This disclosure is directed to generating a set of data elements for more secure encryption or more resilient decryption associated with generating a target set of conditional data elements. The target set of conditional data elements may fulfill a condition. Public keys associated with an encrypted message may be associated with conditional data elements of the target set of conditional data elements. By performing at least one cycle of decryption associated with the public keys, an encrypted message may be decrypted.

In an example a method includes retrieving, from a persistent memory, a previously-identified counter value corresponding to an iteration of a prime number generation procedure that previously produced a verified prime number. The method further includes re-generating, using processing circuitry implementing a deterministic prime number calculator and with the previously-identified counter value as an input to the deterministic prime number calculator, the verified prime number.

Computing devices, methods, and systems for corrections to the “almost” binary extended GCD in a cryptographic operation of a cryptographic process are disclosed. Exemplary implementations may: receive, from a cryptographic process, a command to compute a binary extended greatest common denominator of a first input value and a second input value for a cryptographic operation; compute, by a binary extended GCD algorithm, the binary extended GCD using a multiplication with an inverse of two, instead of a division by two, to obtain a first output value; compute, by the binary extended GCD algorithm, a second output value and a third output value; and return, to the cryptographic process, the first output value, the second output value, and the third output value.

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.

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.

Aspects of the present disclosure involve a method, a system and a computer readable memory to perform a cryptographic operation that includes identifying a first set of mutually coprime numbers, obtaining a second set of input numbers coprime with a corresponding one of the first set of mutually coprime numbers, obtaining an output number that is a weighted sum of the second set of input numbers, each of the second set of input numbers being taken with a weight comprising a product of all of the first set of mutually coprime numbers except the corresponding one of the first set of mutually coprime numbers, and performing the cryptographic operation using the output number.

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.

The present invention relates to a method for generating a prime number and using it in a cryptographic application, comprising the steps of: a) determining at least one binary base B with a small size b=log.sub.2(B) bits and for each determined base B at least one small prime p.sub.i such that B mod p.sub.i=1, with i an integer, b) selecting a prime candidate Y.sub.P, c) decomposing the selected prime candidate Y.sub.P in a base B selected among said determined binary bases : Y.sub.P=Σy.sub.jB.sup.id) computing a residue y.sub.PB from the candidate Y.sub.P for said selected base such that y.sub.PB=Σ.sub.yje) testing if said computed residue y.sub.PB is divisible by one small prime pi selected among said determined small primes for said selected base B, f) while said computed residue y.sub.PB is not divisible by said selected small prime, iteratively repeating above step e) until tests performed at step e) prove that said computed residue y.sub.PB is not divisible by any of said determined small primes for said selected base B, g) when said computed residue y.sub.PB is not divisible by any of said determined small primes for said selected base B, iteratively repeating steps c) to f) for each base B among said determined binary bases, h) when, for all determined bases B, said residue y.sub.PB computed for a determined base is not divisible by any of said determined small primes for said determined base B, executing a known rigorous probable primality test on said candidate Y.sub.P, and when the known rigorous probable primality test is a success, storing said prime candidate Y.sub.P and using said stored prime candidate Y.sub.P in said cryptographic application.

Aspects of the present disclosure involve a method, a system and a computer readable memory to generate and use prime numbers in cryptographic operations by determining one or more polynomial functions that have no roots modulo each of a predefined set of prime numbers, selecting one or more input numbers, generating a candidate number by applying one or more instances of the one or more polynomial functions to the one or more input numbers, determining that the candidate number is a prime number, and using the determined prime number to decrypt an input into the cryptographic operation.

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.