### The Ugly Duckling in factoring aka the filtering steps part I

People that knows me well are well aware that prime numbers have been my obsession since my childhood and they are source of continue interest for me. Actually thanks to cryptography they are a relevant part of my everyday life.
One of the most important problem in cryptography since the discovery of RSA is factoring.
The factoring problem consists of finding the prime numbers p and q given a large number , N = p x q.
Unless you are still convinced that factoring is an easy peasy problem, you should know that, while probably not NP-complete, factoring is indeed reaaally hard.
The faster known method for factoring is currently NFS (Number Field Sieve) and if you are interested in the topic I suggest you to read  this beautiful article from the great Carl Pomerance titled "A Tale of Two Sieves" . But it is not what I wanted to talk about today, mainly because the complete algorithm and all its shades go well beyond my current knowledge.
Today instead I want to talk you about one of the IMHO too often underappreciated (but crucial) steps of NFS that goes under the name of filtering. I became more familiar about this topic while reading Topics in Computational Number Theory Inspired by Peter L. Montgomery where I discovered that the legendary Peter L. Montgomery played a big role in the concrete development of these techniques.
One of the most intriguing part of filtering is that at the first sight these methods look trivial (because well, they are) but my main point, that drove me to write this blog post, is that they are probably one of the most important part of the all  NFS algorithm.
The current NFS record factorization is RSA-768, an integer of 768 bits (232 digits). At the beginning of the filtering step, the matrix had about 47 billion rows and 35 billion columns. After the first part of the filtering step, the matrix had about 2.5 billion rows and 1.7 billions columns. At the end of the filtering step, the matrix used in the linear algebra had about 193 million rows and columns. So the filtering step reduced the size of the matrix by more than 99%!!!
But let's go in order...

## ..What is this filtering about?

All the modern factorization procedures consist basically of 3 steps:
1. Relation Building
2. Elimination
3. GCD Computation
Filtering is part of step 2 (Elimination). The main goal of the filtering step is to reduce the size of a large sparse matrix over a finite field in order to being able to compute its kernel (we will see that for the factorization problem what we actually need is the left kernel). We will see that filtering per see is a 4 steps process:
1. Removing duplicates
2. Removing singletons
3. Removing cliques
4. Merging
As we said in this example we focus on filtering that is part of Elimination so in order to showcase it, we borrow a simple Relation Building example from "An Introduction to Mathematical Cryptography"

 from "An Introduction to Mathematical Cryptography"

Again, without going to much in details, the goal here is to find a subset of all these relations whose product is a square on each side of the equality.  This is equivalent to: the sum of the exponents in all chosen relations must be even. If you wonder why, we are trying to leverage one of the simplest of the identities in all of the mathematics:

X^2 - Y^2 = (X+Y)(X-Y)

But let's not diverge from out today's topic and let's focus on filtering. The problem above can be translated in a linear algebra problem. If one sees the relations as rows of a matrix where each column corresponds to an ideal, the coefficients of the matrix are the exponents of the ideals in the relations. As we are looking for even exponents, one can consider the matrix over GF(2).

Finding a linear combination of relations such as every exponent is even is equivalent to
computing the left kernel of the matrix. So lets translate A to B = A'

As we mentioned before, the matrix that comes out from factoring problem is huge (in the order of billions of row/columns). Our goal is to minimize the size of the matrix before the kernel computation in order to make this operation possible. So let's filter

## Removing duplicates

The first step of filtering is extremely trivial. It is simply about removing the duplicate rows in the matrix. As weird at it sounds having those rows is inevitable though in the factorization context (because lattice sieving with many distinct special q-primes will produce identical relations).

## Removing singletons

The rule of removing singleton is as simple as  the removing duplicates above: if there is a row in B that contains only a single entry that is non-zero module 2 them the column containing non-zero entry can not occur in a dependency. Such column, called singleton, can be removed from B (together with the respective row). In order to give an example let's use a simpler matrix then B above called M

Now let's apply the removing singleton filter:

So we removed the first row and column of M. But we are not done yet as you can see this removal generated a new singleton (the first column of the new matrix) so several passes are normally requited before all singleton are normally removed from M. Continuing until the very end may not be worth the effort though (we will see in the next post how to handle this part).

## Conclusion

Filtering is a really important step in Number Field Sieve and it is implemented also in important integer factorizatiion tools as  CADO-NFS, Msieve and GGNFS. In this blog post we covered the first to phases of filtering as removing duplicates and removing singleton. Stay tuned for the last two steps removing cliques and merging coming in part II of the blog post series.

Unknown said…
Factoring in cryptography is a really passionate topic. Thank you very much for the post. I work in another side of factoring the business area!

### Slack SAML authentication bypass

tl;dr  I found a severe issue in the Slack's SAML implementation that allowed me to bypass the authentication. This has now been solved by Slack.
Introduction IMHO the rule #1 of any bug hunter (note I do not consider myself one of them since I do this really sporadically) is to have a good RSS feed list.  In the course of the last years I built a pretty decent one and I try to follow other security experts trying to "steal" some useful tricks. There are many experts in different fields of the security panorama and too many to quote them here (maybe another post). But one of the leading expert (that I follow) on SAML is by far Ioannis Kakavas. Indeed he was able in the last years to find serious vulnerability in the SAML implementation of Microsoft and Github. Usually I am more an "OAuth guy" but since both, SAML and OAuth, are nothing else that grandchildren of Kerberos learning SAML has been in my todo list for long time. The Github incident gave me the final…

### Bug bounty left over (and rant) Part III (Google and Twitter)

tl;dr in this blog post I am going to talk about some bug bounty left over with a little rant.

Here you can find bug bounty left over part I and II
Here you can find bug bounty rant part I and II
Introduction In one of my previous post I was saying that:

"The rule #1 of any bug hunter... is to have a good RSS feed list."
Well well well allow me in this post to state rule #2 (IMHO)

"The rule #2 of any bug hunter is to DO NOT be to fussy with 'food' specifically with left over"

aka even if the most experience bug hunter was there (and it definitely was my case here, given the fact we are talking about no one less than filedescriptor) do not assume that all the vulnerabilities have been found! So if you want some examples here we go.
Part I - GoogleI have the privilege to receive from time to time Google Vulnerability Research Grant. One of the last I received had many target options to choose from, but one in particular caught my attention, namely Google Issue T…

### How to try to predict the output of Micali-Schnorr Generator (MS-DRBG) knowing the factorization

The article was modified since its publication. Last update was 09/10/2017

See  also Part II and Part III of this series

tl;dr in this post we are going to describe how to try predict the output of Micali-Schnorr Generator (MS-DRBG)  knowing the factorization of the n value. If this sounds like, "why the hell should I care?", you might want to give a look at this great post from Matthew Green about the backdoor in Dual_EC_DRBG. But In a nutshell, quoting Matthew Green : Dual_EC_DRBG is not the only asymmetric random number generator in the ANSI and ISO standards (see at the bottom).   it’s not obvious from the public literature how one would attack the generator even if one knew the factorization of the n values above. What I am NOT claiming in this post though is that there is a backdoor in one of this standard.

Introduction
The first time I heard about this problem is about couple of weeks ago via this Matthew's tweet: As a curiosity, the NSA didn’t just standardize Dua…