from the ECRNG a random number for use in cryptographic operations within a security domain;
using the random number in a cryptographic operation; and
logging an output of the ECRNG to reconstruct the random number with the escrow key.
14. The non-transitory computer readable medium of claim 13 , wherein the pair of points (P, Q) is a pair of elliptic curve points.
15. The non-transitory computer readable medium of claim 13 , wherein one point of the pair of points (P, Q) is obtained from an output of a one way function.
16. The non-transitory computer readable medium of claim 15 , wherein the one way function is a hash function.
17. The non-transitory computer readable medium of claim 15 , wherein the other point of the pair of points (P, Q) is utilized as an input to the one way function.
18. The non-transitory computer readable medium of claim 17 , wherein the one point of the pair of points (P, Q) is Q, and the other point of the pair of points (P, Q) is P.