https://github.com/lcsig/fermat-rsa
Ahmed S. Shatnawi, Mahmoud M. Almazari, Zakarea AlShara, Eyad Taqieddin, Dheya Mustafa, RSA cryptanalysis - Fermat factorization exact bound and the role of integer sequences in factorization problem, Journal of Information Security and Applications, Volume 78, 2023, 103614, ISSN 2214-2126, https://doi.org/10.1016/j.jisa.2023.103614
https://github.com/lcsig/fermat-rsa
fermat-factorization integer-sequences number-theory oeis paper rsa rsa-cryptography
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Ahmed S. Shatnawi, Mahmoud M. Almazari, Zakarea AlShara, Eyad Taqieddin, Dheya Mustafa, RSA cryptanalysis - Fermat factorization exact bound and the role of integer sequences in factorization problem, Journal of Information Security and Applications, Volume 78, 2023, 103614, ISSN 2214-2126, https://doi.org/10.1016/j.jisa.2023.103614
- Host: GitHub
- URL: https://github.com/lcsig/fermat-rsa
- Owner: lcsig
- License: mit
- Created: 2022-02-23T05:42:32.000Z (about 4 years ago)
- Default Branch: main
- Last Pushed: 2025-01-18T13:28:59.000Z (about 1 year ago)
- Last Synced: 2025-06-15T05:34:09.173Z (9 months ago)
- Topics: fermat-factorization, integer-sequences, number-theory, oeis, paper, rsa, rsa-cryptography
- Homepage:
- Size: 1.64 MB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Fermat-RSA
Code For Fermat Factorization Research
## The closest value to 4 * 1303 * 197 at each row
```matlab
p = 1303; q = 197;
fourN = 4 * p * q;
list_Floor = zeros(1, floor(sqrt(fourN)));
list_Ceil = zeros(1, floor(sqrt(fourN)));
for r=1:floor(sqrt(fourN))
c_Floor = floor((fourN - r^2) / (2 * r) + 1);
c_Ceil = ceil((fourN - r^2) / (2 * r) + 1);
list_Floor(r) = (r^2 + 2 * r *(c_Floor - 1));
list_Ceil(r) = (r^2 + 2 * r *(c_Ceil - 1));
end
plot(1:floor(sqrt(fourN)), list_Floor, 1:floor(sqrt(fourN)), list_Ceil);
xlabel("Row Values"); xlim([1, floor(sqrt(fourN))])
ylabel("The Closest Number To 4*N");
legend("Floor(c)", "Ceil(c)");
```
## The closest value to 4 * 1303 * 197 after eliminating the odd rows and the even columns
```matlab
p = 1303; q = 197;
fourN = 4 * p * q;
listR = [];
listValFloor = [];
listValCeil = [];
for r=1:floor(sqrt(fourN))
c_Floor = floor((fourN - r^2) / (2 * r) + 1);
c_Ceil = ceil((fourN - r^2) / (2 * r) + 1);
if mod(r, 2) == 1 || mod(c_Floor, 2) == 0
continue;
end
listR = [listR, r];
listValFloor = [listValFloor, (r^2 + 2*r*(c_Floor - 1))];
listValCeil = [listValCeil, (r^2 + 2*r*(c_Ceil - 1))];
end
plot(listR, listValFloor, listR, listValCeil);
xlabel("Row Values"); xlim([1, floor(sqrt(fourN))]);
ylabel("The Closest Number To 4*N");
legend("Floor(c)", "Ceil(c)");
```
## The closest value to 4 * 1303 * 197 after eliminating the odd rows and the even columns. Also, the row which is not equal to 2 times a prime number is removed
```matlab
p = 1303; q = 197;
fourN = 4 * p * q;
listR = [];
listValFloor = [];
listValCeil = [];
primeList = primes(floor(sqrt(fourN)));
for r=1:floor(sqrt(fourN))
c_Floor = floor((fourN - r^2) / (2 * r) + 1);
c_Ceil = ceil((fourN - r^2) / (2 * r) + 1);
if mod(r, 2) == 1 || mod(c_Floor, 2) == 0
continue;
end
if ismember(r / 2, primeList) == 0
continue;
end
listR = [listR, r];
listValFloor = [listValFloor, (r^2 + 2*r*(c_Floor - 1))];
listValCeil = [listValCeil, (r^2 + 2*r*(c_Ceil - 1))];
end
plot(listR, listValFloor, listR, listValCeil);
xlabel("Row Values"); xlim([1, floor(sqrt(fourN))]);
ylabel("The Closest Number To 4*N");
legend("Floor(c)", "Ceil(c)");
```
## The closest value to 4 * 1303 * 197 by column prespective
```matlab
p = 1303; q = 197;
fourN = 4 * p * q;
list_Floor = zeros(1, floor((fourN - 1) / 2 + 1));
list_Ceil = zeros(1, floor((fourN - 1) / 2 + 1));
for c=1:floor((fourN - 1) / 2 + 1)
r_Floor = floor(sqrt( (c-1)^2 + fourN ) - c + 1);
r_Ceil = ceil(sqrt( (c-1)^2 + fourN ) - c + 1);
list_Floor(c) = (r_Floor^2 + 2 * r_Floor *(c - 1));
list_Ceil(c) = (r_Ceil ^2 + 2 * r_Ceil *(c - 1));
end
plot(1:floor((fourN - 1) / 2 + 1), list_Floor, 'o', ...
1:floor((fourN - 1) / 2 + 1), list_Ceil, 'o');
xlabel("Columns Values"); xlim([1, floor((fourN - 1) / 2 + 1)])
ylabel("The Closest Number To 4*N");
legend("Floor(r)", "Ceil(r)");
```
## The closest value to 4 * 1303 * 197 by column prespective after eliminating the odd rows and the even columns
```matlab
p = 1303; q = 197;
fourN = 4 * p * q;
listC = [];
listValFloor = [];
listValCeil = [];
for c=1:floor((fourN - 1) / 2 + 1)
r_Floor = floor(sqrt( (c-1)^2 + fourN ) - c + 1);
r_Ceil = ceil(sqrt( (c-1)^2 + fourN ) - c + 1);
if mod(r_Floor, 2) == 1 || mod(c, 2) == 0
continue;
end
listC = [listC, c];
listValFloor = [listValFloor, (r_Floor^2 + 2 * r_Floor *(c - 1))];
listValCeil = [listValCeil, (r_Ceil ^2 + 2 * r_Ceil *(c - 1))];
end
plot(listC, listValFloor, 'o', ...
listC, listValCeil, 'o');
xlabel("Columns Values"); xlim([1, floor((fourN - 1) / 2 + 1)])
ylabel("The Closest Number To 4*N");
legend("Floor(r)", "Ceil(r)");
```
## Factoring in O(1)
```java
BigInteger ONE = BigInteger.valueOf(1);
BigInteger FOUR = BigInteger.valueOf(4);
String N = "734465642310298504833940186303906456";
N = N + "22848602022499768136179936493297372335479";
BigInteger n = new BigInteger(N);
BigInteger fourN = n.multiply(FOUR);
BigInteger ceilSqrt = fourN.sqrt().add(ONE);
// By The Quadratic Formula
BigInteger qBySol = ceilSqrt.subtract(ceilSqrt.pow(2).subtract(fourN).sqrt());
qBySol = qBySol.divide(TWO);
System.out.println("The Smallest Prime Factor Is: " + qBySol);
```
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