**ALGEBRAIC FACTORING OF THE
CRYPTOGRAPHY MODULUS** **AND PROOF OF GOLDBACH'S CONJECTURE **

**James Constant**

**math@coolissues.com**

**Solving an Algebraic
Quadratic Equation Factors the Cryptographic Modulus** *p=ab*** of Prime Numbers ***a***
and*** b*** and Proves Goldbach's Conjecture **

**Introduction**

In the RSA cryptographic system a user chooses a pair of prime
numbers *a* and *b* so large that factoring the product
(modulus) *p=ab* is beyond all projected computing
capabilities. As of 1984, the consensus was that *a* and *b*
need to be about 75 decimal digits in size, and so *p* was
roughly a 150-digit number. Since the largest hard numbers that
could then be factored were 80 digits or less in size, and since
the difficulty of factoring grows exponentially with the size of
the number, 150 digits appeared cryptosecure for the foreseeable
future.^{1} In 1996, a 130 digit number was factored and
cryptographers talked about 100 digit prime numbers and 200 digit
moduli.^{2} Today, some suggest 1024 and 2048 digit
moduli.^{3}

**Algebraic Factoring of
the Cryptographic Modulus**

Consider the product *p* and sum*
s* of two prime numbers *a* and *b*

(1) .....*ab=p* ............... *p* known, *a* and *b* unknown

..........*a + b = s *........GC *s*
even *>2.*

from which *a* and *b* are to
be found. Equations (1) lead to the algebraic quadratic equation

Since both *s*^{2 }and
*c*^{2} are even numbers, and since *4p*
is a known number, I can find

simply by adding even numbers *c*^{2}
(*c* even) to *4p *until the sum reaches *s*^{2}*,
*as required by (4).

**Number of Operations Involved in
Finding ***a*** and ***b*

1. Given the product *p* of two
unknown primes *a* and *b* start by computing* 4p*

2. Find the first candidate solution *s*

3. compute *c*^{2}*
=s*^{2} - *4p* for the first candidate
solution *s*^{2}

4. determine if the computed *c*^{2}
for the first candidate solution *s*^{2} is
the square of an even number *c*,
as required by (3). If not reject and repeat steps 2-4 for the
second candidate solution *s*^{2}

5. Compute the sum and difference and
divide by *2*, as required by (3).

**Proof of Goldbach's
Conjecture**

Start from equation (2) in which *a,s,p*
are integers. If I specify *s* is any even number greater
than *2* and *p* is a product of two primes *a*
and *b*, equation (2) devolves into *s = a + b* proving GC. The actual values of *a* and *b*
are provided by equation (3). Proof of GC also proves the
Twin Primes conjecture.**5**

^{1} **See Cryptography at****
http://britannica.com/bcom/eb/article/3/0,5716,117763+7+109639,html?query=rsa%...**

^{2} **See Why do
cryptography experts get excited about prime numbers? at ****http://www.madsci.org/posts/archives/may98/893442660.Cs.r.html****
**

^{3} **SeeThe Mathematical
Guts of RSA Encryption at ****http://world.std.com/~franl/crypto/rsa-guts.html**

**See The New RSA Factoring Challenge at ****http://www.rsasecurity.com/rsalabs/challenges/factoring****
**

**4**** See
Goldbach's Conjecture at ****http://primes.utm.edu/glossary/page.php?sort=GoldbachConjecture****
**

**5 ****See Proof of the Twin Primes
Conjecture at ****http://tprimes.coolissues.com/tprimes.htm**** **

**6 ****See
Cryptographic Algorithms at ****http://www.ssh.fi/tech/crypto/algorithms.html**

**Copyright 2002 by James Constant**

**By same author: ****http://www.coolissues.com/mathematics/sameauthor.htm**