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Old 2020-04-28, 09:01   #1
Alberico Lepore
 
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Default HOW TO BYPASS RSA DIFFICULTY... not!

HOW TO BYPASS RSA DIFFICULTY

N=p*q and p+q-4 = 0 (mod 8) and x > 2

(3*N-1)/8=3*x*(x+1)/2-3*y*(y-1)/2+(3*x+1)*(3*x+2)/2

where p=[2*(3*x+1-(x-y+1))+1-(4*y-2)] and q=[2*(3*x+1-(x-y+1))+1]

BYPASS in this way

y=1/2*(sqrt(4*m^2-8*((3*N-1)/8-1)/3-131)+1)=(Z-1)/2

solve Z^2-4*m^2=-8*((3*N-1)/8-1)/3-131

4*m^2-Z^2=8*((3*N-1)/8-1)/3+131

EXAMPLE

N=187

4*m^2-Z^2=315

factorize 315=3^2*5*7=15*21

2*m=(15+21)/2

Z=(21-15)/2

->

m=9

Z=3

y=(Z-1)/2=1

x=3

p=11

q=17
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Old 2020-04-28, 13:42   #2
Alberico Lepore
 
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in general RSA is of four types
2 types N = 4 * G + 1
2 types N = 4 * H + 3
of the latter 2 types we are interested in p + q-4 = 0 (mod 8)
now to go from 4 * G + 1 to 4 * H + 3 just multiply by 3
to pass from p + q = 0 (mod 8) to p + q-4 = 0 (mod 8) just multiply by 5
so we will have two 2 ^ (to something)
now you can try to factor the new generated number 8 * ((3 * N-1) / 8-1) / 3 + 131
or try to factor in for example the first P 1000 primes
to the remaining number if of form 4 * G + 1 multiply also by 3
therefore we will have 1000 * [2 ^ (log in base r)] where r is a sort of coefficient that goes according to the number N and the choice of P
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Old 2020-04-28, 14:39   #3
VBCurtis
 
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Quote:
Originally Posted by Alberico Lepore View Post
EXAMPLE

N=187
How is your method faster than trial division? I can factor 187 by dividing by 2,3,5,7,11 and presto! I'm finished. Or I could start at the sqrt of 187 rounded down: divide by 13, then 11. Presto! Two steps.

Hint It's not. But if you're really clever, you might stumble onto a quadratic sieve. You might try reading about that one, rather than wandering in the dark like you do.
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Old 2020-04-29, 16:28   #4
Alberico Lepore
 
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Quote:
Originally Posted by VBCurtis View Post
How is your method faster than trial division? I can factor 187 by dividing by 2,3,5,7,11 and presto! I'm finished. Or I could start at the sqrt of 187 rounded down: divide by 13, then 11. Presto! Two steps.

Hint It's not. But if you're really clever, you might stumble onto a quadratic sieve. You might try reading about that one, rather than wandering in the dark like you do.
yes the demostraction was wrong.
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