Abstract: Cryptographic methods and systems for key exchange, digital signature and zero-knowledge proof. In the digital signature scenario, there is provided a method of signing a digital document, comprising: obtaining a private cryptographic key associated with the signer; obtaining a digital asset from the digital document; selecting a base data element; computing a plurality of signature data elements from (i) the digital asset, (ii) the base data element and (iii) the private cryptographic key; and transmitting the digital document and the plurality of signature data elements to a recipient over a data network. Provenance of the digital document is confirmable by the recipient carrying out a predefined computation involving the digital document, the signature data elements, a plurality of noise variables and a public cryptographic key corresponding to the private cryptographic key associated with the signer. In the zero-knowledge proof scenario, the digital asset plays the role of a challenge data element.

Abstract: Cryptographic methods and systems for key exchange, digital signature and zero-knowledge proof. In the digital signature scenario, there is provided a method of signing a digital document, comprising: obtaining a private cryptographic key associated with the signer; obtaining a digital asset from the digital document; selecting a base data element; computing a plurality of signature data elements from (i) the digital asset, (ii) the base data element and (iii) the private cryptographic key; and transmitting the digital document and the plurality of signature data elements to a recipient over a data network. Provenance of the digital document is confirmable by the recipient carrying out a predefined computation involving the digital document, the signature data elements, a plurality of noise variables and a public cryptographic key corresponding to the private cryptographic key associated with the signer. In the zero-knowledge proof scenario, the digital asset plays the role of a challenge data element.

Abstract: 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: [(f0(R0?1 N?0 mod S)+P?+f?(Rn?1 N?n mod S))/(h0(R0?1 N?0 mod S)+Q?+h?(Rn?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) R0, Rn and S are data elements of the private key; (iii) f(·) is a polynomial function defined by coefficients f0, f1, . . . f? that are also data elements of the private key; and (iv) h( ) is a polynomial function defined by coefficients h0, h1, . . . h? 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.

Abstract: Cryptographic methods and systems for key exchange, digital signature and zero-knowledge proof. In the digital signature scenario, there is provided a method of signing a digital document, comprising: obtaining a private cryptographic key associated with the signer; obtaining a digital asset from the digital document; selecting a base data element; computing a plurality of signature data elements from (i) the digital asset, (ii) the base data element and (iii) the private cryptographic key; and transmitting the digital document and the plurality of signature data elements to a recipient over a data network. Provenance of the digital document is confirmable by the recipient carrying out a predefined computation involving the digital document, the signature data elements, a plurality of noise variables and a public cryptographic key corresponding to the private cryptographic key associated with the signer. In the zero-knowledge proof scenario, the digital asset plays the role of a challenge data element.