A cryptographic decision-making of set membership is a method or system which make a secure decision-making for positive membership e∈S or negative membership e.Math.S in an unforgeable and non-repudiation way for any element e and a set S. The proposed method of the present invention comprises: acquire a set U={e.sub.1, . . . , e.sub.n} and map each element e.sub.i in U into a random point v.sub.i in a cryptography space; acquire a set S={e′.sub.1, . . . , e′.sub.m}⊂U, determine a random point v′.sub.i corresponding to each element e′.sub.i in the set S, and construct a function ƒ.sub.S(x) according to all random points v′.sub.i; introduce a random secret γ to generate ƒ.sub.S(γ) by using the function ƒ.sub.S(x), and produce a public parameter mpk according to the random secret γ; and generate the cryptographic representation of set S by using the function ƒ.sub.S(γ) and the public parameter mpk. In the embodiments, we provide two kinds of cryptographic representations of set, including Poles-based Aggregation and Zeros-based Aggregation, to make the decision on positive membership e.sub.i∈S and negative membership e.sub.i.Math.S.

A device may select a first pseudorandom integer within a range of integers. The device may generate a first candidate prime, based on the first pseudorandom integer, for primality testing. Based on determining that the first candidate prime fails a primality test, the device may select a second pseudorandom integer within the range of integers. The device may generate a second candidate prime, based on the second pseudorandom integer, for primality testing. The device may determine whether the second candidate prime satisfies the primality test. The device may selectively: re-perform, based on the second candidate prime failing the primality test, the selecting the second pseudorandom integer, the generating the second candidate prime, and the determining whether the second candidate prime satisfies the primality test, or using, based on the second candidate prime satisfying the primality test, the second candidate prime as a prime integer in a cryptographic protocol.

Disclosed are systems and techniques for enhanced protection of cryptographic key generation in cryptographic applications. In particular, described is a method and a system that performs the method of obtaining input numbers associated with a cryptographic application, generating masking matrix based on at least one random value, obtaining masked numbers using a matrix product of the MM and the input numbers, determining a greatest common divisor (GCD) of the masked numbers, identifying a GCD of the input numbers, and using the identified GCD to generate a cryptographic key.

Disclosed are methods and systems to provide distributed computation within a Fully Homomorphic Encryption (FHE) system by using g-adic properties to separate a ciphertext into multiple ciphertexts for each Hensel digit level. A number t of computation units may individually perform addition and/or multiplication of each Hensel digit level on each of the computation units and then reconstruct the resulting value from the result ciphertext of each computation unit using p-adic and g-adic operations. Accordingly, computation burdens may be distributed to several computation units.

The present invention relates to a method to generate prime numbers on board a portable device, said method comprising the steps of, each time at least one prime number is requested: when available, retrieve results from previously performed derivation calculation or, if not, select a start point for derivation; process derivation calculation to converge towards a prime number; if a prime number is found, store it and restart derivation calculation from a new start point; stop the derivation calculation after a predetermined amount of time; store intermediate results to be used a next time a prime number will be requested; output a stored prime number.

One feature pertains to a method for generating a prime number by repeatedly generating a random number seed S having k bits, generating a random number R having n bits based on the seed S, where k is less than n, and determining whether the random number R is prime. The steps are repeated until it is determined that the random number R generated is prime, upon which the random number seed S used to generate the random number R is stored in a memory circuit. Later, the stored random number seed S may be retrieved from the memory circuit, and the prime number is regenerated based on the random number seed S. In one example, the random number R generated is further based on a secret key k.sub.S that may be stored in a secure memory circuit.

Methods and systems for secure authentication in an extended reality (XR) environment are described herein. An XR environment may be output by a computing device and for display on a device configured to be worn by a user. A first plurality of images may be determined via the XR environment. The first plurality of images may be determined based on a user looking at a plurality of objects, real or virtual, in the XR environment. The first plurality of images may be sent to a server, and the server may return a second plurality of images. A public key and private key may be determined based on different portions of each of the second plurality of images. The public key may be sent to the server to register and/or authenticate subsequent communications between the computing device and the server.

Methods and systems for secure authentication in an extended reality (XR) environment are described herein. An XR environment may be output by a computing device and for display on a device configured to be worn by a user. A first plurality of images may be determined via the XR environment. The first plurality of images may be determined based on a user looking at a plurality of objects, real or virtual, in the XR environment. The first plurality of images may be sent to a server, and the server may return a second plurality of images. A public key and private key may be determined based on different portions of each of the second plurality of images. The public key may be sent to the server to register and/or authenticate subsequent communications between the computing device and the server.

Disclosed herein is a method for generating a composite cryptographic signature. The method comprises receiving a message and a first part of a first party signature, wherein the first part of the first party signature is derived from the message and a first share of a first private key. The method further comprises generating a first party signature from the first part of the first party signature and a second share of the first private key and generating a second party signature from the message and a second private key. The method further comprises combining the first party signature and the second party signature to generate a composite cryptographic signature. An apparatus, a computer-readable medium for implementing this method are also disclosed.

An electronic generation device (100) arranged to generate parameters for digital obfuscated arithmetic is provided. The generation device includes a prime number unit (110) arranged to generate a prime modulus (p) a base element unit (120) arranged to generate a prime modulus and a base element such that each ring-element modulo the prime modulus may be expressed as a difference between two powers of the potential base element.