Keys for elliptic curve cryptography
Cryptographic circuitry, in operation, generates N first pairs of elliptic curve cryptography (ECC) keys r(i), R(i), with i varying from 1 to N, using K second pairs of ECC keys p(k), P(k), with k varying from 1 to K, wherein K is smaller than N. Each pair r(i), R(i) of the first pairs of keys is a linear combination of pairs of the second pairs of ECC keys according to: ∀ i ∈ [ 1 ; N ] { r ( l ) = ∑ j = 1 K A ( i , j ) * p ( j ) R ( i ) = ∑ j = 1 K A ( i , j ) * P ( j ) , wherein A(i,j) designates a general term of a matrix A of size N*K, and all the sub-matrices of size K*K are invertible. The cryptographic circuitry, in operation, executes cryptographic operations using one or more pairs of the first pairs of ECC keys.
1. A method, comprising:
generating, using cryptographic circuitry, N first pairs of elliptic curve cryptography (ECC) keys r(i), R(i), with i varying from 1 to N, using K second pairs of ECC keys p(k), P(k), with k varying from 1 to K, wherein K is smaller than N, and each pair r(i), R(i) of the first pairs of keys is a linear combination of pairs of the second pairs of ECC keys according to:
∀
i
∈
[
1
;
N
]
{
r
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
p
(
j
)
R
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
P
(
j
)
,
wherein A(i,j) designates a general term of a matrix A of size N*K, and all sub-matrices of size K*K are invertible; and
executing, using the cryptographic circuitry, cryptographic operations using one or more pairs of the first pairs of ECC keys.
2. The method according to claim 1 , comprising initialization of the N first pairs of keys before the application of the linear combination.
3. The method according to claim 2 , wherein during the initialization, the N first pairs of ECC keys are set to zero and to the point at infinity, respectively.
4. The method according to claim 2 , wherein during the initialization, the N first pairs of keys are initialized to the value taken by the last generated first pair of keys.
5. The method according to claim 2 , wherein during the initialization, the N first pairs of keys (r(i), R(i)) are initialized to the value taken by one of the last W generated pair of keys (r(i−j), R(i−j)), where j<=W, j<i and j>=1.
6. The method according to claim 1 , wherein matrix A is a Vandermonde matrix.
7. The method according to claim 1 , wherein the first and second pairs of keys are formed of a scalar and of a point on an elliptic curve.
8. The method of claim 1 , wherein the cryptographic operations include:
generating a digital signature;
generating a key agreement; or
combinations thereof.
9. The method of claim 1 , comprising:
generating the K second pairs of ECC keys; and
storing the generated K pairs of ECC keys.
10. A device, comprising:
a memory; and
cryptographic circuitry coupled to the memory, wherein the cryptographic circuitry, in operation:
generates N first pairs of elliptic curve cryptography (ECC) keys r(i), R(i), with i varying from 1 to N, using K second pairs of ECC keys p(k), P(k), with k varying from 1 to K, wherein K is smaller than N, and each pair r(i), R(i) of the first pairs of keys is a linear combination of pairs of the second pairs of ECC keys according to:
∀
i
∈
[
1
;
N
]
{
r
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
p
(
j
)
R
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
P
(
j
)
,
wherein A(i,j) designates a general term of a matrix A of size N*K, and all sub-matrices of size K*K are invertible; and
executes cryptographic operations using one or more pairs of the first pairs of ECC keys.
11. The device according to claim 10 , wherein the cryptographic circuitry, in operation, initializes the N first pairs of keys before performing the linear combinations.
12. The device according to claim 11 , wherein during the initialization, the N first pairs of ECC keys are set to zero and to the point at infinity, respectively.
13. The device according to claim 11 , wherein during the initialization, the N first pairs of keys are initialized to the value taken by the last generated first pair of keys.
14. The device according to claim 11 , wherein during the initialization, the N first pairs of keys (r(i), R(i)) are initialized to the value taken by one of the last W generated pair of keys (r(i−j), R(i−j)), where j<=W, j<i and j>=1.
15. The device according to claim 10 , wherein matrix A is a Vandermonde matrix.
16. The device of claim 10 , wherein the cryptographic operations include:
generating a digital signature;
generating a key agreement; or
combinations thereof.
17. The device of claim 10 , wherein the memory, in operation, stores the generated K pairs of ECC keys.
18. A system, comprising:
functional circuitry; and
cryptographic circuitry coupled to the functional circuitry, wherein the cryptographic circuitry, in operation:
generates N first pairs of elliptic curve cryptography (ECC) keys r(i), R(i), with i varying from 1 to N, using K second pairs of ECC keys p(k), P(k), with k varying from 1 to K, wherein K is smaller than N, and each pair r(i), R(i) of the first pairs of keys is a linear combination of pairs of the second pairs of ECC keys according to:
∀
i
∈
[
1
;
N
]
{
r
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
p
(
j
)
R
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
P
(
j
)
,
wherein A(i,j) designates a general term of a matrix A of size N*K, and all sub-matrices of size K*K are invertible; and
executes cryptographic operations using one or more pairs of the first pairs of ECC keys.
19. The system according to claim 18 , wherein the cryptographic circuitry, in operation, initializes the N first pairs of keys before performing the linear combinations.
20. The system according to claim 18 , wherein matrix A is a Vandermonde matrix.
21. The system according to claim 18 , comprising a processor coupled to the functional circuitry and the cryptographic circuitry, wherein the processor, in operation, exchanges data with the cryptographic circuitry.
22. A non-transitory computer-readable medium having contents which configure cryptographic circuitry to perform a method, the method comprising:
generating N first pairs of elliptic curve cryptography (ECC) keys r(i), R(i), with i varying from 1 to N, using K second pairs of ECC keys p(k), P(k), with k varying from 1 to K, wherein K is smaller than N, and each pair r(i), R(i) of the first pairs of keys is a linear combination of pairs of the second pairs of ECC keys according to:
∀
i
∈
[
1
;
N
]
{
r
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
p
(
j
)
R
(
i
)
=
∑
j
=
1
K
A
(
i
,
j
)
⋆
P
(
j
)
,
wherein A(i,j) designates a general term of a matrix A of size N*K, and all sub-matrices of size K*K are invertible; and
executing cryptographic operations using one or more pairs of the first pairs of ECC keys.
23. The non-transitory computer-readable medium of claim 22 , wherein the method comprises initializing the N first pairs of keys before application of the linear combination.
24. The non-transitory computer-readable medium of claim 22 , wherein the contents comprise instructions executable by the cryptographic circuitry.