Patent•
Prime number generation apparatus B-smoothness judgement apparatus and computer memory product
10 Mar 1998-
TL;DR: One or a plurality of prime numbers pi which are generated and a generated random number are used to calculate a larger prime number candidate, and when the judgment is made that the candidate is a prime number, the prime number p is outputted.
Abstract: One or a plurality of prime numbers pi which are generated and a generated random number are used to calculate a larger prime number candidate, and a judgment is made as to whether or not the prime number candidate is a prime number by using a provable prime number judging method, and when the judgment is made that the candidate is a prime number, the prime number p is outputted. As for at least three polynomials F(p) which are factors of ps−1 (s: arbitrary natural number) by a prime number p, a measure against prime factorization is taken. Moreover, when the prime number p is used for a secret key of RSA cryptosystem, a strong prime number p against the iterated-encryption attack on RSA cryptosystem is generated.
Citations
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Patent•
26 Mar 2001
TL;DR: In this paper, a process for searching in parallel for a plurality of prime number values simultaneously includes the steps of: randomly generating k random odd numbers (wherein k is preferably more than 2, but could also be one or more) expressed as n 0,0, n 1,0,... n ((k−1)),0, each number providing a prime number candidate; determining a plurality y additional odd numbers based on each one of the randomly generated odd numbers n 0.
Abstract: A process is provided for searching in parallel for a plurality of prime number values simultaneously includes the steps of: randomly generating a plurality of k random odd numbers (wherein k is preferably more than 2, but could also be one or more) expressed as n 0,0 , n 1,0 , . . . n ((k−1)),0 , each number providing a prime number candidate; determining a plurality of y additional odd numbers based on each one of the randomly generated odd numbers n 0,0 , n 1,0 , . . . n (k−1),0 to provide additional prime number candidates thereby yielding a total number of prime number candidates; sieving the total number of prime number candidates by performing a small divisor test on each of the candidates in order to eliminate candidates revealed to be composite numbers by the small divisor test thereby yielding a sieved number s of candidates; and performing a first probabilistic primality test on each of the sieved number s of candidates, each of the plurality of s first primality tests including an associated exponentiation operation executed by an associated one of a plurality of s of the exponentiation units, the exponentiation operations being performed by the plurality of s exponentiation units substantially simultaneously in order to eliminate candidates revealed to be composite numbers by the primality test thereby yielding a remaining number r of candidates.
45 citations
Patent•
28 Sep 1999TL;DR: In this paper, a method and apparatus for testing processes in a computer system is described, where the selected test points are based on prime numbers and varied for successive execution iterations, and an efficient distribution of evaluated test points is achieved, with all possible test points being evaluated within a small number of execution iterations.
Abstract: A method and apparatus for testing processes in a computer system are described. In a software process, there exist many test points in the execution of the process where stress testing may be applied. The process is executed with stress testing applied at selected test points and test intervals. The selected test points are based on prime numbers and varied for successive execution iterations. An efficient distribution of evaluated test points is achieved, and all possible test points are ultimately evaluated within a small number of execution iterations. In one embodiment, the total number of test points is first determined. A first execution run is evaluated at selected test points that correspond to prime numbers greater than the square root of the total number of test points. Subsequent execution iterations are then performed evaluating test points at selected test intervals, where the test intervals for respective execution iterations correspond to prime numbers less than or equal to the square root of the total number of test points. The prime numbers for the selected test intervals are chosen in decreasing order, for example.
44 citations
Patent•
21 Jun 2002TL;DR: In this paper, the authors present a program that provides fast generation and testing of probable prime numbers for cryptographic applications, including a smart increment program that finds successive candidates using a table of congruent values that are relatively prime to a selected set of very small primes.
Abstract: A computer program provides fast generation and testing of probable prime numbers for cryptographic applications. The program instructions executed on computer hardware execute steps that include a smart increment program function that finds successive candidates using a table of congruent values that are relatively prime to a selected set of very small primes do identify an increment to the next candidate, thereby sieving out about ¾ths of the really obvious components that don't need to be subjected to trial division. The program instructions also include a small primes testing program function that speeds trial division against a list of small primes by carrying out the division on modular reduced values rather than the very large candidates themselves. Only the about 10% of the candidates that pass the small primes test will then be subjected to more rigorous, but time consuming, probable primality tests.
28 citations
Patent•
16 Sep 2003
TL;DR: In this paper, a system and method of actuating a remote control access system in a motor vehicle non-invasively detects the occurrence of an event involving an actuation of at least one component of a vehicle.
Abstract: A system and method of actuating a remote control access system in a motor vehicle non-invasively detects the occurrence of an event involving an actuation of at least one component of a motor vehicle. A control signal is transmitted to a remote control access system as a result of detecting the event.
22 citations
Patent•
21 Dec 2004TL;DR: In this article, the authors propose a prime calculating apparatus that stores a known prime q and management information unique in the use range of primes, which can be used to calculate prime candidates from unique management information.
Abstract: The present invention offers a prime calculating apparatus for achieving prime calculation where producing identical primes is avoided by simple management techniques. The prime calculating apparatus stores a known prime q and management information unique in the use range of primes. The prime calculating apparatus reads the management information; generates random information R based on the read management information; reads prime q; calculates prime candidate N, according to N=2×random information R×prime q+1, using the read prime q and generated random information R; tests whether the calculated prime candidate N is a prime; and outputs the calculated prime candidate N as a prime when the primality of the calculated prime candidate N is determined. Herewith, the prime calculating apparatus is able to calculate prime candidates from unique management information while avoiding producing identical primes.
17 citations
References
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Book•
01 Jan 1996TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.
Abstract: From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.
13,597 citations
Patent•
15 Dec 1980
TL;DR: A public-key data encryption system employing RSA public key data encryption including a message encrypter capable of encrypting messages using a non-secret encryption key, and an encryption-decryption key generator, including a microprocessor or other large-scale integrated circuit or circuits formed to generate a sequence of prime numbers as discussed by the authors.
Abstract: A public-key data encryption system employing RSA public-key data encryption including a message encrypter capable of encrypting messages using a non-secret encryption key, a transmitter-receiver coupled to the message encrypter which transmits or receives an encrypted message to or from a remote location, the transmitter-receiver also being coupled to a decrypter capable of decrypting a received encrypted message using a decryption key which is a secret input to the decrypter, and an encryption-decryption key generator, including a microprocessor or other large-scale integrated circuit or circuits formed to generate a sequence of prime numbers beginning with a selected known prime number having a length relatively short with respect to the desired length of the last in the sequence of prime numbers, and which is constructed to form the sequence of prime numbers in the form hP+1 where P is the preceding prime number in the sequence, and to test hP+1 for primality by first determining if hP+1 has a GCD of 1 with x, wherein x is a composite number consisting of the product of all known prime numbers less than or equal to a pre-selected known prime number and if the GCD is not equal to 1, incrementing h to form a new hP+1 to be tested for a GCD equal to 1, and when a GCD is found to be 1, performing the primality tests to determine whether 2 hP ≡1 [mod (hP+1)] and 2 h ≢1 [mod (hP+1)], and if either 2 hP ≢1 [mod (hP+1)] or 2 h ≡1 [mod (hP+1)] further incrementing h and so on until a prime is found in this manner and then determining if the length of the prime number is of or greater than the desired length. If the hP+1 which has been determined to be prime is not of the desired length, hP+1 is placed in the sequence of prime numbers and a new h selected to be used to find the next prime number in the sequence in accordance with the above described procedure by forming a new hP+1 in which P is the previously determined prime number in the sequence of prime numbers. When a prime number in the sequence of prime numbers is found which is of the desired length it is input into the encryption-decryption key generator for generating the RSA public-key encryption and decryption keys.
124 citations
TL;DR: A very efficient recursive algorithm for generating nearly random provable primes and a detailed analysis of the iterated encryption attack on the RSA public-key cryptosystem is presented.
Abstract: A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the Miller-Rabin test for only one base. Therefore our algorithm is even faster than algorithms presently used for generating only pseudoprimes because several Miller-Rabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval.
Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA public-key cryptosystem is presented. The prime-generation algorithm can easily be modified to generate nearly random primes or RSA-moduli that satisfy these security constraints. Further results described in this paper include an analysis of the optimal upper bound for trial division in the Miller-Rabin test as well as an analysis of the distribution of the number of bits of the smaller prime factor of a random k-bit RSA-modulus, given a security bound on the size of the two primes.
94 citations
Patent•
31 May 1984TL;DR: In an encryption scheme based on the use of a public key having secret factors p and q, additional requirements on p and Q are invoked in order to ensure a high level of security as mentioned in this paper.
Abstract: In an encryption scheme based on the use of a public key having secret factors p and q, additional requirements on p and q are invoked in order to ensure a high level of security. In particular, it is additionally required that a value p+1 have a large prime factor r and that the value r-1 also have a large prime factor r'.
63 citations
01 Nov 1990
TL;DR: This method exploits a well-known result due to Pocklington that allows one to prove the primality of p when only a partial factorization of p − 1 is known, and is used to generate an RSA-modulus m = pq that is nearly uniformly distributed over all secure RSA- moduli in a given interval I.
Abstract: This paper describes a new method for generating primes together with a proof of their primality that is extremely efficient (for 100-digit primes the average running time is equal to the average time required for finding a “strong pseudoprime” of the same size that passes the Miller-Rabin test for only four bases), that yields primes that are nearly uniformly distributed over the set of all primes in a given interval, and that is easily modified to yield (with no additional computational effort) primes that are nearly uniformly distributed over the subset of these primes that satisfy certain security constraints for use in the RSA cryptosystem. This method is used to generate, for a given encryption exponent e, an RSA-modulus m = pq that is nearly uniformly distributed over all secure RSA-moduli in a given interval I, i.e., over the set of all integers in I that are (1) the product of exactly two primes p and q none of which is smaller than a given limit L, where (2) (p − 1, e) = (q − 1, e) = 1 and (3) p − 1 and q − 1 each contain a prime factor greater than a given limit L′, and where (4) for all but a provably (given) small fraction of plaintexts in Z*m, the minimum number of iterated encryptions with exponent e required to recover the plaintext, is provably greater than a given limit M. Our method exploits a well-known result due to Pocklington [20] that allows one to prove the primality of p when only a partial factorization of p − 1 is known. These prime factors of p − 1 are generated by recursive application of the prime generating procedure. Although the discussion is centered on the RSA system, our method can of course be used in other cryptographic systems, such as the Diffie-Hellman public key distribution system, that require large primes satisfying certain security constraints.
41 citations