Factorization by power P-Q differences

RSA POISONING ATTACK - Prime grimorie vol 3

A final description of NSIF group, code, and demo

The NSI group is the smallest distance among n^2 and multiple of p or q -1 , or p-q, or p , or q multiple to the nearest power.

NSI = n^2 - $ minimum ( mod (n^2) (p-q) , mod (n^2) ((p-1) or (q-1)) )

FACTOR N = gcd N $ (powMod 2 (ex * modular_inverse ex (NSI) - 1) n) - 1

Conjecture

If n is product of 2 primes :

mod (n^2) (p-q) < (p-q) , easy factorize from distance than find a factor

mod ((p*q)^2) (p-q) = 1 or prime or square

4PQ = (P+Q)^2-(P-Q)^2

gcd (pq) (mod ((p+q)^3-(p-q)^3) (pq) ) = factor

FACTOR N = gcd N $ (powMod 2 (ex * modular_inverse ex (p-q || p+q multiples ) - 1) n) - 1

sigma(n)^2 = 4n + (P - Q)^2

n^2-1 = x6 + y6 + (x6 * y6)

n^2 / 6 = (y + 1) * x6

1 = ((sqrt (x * 6+1) * (sqrt (y * 6+1)))^2) - ( (x * 6+y * 6) + ((x * 6)*(y * 6)))

1 = ((p * q)^2) - ( (p^2-1+q^2-1) + ((q^2-1) * (p^2-1)))

Implementation in RsaCTFTool

RsaCtfTool/RsaCtfTool#266

The tool allow you to compare the result of Power Modular Factorization / Rsa Poisoning versus other factorization methods .

Factorization from p-q

SIMPLE ALGORITHM FOR N BITS (Haskell)

ex = 1826379812379156297616109238798712634987623891298419

{- | For result 0 is the period , if period is NSI is common factor of N.  -}
tryperiod n period _ = (powMod 2 (ex * (modular_inverse ex period) - 1) n) - 1


{- | N is the pubkey, tries is the number of tries in the loop, until when try to sum 1, distance is the starting distance  -}
nsif n tries distance
	| d /=1 && d /= n = (div n d,d,divcar)
	| otherwise = (0,0,0)
	where
	out2 = reverse ((1,1): take 1 (dropWhile (\(r,u)-> r==1 && r /= n ) $ map (\x-> (gcd (n) ((tryperiod ((n)) ((n)^2-x^2) x)  ),x)) $ [distance..distance+tries]))
	d = fst $ head out2
	divcar = snd $ head out2

Usage

-- N, loops, distance

./nsif 377 0 3

(29,13,3)

LICENSE

100 % Pure dark math of Mas de Pascualet

Institutional investigation of this paper is forbidden

All commercial rights of this document or derivates, are reserved .

Blackhole Consulting

Authors

Enrique Santos (Intellectual author)

Vicent Nos Ripolles (Hacker)

Francisco Blas Izquierdo (Hacker) Phd Student , Chalmers Institute

Maestrazgo , computation center