Quantum-safe cryptographic method and system
A cryptographic method and system. A plurality of ciphers is identified in a message received by a recipient, such message encrypting a digital asset. A private key associated with the recipient is obtained. The private key corresponds to a public key associated with the recipient. The method includes solving for x in the equation: [(f 0 (R 0 −1 N ′ 0 mod S)+ P ′+f λ (R n −1 N ′ n mod S))/(h 0 (R 0 −1 N ′ 0 mod S)+ Q ′+h λ (R n −1 N ′ n mod S))]*h(x)−f(x)=0 mod p, where (i) P ′, Q ′, N ′ 0 , and N ′ n correspond to the ciphers in the received message; (ii) R 0 , R n and S are data elements of the private key; (iii) f(·) is a polynomial function defined by coefficients f 0 , f 1 , . . . f λ that are also data elements of the private key; and (iv) h(·) is a polynomial function defined by coefficients h 0 , h 1 , . . . h λ that are also data elements of the private key. The value of x is assigned to the digital asset, which is then stored in non-transitory memory or packaged in a message sent over the data network.
1. A method of operating a first computing apparatus of a recipient for cryptographic communication with a second computing apparatus, the method comprising:
receiving a message from the second computing apparatus;
identifying a plurality of ciphers in the received message;
retrieving from a memory of the first computing apparatus a private cryptographic key associated with the recipient, the private cryptographic key corresponding to a public cryptographic key associated with the recipient, the private cryptographic key comprising a plurality of private cryptographic key data elements;
solving for x in an equation:
[( f 0 ( R 0 −1 N ′ 0 mod S )+ P ′+f λ ( R n −1 N ′ n mod S ))/( h 0 ( R 0 −1 N ′ 0 mod S )+ Q ′+h λ ( R n −1 N ′ n mod S ))]* h ( x )− f ( x )=0 mod p , where:
p is a predetermined integer stored in the memory of the first computing apparatus;
P ′, Q ′, N ′ 0 , and N ′ n correspond to the ciphers in the received message;
R 0 , R n and S are included in the private cryptographic key data elements;
f(·) is a first polynomial function defined by a set of coefficients f 0 , f 1 , . . . f λ included in the private cryptographic key data elements, with λ being a predetermined integer; and
h(·) is a second polynomial function defined by a set of coefficients h 0 , h 1 , . . . h λ included in the private cryptographic key data elements;
assigning the value of x to a digital asset; and
storing the digital asset in non-transitory memory or packaging the digital asset in a message sent over a data network.
2. The method defined in claim 1 , further comprising receiving the private cryptographic key from a key generation server over an out-of-bank link.
3. The method defined in claim 1 , further comprising receiving the private cryptographic key from a key generation server not over the data network.
4. The method defined in claim 1 , further comprising generating the public cryptographic key and the private cryptographic key.
5. The method defined in claim 4 , further comprising withholding the private cryptographic key from a sender of the message.
6. The method defined in claim 4 , further comprising sending the public cryptographic key to a sender of the message over the data network.
7. The method defined in claim 1 , wherein λ is selected to be no greater than 3.
8. The method defined in claim 1 , wherein when solving for x yields a plurality of candidate solutions, the method further comprises selecting one of the candidate solutions as the digital asset.
9. The method defined in claim 8 , wherein selecting one of the candidate solutions as the digital asset comprises:
comparing a portion of each of the candidate solutions against a predetermined flag or code and selecting as the digital asset the candidate solution for which the compared portion matches the predetermined flag or code; or
performing a checksum on each candidate solution and comparing the checksum to a received checksum associated with each candidate solution and selecting as the digital asset the candidate solution for which the checksums are matching.
10. The method defined in claim 1 , wherein the public cryptographic key is formed from:
a. a first set of public key coefficients, being the coefficients of a first product polynomial in x obtained as the product of (i) a base polynomial in x of order n and m noise variables and (ii) a first entanglement polynomial f(x), minus the coefficient of x 0 and the coefficient of x n+λ , with m and n being predetermined integers stored in the memory of the first computing apparatus;
b. a second set of public key coefficients, being the coefficients of a second product polynomial in x obtained as the product of (i) said base polynomial in x of order n and said m noise variables and (ii) a second entanglement polynomial h(x), minus the coefficient of x 0 and the coefficient of x n+λ ;
c. a third set of public key coefficients, being the coefficients of a first linear combination of the noise variables, each multiplied by R 0 and then mod S; and
d. a fourth set of public key coefficients, being the coefficients of a second linear combination of the noise variables, each multiplied by R n and then mod S.
11. The method defined in claim 10 , wherein successful decryption of the digital asset occurs if the public cryptographic key was used by a sender to encrypt the digital asset by:
a. computing the cipher P ′ as the result of using the first set of public key coefficients as the coefficients of a first polynomial of the digital asset;
b. computing the cipher Q ′ as the result of using the second set of public key coefficients as the coefficients of a second polynomial of the digital asset;
c. computing the cipher N ′ 0 as the result of using the third set of public key coefficients as the coefficients of the terms of a linear combination of the noise variables with values selected by the sender; and
d. computing the cipher N ′ n as the result of using the fourth set of public key coefficients as the coefficients of the terms of a linear combination of said noise variables multiplied by the digital asset to the power n+λ.
12. The method defined in claim 10 , wherein p is selected to be at least as great as 2 6 , wherein n is selected to be at least as great as 3, or wherein m is selected to be at least as great as 2.
13. The method defined in claim 10 , wherein p is selected to be at least as great as 2 6 , wherein n is selected to be equal to 1, and wherein m is selected to be equal to 1.
14. The method defined in claim 1 , wherein p is represented by X bits and is selected to be the largest prime number less than 2 X .
15. A non-transitory computer-readable storage medium comprising computer-readable instructions which, when executed by a processing entity of a first computing apparatus, cause the first computing apparatus to carry out operations for cryptographic communication with a second computing apparatus, the operations including:
receiving a message from the second computing apparatus;
identifying a plurality of ciphers in the received message;
retrieving from a memory of the first computing apparatus a private cryptographic key associated with a recipient, the private cryptographic key corresponding to a public cryptographic key associated with the recipient, the private cryptographic key comprising a plurality of private cryptographic key data elements;
solving for x in an equation:
[( f 0 ( R 0 −1 N ′ 0 mod S )+ P ′+f λ ( R n −1 N ′ n mod S ))/( h 0 ( R 0 −1 N ′ 0 mod S )+ Q ′+h λ ( R n −1 N ′ n mod S ))]* h ( x )− f ( x )=0 mod p , where:
p is a predetermined integer stored in the memory of the first computing apparatus;
P ′, Q ′, N ′ 0 , and N ′ n correspond to the ciphers in the received message;
R 0 , R n , S p and S q are included in the private cryptographic key data elements;
f(·) is a first polynomial function defined by a set of coefficients f 0 , f 1 , . . . f λ included in the private cryptographic key data elements; and
h(·) is a second polynomial function defined by a set of coefficients h 0 , h 1 , . . . h λ included in the private cryptographic key data elements;
assigning the value of x to a digital asset; and
storing the digital asset in non-transitory memory or packaging the digital asset in a message sent over a data network.
16. A method of operating a first computing apparatus of a recipient server for cryptographic communication with a second computing apparatus, the method comprising:
receiving a message from the second computing apparatus;
identifying a plurality of ciphers in the received message;
retrieving from a memory of the first computing apparatus a private cryptographic key associated with the recipient, the private cryptographic key corresponding to a public cryptographic key associated with the recipient, the private cryptographic key comprising a plurality of private cryptographic key data elements;
solving for x in an equation:
[( R p −1 P ′ mod S p )/( R q −1 Q ′ mod S q )]* h ( x )− f ( x )=0 mod p , where:
p is a predetermined integer stored in the memory of the first computing apparatus;
P ′ and Q ′ correspond to the ciphers in the received message;
R p , R q , S p and S q are included in the private cryptographic key data elements;
f(·) is a first polynomial function defined by a set of coefficients f 0 , f 1 , . . . f λ included in the private cryptographic key data elements; and
h(·) is a second polynomial function defined by a set of coefficients h 0 , h 1 , . . . h λ included in the private cryptographic key data elements;
assigning the value of x to a digital asset; and
storing the digital asset in non-transitory memory or packaging the digital asset in a message sent over a data network.
17. The method defined in claim 16 , further comprising receiving the private cryptographic key from a key generation server over an out-of-bank link.
18. The method defined in claim 16 , further comprising receiving the private cryptographic key from a key generation server not over the data network.
19. The method defined in claim 16 , further comprising generating the public cryptographic key and the private cryptographic key.
20. The method defined in claim 19 , further comprising withholding the private cryptographic key from a sender of the message.
21. The method defined in claim 19 , further comprising sending the public cryptographic key to a sender of the message over the data network.
22. The method defined in claim 16 , wherein λ is selected to be no greater than 3.
23. The method defined in claim 16 , wherein when solving for x yields a plurality of candidate solutions, the method further comprises selecting one of the candidate solutions as the digital asset.
24. The method defined in claim 16 , wherein S p and S q are selected to be identical.
25. The method defined in claim 16 , wherein the public cryptographic key is formed from:
a. a first set of public key coefficients, being the coefficients of a first product polynomial in x obtained as the product of (i) a base polynomial in x of order n and m noise variables and (ii) a first entanglement polynomial f(x), minus the resulting coefficient of x 0 and the resulting coefficient of x n+λ , each remaining coefficient multiplied by R p and then mod S p , with m and n being predetermined integers stored in the memory of the first computing apparatus; and
b. a second set of public key coefficients, being the coefficients of a second product polynomial in x obtained as the product of (i) said base polynomial in x of order n and said m noise variables and (ii) a second entanglement polynomial h(x), minus the resulting coefficient of x 0 and the resulting coefficient of x n+λ , each remaining coefficient multiplied by R q and then mod S q .
26. The method defined in claim 25 , wherein successful decryption of the digital asset occurs if the public cryptographic key was used by a sender to encrypt the digital asset by:
a. computing the cipher P ′ as the result of using the first set of public key coefficients as the coefficients of a first polynomial of the digital asset; and
b. computing the cipher Q ′ as the result of using the second set of public key coefficients as the coefficients of a second polynomial of the digital asset.
27. The method defined in claim 25 , wherein p is selected to be at least as great as 2 32 , wherein n is selected to be at least as great as 1, and wherein and m is selected to be at least as great as 1.
28. The method defined in claim 25 , wherein p is selected to be at least as great as 2 32 , wherein n is selected to be at least as great as 1, and wherein and m is selected to be at least as great as 2.
29. A non-transitory computer-readable storage medium comprising computer-readable instructions which, when executed by a processing entity of a first computing apparatus, cause the computing apparatus to carry out operations for cryptographic communication with a second computing apparatus, the operations including:
receiving a message from the second computing apparatus;
identifying a plurality of ciphers in the received message;
retrieving from a memory of the first computing apparatus a private cryptographic key associated with a recipient, the private cryptographic key corresponding to a public cryptographic key associated with the recipient, the private cryptographic key comprising a plurality of private cryptographic key data elements;
solving for x in an equation:
[( R p −1 P ′ mod S p )/( R q −1 Q ′ mod S q )]* h ( x )− f ( x )=0 mod p , where:
p is a predetermined integer stored in the memory of the first computing apparatus;
P ′ and Q ′ correspond to the ciphers in the received message;
R p , R q , S p and S q are included in the private cryptographic key data elements;
f(·) is a first polynomial function defined by a set of coefficients f 0 , f 1 , . . . f λ included in the private cryptographic key data elements; and
h(·) is a second polynomial function defined by a set of coefficients h 0 , h 1 , . . . h λ included in the private cryptographic key data elements;
assigning the value of x to a digital asset; and
storing the digital asset in non-transitory memory or packaging the digital asset in a message sent over a data network.
30. A method of operating an encryption server of a first computing apparatus to encrypt a digital asset for transmission to a second computing apparatus over a data network, the method comprising:
identifying a public key associated with a recipient, the public key comprising a first, a second, a third and a fourth set of public key coefficients, the public key being part of a public-private key pair associated with the recipient;
selecting m noise variables, wherein m is an integer greater than or equal to 1;
creating a plurality of ciphers based on the public key, the digital asset and the noise variables, wherein a ciphertext comprises:
a first cipher computed as the result of using the first set of public key coefficients as the coefficients of a first polynomial of the digital asset;
a second cipher computed as the result of using the second set of public key coefficients as the coefficients of a second polynomial of the digital asset;
a third cipher computed as the result of using the third set of public key coefficients as the coefficients of terms of a linear combination of the noise variables; and
a fourth cipher computed as the result of using the fourth set of public key coefficients as the coefficients of the terms of a linear combination of the noise variables multiplied by the digital asset to the power n+λ, where n and λ are integers at least as great as one; and
sending a message comprising the plurality of ciphers over the data network towards the recipient server in possession of the private key.