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In the following article, we propose an integer factorization algorithm, based on sieving quadrad... more In the following article, we propose an integer factorization algorithm, based on sieving quadradic numbers. The algorithm does not use Fermat's factorization method based on finding a congruence of squares modulo the integer N which we intend to factor. The present approach is based on the difference of squares which are "close" to the integer N. The proposed method could be used to attack the RSA public-key cryptosystem. Method Suppose we are trying to factor the composite number N = pq with p and q prime numbers. We start to look for a number M such that í µí± 2 =(í µí±+í µí± 2) 2 , (1) on this purpose we select a square number í µí± 2 as close as possible to N, but bigger than N according to condition (1). Now we make the difference with the adjacent, lower square number considering that every quadradic number can be expressed as the sum of the first 2í µí± − 1 odd numbers: í µí± 2 − (í µí± −1) 2 = 2í µí± − 1 (2) the difference (2) will be always equal to the odd number 2í µí± − 1. We are looking for a solution of this simple equation: 2í µí± − 1 = í µí± + í µí± − 1 = í µí±í µí± − φ(í µí±) (3) Where φ(N) is the Euler totient function φ(N) = (p − 1)(q − 1). We have to verify if we have found the right difference: 2í µí± − 1 = í µí± + í µí± − 1 so we make a verification based on Euler's totient theorem. Given an integer í µí± coprime with N , we have to check if we have found Euler totient function φ(í µí±) : í µí± φ(í µí±) ≡ 1 mod N (4) Or similarly í µí± pq+1 ≡ í µí± p+q mod N (4.1) To simplify calculations, í µí± ,in our examples, will be equal to: í µí± = 2. If condition (4) is verified we have found φ(í µí±) and also p+q; therefore, we are able to factorize N. If p and q are close to each other, and their difference is small, then equation (3) will soon be valid.
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In the following article, we propose an integer factorization algorithm, based on sieving quadrad... more In the following article, we propose an integer factorization algorithm, based on sieving quadradic numbers. The algorithm does not use Fermat's factorization method based on finding a congruence of squares modulo the integer N which we intend to factor. The present approach is based on the difference of squares which are "close" to the integer N. The proposed method could be used to attack the RSA public-key cryptosystem. Method Suppose we are trying to factor the composite number N = pq with p and q prime numbers. We start to look for a number M such that í µí± 2 =(í µí±+í µí± 2) 2 , (1) on this purpose we select a square number í µí± 2 as close as possible to N, but bigger than N according to condition (1). For the algorithm we have N∈ ℕ and ⌈√í µí± ⌉ = í µí± Now we make the difference with the adjacent, lower square number considering that every quadradic number can be expressed as the sum of the first 2í µí± − 1 odd numbers: í µí± 2 − (í µí± −1) 2 = 2í µí± − 1 (2) the difference (2) will be always equal to the odd number 2í µí± − 1. We are looking for a solution of this simple equation: 2í µí± − 1 = í µí± + í µí± − 1 = í µí±í µí± − φ(í µí±) (3) Where φ(N) is the Euler totient function φ(N) = (p − 1)(q − 1). We have to verify if we have found the difference: 2í µí± − 1 = í µí± + í µí± − 1 that satisfies condition (1) for this reason we make a verification based on Euler's totient theorem. Given an integer í µí± coprime with N , we have to check if we have found Euler totient function φ(í µí±) : í µí± φ(í µí±) ≡ 1 mod N (4) Or similarly í µí± pq+1 ≡ í µí± p+q mod N (4.1) To simplify calculations, í µí± ,in our examples, will be equal to: í µí± = 2. If condition (4) is verified we have found φ(í µí±) and also p+q; therefore, we are able to factorize N.
In the following article, we propose an integer factorization algorithm, based on sieving quadrad... more In the following article, we propose an integer factorization algorithm, based on sieving quadradic numbers. The algorithm does not use Fermat's factorization method based on finding a congruence of squares modulo the integer N which we intend to factor. The present approach is based on the difference of squares which are "close" to the integer N. The proposed method could be used to attack the RSA public-key cryptosystem. Method Suppose we are trying to factor the composite number N = pq with p and q prime numbers. We start to look for a number M such that í µí± 2 =(í µí±+í µí± 2) 2 , (1) on this purpose we select a square number í µí± 2 as close as possible to N, but bigger than N according to condition (1). Now we make the difference with the adjacent, lower square number considering that every quadradic number can be expressed as the sum of the first 2í µí± − 1 odd numbers: í µí± 2 − (í µí± −1) 2 = 2í µí± − 1 (2) the difference (2) will be always equal to the odd number 2í µí± − 1. We are looking for a solution of this simple equation: 2í µí± − 1 = í µí± + í µí± − 1 = í µí±í µí± − φ(í µí±) (3) Where φ(N) is the Euler totient function φ(N) = (p − 1)(q − 1). We have to verify if we have found the right difference: 2í µí± − 1 = í µí± + í µí± − 1 so we make a verification based on Euler's totient theorem. Given an integer í µí± coprime with N , we have to check if we have found Euler totient function φ(í µí±) : í µí± φ(í µí±) ≡ 1 mod N (4) Or similarly í µí± pq+1 ≡ í µí± p+q mod N (4.1) To simplify calculations, í µí± ,in our examples, will be equal to: í µí± = 2. If condition (4) is verified we have found φ(í µí±) and also p+q; therefore, we are able to factorize N. If p and q are close to each other, and their difference is small, then equation (3) will soon be valid.
In the following article, we propose an integer factorization algorithm, based on sieving quadrad... more In the following article, we propose an integer factorization algorithm, based on sieving quadradic numbers. The algorithm does not use Fermat's factorization method based on finding a congruence of squares modulo the integer N which we intend to factor. The present approach is based on the difference of squares which are "close" to the integer N. The proposed method could be used to attack the RSA public-key cryptosystem. Method Suppose we are trying to factor the composite number N = pq with p and q prime numbers. We start to look for a number M such that í µí± 2 =(í µí±+í µí± 2) 2 , (1) on this purpose we select a square number í µí± 2 as close as possible to N, but bigger than N according to condition (1). For the algorithm we have N∈ ℕ and ⌈√í µí± ⌉ = í µí± Now we make the difference with the adjacent, lower square number considering that every quadradic number can be expressed as the sum of the first 2í µí± − 1 odd numbers: í µí± 2 − (í µí± −1) 2 = 2í µí± − 1 (2) the difference (2) will be always equal to the odd number 2í µí± − 1. We are looking for a solution of this simple equation: 2í µí± − 1 = í µí± + í µí± − 1 = í µí±í µí± − φ(í µí±) (3) Where φ(N) is the Euler totient function φ(N) = (p − 1)(q − 1). We have to verify if we have found the difference: 2í µí± − 1 = í µí± + í µí± − 1 that satisfies condition (1) for this reason we make a verification based on Euler's totient theorem. Given an integer í µí± coprime with N , we have to check if we have found Euler totient function φ(í µí±) : í µí± φ(í µí±) ≡ 1 mod N (4) Or similarly í µí± pq+1 ≡ í µí± p+q mod N (4.1) To simplify calculations, í µí± ,in our examples, will be equal to: í µí± = 2. If condition (4) is verified we have found φ(í µí±) and also p+q; therefore, we are able to factorize N.