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Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
Note on Integer Factoring Methods I
"... Abstract. This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations. The estimated theoretical running time complexities of the corresponding algorithms are encouraging and improve the current ones. The work is presente ..."
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Abstract. This note presents the basic mathematical structure of a new integer factorization method based on systems of linear Diophantine equations. The estimated theoretical running time complexities of the corresponding algorithms are encouraging and improve the current ones. The work is presented as a theoretical contribution to the theory of integer factorization.
Preprint, arXiv:1010.2489 PROOF OF THREE CONJECTURES ON CONGRUENCES
, 1010
"... Abstract. In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a> 1 then ⌊ 3 4 pa ⌋ k=0 ..."
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Abstract. In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a> 1 then ⌊ 3 4 pa ⌋ k=0
Preprint, arXiv:0911.3060 FIBONACCI NUMBERS MODULO CUBES OF PRIMES
, 911
"... Abstract. Let p be an odd prime. It is well known that F p p− () 5 0 (mod p) where {Fn} n�0 is the famous Fibonacci sequence and (−) is the Jacobi symbol. In this paper we show that if p ̸ = 5 then we may determine F p p−( 5) mod p3 in the following way: (p−1)/2 k=0 ..."
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Abstract. Let p be an odd prime. It is well known that F p p− () 5 0 (mod p) where {Fn} n�0 is the famous Fibonacci sequence and (−) is the Jacobi symbol. In this paper we show that if p ̸ = 5 then we may determine F p p−( 5) mod p3 in the following way: (p−1)/2 k=0
GENERALISED MERSENNE NUMBERS REVISITED
"... Abstract. Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 1862) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplica ..."
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Abstract. Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 1862) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked. Asymptotically, using a cyclic rather than a linear convolution, residue multiplication modulo a Mersenne number is twice as fast as integer multiplication; this property does not hold for prime GMNs, unless they are of Mersenne’s form. In this work we exploit an alternative generalisation of Mersenne numbers for which an analogue of the above property — and hence the same efficiency ratio — holds, even at bitlengths for which schoolbook multiplication is optimal, while also maintaining very efficient reduction. Moreover, our proposed primes are abundant at any bitlength, whereas GMNs are extremely rare. Our multiplication and reduction algorithms can also be easily parallelised, making our arithmetic particularly suitable for hardware implementation. Furthermore, the field representation we propose also naturally protects against sidechannel attacks, including timing attacks, simple power analysis and differential power analysis, which is essential in many cryptographic scenarios, in constrast to GMNs. 1.
−1/2
"... Abstract. In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a> 1 then ⌊ 3 4 pa ⌋ k=0 ..."
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Abstract. In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a> 1 then ⌊ 3 4 pa ⌋ k=0
DETERMINISTIC ELLIPTIC CURVE PRIMALITY PROVING FOR A SPECIAL SEQUENCE OF NUMBERS
"... Abstract. We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/ ..."
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Abstract. We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasiquadratic in log N. Notably, neither of the classical “N − 1 ” or “N + 1 ” primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. We believe that this is the largest proven prime N for which no significant partial factorization of N − 1 or N + 1 is known. 1.