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.sub.0(R.sub.0.sup.−1 N′.sub.0 mod S)+P′+f.sub.λ(R.sub.n.sup.−1 N′.sub.n mod S))/(h.sub.0(R.sub.0.sup.−1 N′.sub.0 mod S)+Q′+h.sub.λ(R.sub.n.sup.−1 N′.sub.n mod S))]*h(x)−f(x)=0 mod p, where (i) P′, Q′, N′.sub.0, and N′.sub.n correspond to the ciphers in the received message; (ii) R.sub.0, R.sub.n and S are data elements of the private key; (iii) f(.Math.) is a polynomial function defined by coefficients f.sub.0, f.sub.1, . . . f.sub.λ that are also data elements of the private key; and (iv) h( ) is a polynomial function defined by coefficients h.sub.0, h.sub.1, . . . h.sub.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.

Some embodiments are directed to a second cryptographic device (20) and a first cryptographic device (10). The first and second cryptographic devices may be configured to transfer a key seed. The key seed may be protected using a public key from one party and a private key from the other party. For example, a public key may be obtained from a private key through a noisy multiplication. At least one of the first and second cryptographic device may validate an obtained public key, e.g., to avoid leakage of the key seed or of a private key.

Various embodiments relate to a method for multiplying a first and a second polynomial in a ring .sub.q[X]/(X.sup.n+1) to perform a cryptographic operation in a data processing system where q is a positive integer, the method for use in a processor of the data processing system, comprising: receiving the first polynomial and the second polynomial by the processor; mapping the first polynomial into k smaller third polynomials over k smaller rings based upon primitive roots of unity, where k is a positive integer; mapping the second polynomial into k smaller fourth polynomials over the k smaller rings based upon primitive roots of unity; applying an isomorphism to the k third polynomials resulting in k fifth polynomials; applying the isomorphism to the k fourth polynomials resulting in k sixth polynomials; applying a Kronecker substitution on the k fifth polynomials and the k sixth polynomials and perform the multiplication of the k fifth polynomials and the k sixth polynomials to produce a multiplication result; applying an inverse of the isomorphism to the multiplication result to obtain the multiplication of the first polynomial and the second polynomial; and mapping the k inverted polynomials to a single polynomial in the ring mapping the k inverted polynomials to a single polynomial in the ring .sub.q[X]/(X.sup.n+1.

Provided are a random number generation device and the like capable of calculating a high precision random number using a memory capacity selected irrespective of the precision of the random number. A random number calculation device is configured to generate first random numbers based on given number and specify, for the given number of second random numbers in a target numeric extent, bin range depending on the first random numbers based on frequency information representing cumulative frequency regarding a frequency of numeric extent including respective second random numbers among given numeric extents, the numeric extent being determined in accordance with a desirable precision.

An electronic watermark embedding apparatus according to an embodiment is an electronic watermark embedding apparatus capable of embedding an electronic watermark into a decoding circuit of secret-key encryption, and includes an embedding unit configured to generate the decoding circuit. The decoding circuit is embedded with the electronic watermark by being input with a common parameter generated in a setup of the secret-key encryption, a secret key of the secret-key encryption, and the electronic watermark, and is capable of decoding an encrypted text encrypted using the secret-key encryption.

Systems, apparatuses, methods, and computer program products are disclosed for post-quantum cryptography (PQC). An example method includes transmitting a first portion of an electronic communication to a client device over a non-PQC communications channel. The example method further includes transmitting a second portion of the electronic communication to the client device over a PQC communications channel. In some instances, the first portion of the electronic communication may comprise overhead data, and the second portion of the electronic communication may comprise payload data.

A multivariate signature method for resisting key recovery attack, which establishes a new signature verification condition by adding additional value of signature. The verification condition implies verification of internal information x and y, thereby effectively resisting key recovery attack generated by the existence of equivalence key. Specifically, the method includes the three stages of data preprocessing, signature generation and signature verification. The invention is a signature authentication method based on polynomial equations of a plurality of variables in a finite field, which can effectively resist the key recovery attack, provide the basic technical support for the information security and the establishment of the trust system in the quantum computer era, and provide a secure digital signature option in the quantum era. The present invention is especially suitable for use under application condition which has limited storage and processing time, such as smart cards, wireless sensor networks and dynamic RFID tags.

Some embodiments relate to a first electronic network node is provided (110) configured for a cryptographic operation. The first network node is configured to receive as input a difficulty parameter (d), and a structure parameter (n), and to obtain a shared matrix (A), the shared matrix being shared a second network node through a communication interface, entries in the shared matrix (A) being selected modulo a first modulus (q), the shared matrix (A) being a square matrix (k×k) of dimension (k) equal to the difficulty parameter (d) divided by the structure parameter (n), the entries in the shared matrix (A) being polynomials modulo a reduction polynomial (ƒ) of degree equal to the structure parameter (n), said cryptographic operation using the shared matrix.

Bitcoins and the underlying blockchain technology are one of the main innovations in building decentralized applications. The effects of quantum computing on this technology are analyzed in general. Provided herein are effective solutions to address security vulnerabilities in a blockchain-based system that can be exploited by a quantum attacker.

Establishing secure communications by sending a server certificate message, the certificate message including a first certificate associated with a first encryption algorithm and a second certificate associated with a second encryption algorithm, the first certificate and second certificate bound to each other, signing a first message associated with client-server communications using a first private key, the first private key associated with the first certificate, signing a second message associated with the client-server communications using a second private key, the second private key associated with the second certificate, the second message including the signed first message, and sending a server certificate verify message, the server certificate verify message comprising the signed first message and the signed second message.