Hey! I'm David, the author of the Real-World Cryptography book. I'm a crypto engineer at O(1) Labs on the Mina cryptocurrency, previously I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

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# Implementation of Coppersmith attack (RSA attack using lattice reductions) posted February 2015

I've implemented the work of Coppersmith (to be correct the reformulation of his attack by Howgrave-Graham) in Sage.

You can see the code here on github.

I won't go too much into the details because this is for a later post, but you can use such an attack on several relaxed RSA models (meaning you have partial information, you are not totally in the dark).

I've used it in two examples in the above code:

## Stereotyped messages

For example if you know the most significant bits of the message. You can find the rest of the message with this method.

The usual RSA model is this one: you have a ciphertext c a modulus N and a public exponent e. Find m such that m^e = c mod N.

Now, this is the relaxed model we can solve: you have c = (m + x)^e, you know a part of the message, m, but you don't know x. For example the message is always something like "the password today is: [password]". Coppersmith says that if you are looking for N^1/e of the message it is then a small root and you should be able to find it pretty quickly.

let our polynomial be f(x) = (m + x)^e - c which has a root we want to find modulo N. Here's how to do it with my implementation:

dd = f.degree()
beta = 1
epsilon = beta / 7
mm = ceil(beta**2 / (dd * epsilon))
tt = floor(dd * mm * ((1/beta) - 1))
XX = ceil(N**((beta**2/dd) - epsilon))
roots = coppersmith_howgrave_univariate(f, N, beta, mm, tt, XX)

You can play with the values until it finds the root. The default values should be a good start. If you want to tweak:

• beta is always 1 in this case.
• XX is your upper bound on the root. The bigger is the unknown, the bigger XX should be. And the bigger it is... the more time it takes.

## Factoring with high bits known

Another case is factoring N knowing high bits of q.

The Factorization problem normally is: give N = pq, find q. In our relaxed model we know an approximation q' of q.

Here's how to do it with my implementation:

let f(x) = x - q' which has a root modulo q.
This is because x - q' = x - ( q + diff ) = x - diff mod q with the difference being diff = | q - q' |.

beta = 0.5
dd = f.degree()
epsilon = beta / 7
mm = ceil(beta**2 / (dd * epsilon))
tt = floor(dd * mm * ((1/beta) - 1))
XX = ceil(N**((beta**2/dd) - epsilon)) + 1000000000000000000000000000000000
roots = coppersmith_howgrave_univariate(f, N, beta, mm, tt, XX)

What is important here if you want to find a solution:

• we should have q >= N^beta
• as usual XX is the upper bound of the root, so the difference should be: |diff| < XX
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# Nothing up my sleeve numbers posted February 2015

Why does the round constants in sha1 have those specific values?

Kt is the round constant of round t;

from SH1-1 - Wikipedia:

The constant values used are chosen to be nothing up my sleeve numbers: the four round constants k are 230 times the square roots of 2, 3, 5 and 10. The first four starting values for h0 through h3 are the same with the MD5 algorithm, and the fifth (for h4) is similar.

In cryptography, nothing up my sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. They are used in creating cryptographic functions such as hashes and ciphers. These algorithms often need randomized constants for mixing or initialization purposes. The cryptographer may wish to pick these values in a way that demonstrates the constants were not selected for a nefarious purpose, for example, to create a backdoor to the algorithm. These fears can be allayed by using numbers created in a way that leaves little room for adjustment. An example would be the use of initial digits from the number π as the constants. Using digits of π millions of places into its definition would not be considered as trustworthy because the algorithm designer might have selected that starting point because it created a secret weakness the designer could later exploit.

# gwern trying to crack DPR's PGP posted February 2015

After some evidences of the Silk Road trial got out, Gwern noticed a PGP key was in here...

This is the ASCII-armored private key of the main DPR public key, the one he signed forum posts with and messaged with people. I was surprised to see it screenshotted like that, and I thought it would be hilarious if I could take the private key and announce that I was actually the real DPR by signing it with his key (since I've occasionally been accused of it).

more on the story here

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# Vim cheatsheets posted February 2015

I'm using cmder on windows, it's pretty and it comes with a lot of unix tools (cat, ls, bash, ssh, more, grep...) and pipes and streams and... I can use vim in the console. Not emacs, vim. I do have emacs on windows but I don't think I can do a emacs -nw to just use it from the console. So let's go back to learn vim, because I hate being slow. And here is a nice way of doing it!

http://www.viemu.com/a_vi_vim_graphical_cheat_sheet_tutorial.html

you can find several pictures of a keyboard aiming at teaching you step by step how vim works. This is all I needed!

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# Flint vs NTL posted February 2015

I'm digging into the code source of Sage and I see that a lot of functions are implemented with Shoup's NTL. There is also FLINT used. I was wondering what were the differences. I can see that NTL is in c++ and FLINT is in C. On wikipedia:

It is developed by William Hart of the University of Warwick and David Harvey of Harvard University to address the speed limitations of the Pari and NTL libraries.

Although in the code source of Sage I'm looking at they use FLINT by default and switch to NTL when the modulus is getting too large.

By the way, all of that is possible because Sage uses Cython, which allows it to use C in python. I really should learn that...

EDIT:

This implementation is generally slower than the FLINT implementation in :mod:~sage.rings.polynomial.polynomial_zmod_flint, so we use FLINT by default when the modulus is small enough; but NTL does not require that n be int-sized, so we use it as default when n is too large for FLINT.

So the reason behind it seems to be that NTL is better for large numbers.

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# Dread Pirate Roberts' journal? posted February 2015

Silk Road's trial just closed and I ran into this old (?) journal of Ross Ulbricht that contains quite a bunch of interesting passages. I think this will turn into a movie.

03/25/2013
server was ddosed, meaning someone knew the real IP. I assumed they obtained it by becoming a guard node. So, I migrated to a new server and set up private guard nodes. There was significant downtime and someone has mentioned that they discovered the IP via a leak from lighttpd.

03/28/2013
being blackmailed with user info. talking with large distributor (hell's angels).

03/29/2013
commissioned hit on blackmailer with angels

04/01/2013
got word that blackmailer was excuted created file upload script started to fix problem with bond refunds over 3 months old

04/02/2013
got death threat from someone (DeathFromAbove)

04/04/2013
withdrawals all caught up made a sign error when fixing the bond refund bug, so several vendors had very negative accounts. switched to direct connect for bitcoin instead of over ssh portforward received visual confirmation of blackmailers execution

04/06/2013
gave angels go ahead to find tony7

04/08/2013
sent payment to angels for hit on tony76 and his 3 associates

04/21 - 04/30/2013
market and forums under sever DoS attack. Gave 10k btc ransom but attack continued.

05/04/2013
attacker agreed to stop if I give him the first $100k of revenue and$50k per week thereafter. He stopped, but there appears to be another DoS attack still persisting

05/07/2013
paid $100k to attacker 05/22/2013 paid the attacker$50k

05/29/2013
rewrote orders page paid attacker $50k weekly ransom$2M was stolen from my mtgox account by DEA

09/19 - 09/25/2013
red got in a jam and needed $500k to get out. ultimately he convinced me to give it to him, but I got his ID first and had cimon send harry, his new soldier of fortune, to vancouver to get$800k in cash to cover it. red has been mainly out of communication, but i haven't lost hope. Atlantis shut down. I was messaged by one of their team who said they shut down because of an FBI doc leaked to them detailing vulnerabilities in Tor.

09/30/2013
Had revelation about the need to eat well, get good sleep, and meditate so I can stay positive and productive.

All of this sounds so surreal. He is making a huge amount of money for sure. A million dollars doesn't seem much for him. He is constantly buying servers and he seems to be coding a lot. He also seem like a normal dude.

And here's a funny thread on who's Variety Jones

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# Magma vs Sage vs Pari posted February 2015

I was looking for a way to know what are the real differences between magma, sage and pari. I only worked with sage and pari (and by the way, pari was invented at my university!) but heard of magma from sage contributors.

From the sage website: The Sage-Pari-Magma ecosystem

The biggest difference between Sage and Magma is that Magma is closed source, not free, and difficult for users to extend. This means that most of Magma cannot be changed except by the core Magma developers, since Magma itself is well over two million lines of compiled C code, combined with about a half million lines of interpreted Magma code (that anybody can read and modify). In designing Sage, we carried over some of the excellent design ideas from Magma, such as the parent, element, category hierarchy.

Any mathematician who is serious about doing extensive computational work in algebraic number theory and arithmetic geometry is strongly urged to become familiar with all three systems, since they all have their pros and cons. Pari is sleek and small, Magma has much unique functionality for computations in arithmetic geometry, and Sage has a wide range of functionality in most areas of mathematics, a large developer community, and much unique new code.

I also noticed that Sage provides an interface to Shoup's NTL through ntl. functions. Good to know!

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