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:
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
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
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
The Factorization problem normally is: give
N = pq, find
q. In our relaxed model we know an approximation
Here's how to do it with my implementation:
f(x) = x - q' which has a root
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
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.
from Nothing up my sleeve number:
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.
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...
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
`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.
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!
A step by step tutorial showing you how to get admin credentials on a windows machine: http://imgur.com/gallery/H8obU
- reboot in start-up repair mode
- read privacy statement of one menu should open notepad
- thanks to notepad replace
sethc 1 with
- so now when you press 5 times on
shift it will call
cmd instead of
- you know have a shell, use command
net localgroup Administrators to get a list of the admins
net user <ACCOUNT NAME HERE> * to change one account's password.
If you're looking for a "fix", microsoft advise you to turn off sticky keys all completely
And here's another exploit for windows 98
Note that as soon as you can access the hard drive, you don't need to use the first trick and can switch around programs in system32 as you wish (except if windows is encrypted with bitlocker). For example you can do this with an ubuntu live cd and swap
cmd with the
magnifier tool and you will be able to do the same thing.
I posted a tutorial of awk a few posts bellow. But this one is easier to get into I found. It says Awk in 20 minutes but I would say it takes way less than 20 minutes and it's concise and straight to the point, that's how I like it.
EDIT1: And here's a video of someone using a bunch of unix tools (awk, grep, cut, sed, sort, curl...) to parse a log file, pretty impressive and informative: https://vimeo.com/11202537
EDIT2: Here's another post playing with grep, sort, uniq, awk, xargs, find... http://aadrake.com/command-line-tools-can-be-235x-faster-than-your-hadoop-cluster.html
I wanted to get into educational videos and this is my first big try (of 13minutes). I made some quick animations in Flash and some slides and I recorded it. I didn't want to spend too much time on it. It doesn't feel that clear, my English kinda got stuck sometimes and my animations were... crappy, let's say that :D but it's a first try, I will release other videos and improve on the way hopefully :) (I really need to get more pedagogical). So I hope this will at least help some of my fellow students (or people interested in the subject) in understanding Differential Power Analysis
Note: I made a mistake at the start of the video, DPA is non-invasive (source)
I've always wondered how TOR (The Onion Router) worked and was a bit scared of digging into it. After all, bitcoin is pretty hard to grasp, how would TOR be different? But I found out that TOR was actually a pretty simple concept!
The official explanation is top notch. To sum up, instead of sending a packet to the destination (google.com for example), you choose a route of TOR nodes that will lead to that destination (usually 3 nodes). And for efficiency purpose you will keep that route for 10 minutes)
The idea is similar to using a twisty, hard-to-follow route in order to throw off somebody who is tailing you
- The first node only see who's sending the packet (you) and who it is for (the second node). It decrypts the payload and send it to the second node.
- The second node only sees it came from the first node, decrypts the payload and send it to the third node
- The third node sees it came from the second node, decrypts the payload and send it to the destination (in clear if you don't use ssl)
From TOR's FAQ:
Can't the third server see my traffic?
Possibly. A bad third of three servers can see the traffic you sent into Tor. It won't know who sent this traffic. If you're using encryption, such as visiting a bank or e-commerce website, or encrypted mail connections, etc, it will only know the destination. It won't be able to see the data inside the traffic stream. You are still protected from this node figuring out who you are and if using encryption, what data you're sending to the destination.
To be able to do this, this is where the encapsulation or rather onion routing technique is used.
As we know the route we are going to take, we can encrypt several time our packet. For example: we'll encrypt the packet router B will have to send to router C with the public key of router A. So when router A opens the packet and decrypts it with its private key, he only sees the encrypted payload destined to router B. He can then send it to router B and the latter will decrypt it and send the payload to router C and on and on. I think the picture is clearer than my explanations.
Voilà! Not that hard huh?
PS: here's a list of potential attacks on TOR in their design paper
I just ran into amazing tutorials for sed and awk, two famous text processing tools on unix.
If you are on firefox I advise you to disable CSS (View > Page Style > No Style)
$ echo day | sed s/day/night/
There are still many text processing tools I need to learn, like strip, strings, xdd, grep, tr, xargs, cut, sort, tail, split, wc, uniq...
If you are a student looking for an internship in Cryptography, that might be the right post for you.
Those past few months, before landing an internship at Cryptography Services, I applied in many places around the world and went through several interviews. Be it irl interviews, phone interviews, video interviews... I traveled to some remote places and even applied to companies I really did not want to work at. But those are important to get you the experience. You will be bad at your first interviews, and you don't want your first interviews to be the important ones.
In this post I'll talk about how I got interviews, how I prepared and dealt with them, how I got propositions. I might be wrong on some stuff but I'm sure this will help some students anyway :)
Think about them
Put yourself in the employer's shoes, he doesn't want to read a cover letter, he doesn't want to spend more than a few seconds on a resume, he doesn't want to hurt his eyes on horrific fonts, too many colors and typos.
- Ditch all those premade themes.
- Learn how to do something pretty with Word/LaTeX/Indesign/CSS...
- Okay, if you're really bad at building a nice layout, maybe you should use a premade theme, but then use a very simple/plain one.
- No more than one page (you're a student, don't be cocky).
- Use keywords.
- Be simple. Don't write something that belongs in the cover letter.
There is a lot of theory on C.V. making. I'm sure you can find better resources on how fabricate your resume.
- Be concise (I use bullet points for my cover letters and it saves the reader time)
- Write well, make your friends read it, take a break and re-read it.
- Be formal, polite, etc...
- Don't hesitate to contact them again if they don't answer
- Use my application 3pages to write your cover letter (shameless plug :D)
Side projects are important, like really important. If you don't have anything to show you're just as good as the next student, (almost) no one will ask for your grades.
If you don't know what side project to work on, check Cryptopals (I'll be working with the folks who made this by the way :P)
Also if you're looking for something in development, applied crypto, get a github. Many employers will check for your github.
And, you know, you could... you could write a blog. That's a nice way to write down what you're learning, to motivate you into reading more about crypto and to be able to show what you're doing.
Where to apply?
Check with your professors, they usually know a list of known companies that do crypto. In France there are Thales, Morpho, Airbus and many big companies that you might want to avoid, and also very good companies/start-ups like Cryptoexpert, Quarkslab, ... that you should aim for.
You can also ping universities. They will usually accept you without an interview but will rarely pay you. If you really want to do academic researches, you should check some good universities in Finland, France, California, Sydney, India... wherever you want to go.
If you want to do crypto or applied crypto you might want to look for a start-up/smaller companies as they are not too big and you might learn more with them. But even in big companies you might find sizable crypto teams (rarely more than 2 or 3), check Rootlabs, Cloudflare, Matasano...
Preparing for an interview is a really good opportunity to learn many things! If you know what the subject is about read more about it beforehand. If you don't know anything about the internship subject, read more about what your interviewers have done. If you don't know anything because you have been talking to a PR all along then you might want to rethink applying there.
In all the interviews I've done I just had one PR interview. I would advise you to refuse/avoid those, unless you really want to work for the company, as it's most always a waste of time /rant.
Most interviews in France were focusing on my side projects, whereas the ones I had in the US were way more technical. Younger and smaller companies did ask me things about me and my hobbies (one even asked me if I was doing some sport! I thought that was a neat question :))
Now, here's an unsorted list of questions I got asked:
- Explain whitebox cryptography
- How do you obfuscate a program?
- How can you tell a cipher is secure?
- How do stream ciphers work?
- What is a LFSR?
- name different stream/block ciphers
- Explain Simple Power Analysis
- Explain Differential Power Analysis
- Explain Differential Fault Analysis
- Any counter measures?
- Explain the Chinese Remainder Theorem
- How are points on an Elliptic Curve represented?
- What prevents me from signing a bad certificate?
- Imagine a system to send encrypted messages between two persons
- How to do it with Perfect Forward Secrecy?
- How does a compiler works?
- What is XSS?
- Explain Zero-knowledge with the discrete logarithm example
- You have a smartcard you can inject code on, what do you do to perform a DPA?
- You've recorded traces from the smartcard, do you have to do some precomputations on those traces before doing a DPA?
- Techniques to multiply points on an Elliptic Curve
- Example of homomorphic encryption
- Knapsack problem
An interview should be an interaction. You can cheat your way through by trying not to talk too much, but it still should be a conversation because eventually, if you get the job, you guys will be having real work conversations.
You should be a nice dude. Because nobody wants to work with a boring, elitist dude.
You should never correct the interviewer. This feels like a stupid advice but correcting someone that knows more than you, even if you are right, might lead to bad things... Wait to be hired for that.
You should never let the interviewer tell you something you already know. This is an occasion for you to shine.
If you can't answer a question, ask the interviewer how he would have answer, this is a good opportunity to learn something.
Eventually, ask questions about the job, the workplace, the city. Not only they will appreciate it, it is showing that you are interested in their job, but it's also nice for you to see if the company is the kind of place you would like to work at (it also makes the table turns).
I don't know if any of you were planning on bullshiting but we're in a technical field, avoid bullshiting!
I finally chose where I'll be doing my internship to finish my master of Cryptography: it will be at Cryptography Services (a new crypto team, part of NCC Group (iSEC Partners, Matasano, IntrepidusGroup)). I will be conducting researches and audits in the offices of Matasano in Chicago. I'm super excited and the next 6 months of my life should be full of surprise!
Cryptography Services is a dedicated team of consultants from iSEC Partners, Matasano, Intrepidus Group, and NCC Group focused on cryptographic security assessments, protocol and design reviews, and tracking impactful developments in the space of academia and industry.
I guess I'll soon have to create a "life in Chicago" category =)
I was trying to access the Journal of Cryptology on Springer but I had to pay. Thanks to __x86 I realized I had free access to Springer thanks to my university!
So this post is oriented to my fellow classmates. If any of you want to check something there, it's free for us! (Well until we graduate).
I was trying to check the papers that got Dan Boneh and Antoine Joux their Gödel prize:
I'm reading through A Key Recovery Attack on Discrete Log-based Schemes Using a Prime Order Subgoup which is a Small subgroup confinement attack.
It deals with stuff I had no knowledge of, like Schnorr's Signature that I talk about in a previous post, or like what I'm going to talk about now:
The Pohlig-Hellman Algorithm is a method to compute a Discrete Logarithm (which is a difficult problem) on a multiplicative group whose order is a smooth number (also called friable). Meaning its order can be factorized into small primes.
y = g^x mod p
ord_p(g) = p - 1
p - 1 = q_1^(i_1) * ... * q_j^(i_j)
Here y is the public key,
x is the secret key we're trying to compute.
The order of
g, our generator, is
p - 1 since p is prime.
p - 1 is smooth so it can be factorized into something like 2^2 * 3 * 7 (extremely convenient example!)
Following is an overview of the method, if you read an equation and feel like it comes from nowhere (and it should feel like that), I posted a very short paper containing the simple proofs of those bellow.
The idea that should come to your mind, if you're used to seeing that kind of problem, is that there might be a way to use the Chinese Remainder Theorem (abbreviated CRT) to our advantage. What if we could write
x modulo the factors of
p - 1 and then reconstruct the real
x with the CRT? Well, let's do just that!
x modulo a factor
p - 1 we can write
y^((p-1) / q) which we know and can compute, and is also equal to
g^(x_q * (p-1) / q)
(If you have a factor that appears multiple times in the prime decomposition of
p - 1 (for example
p - 1 = 2^5, then there is also a way to ease the computations by finding multiple unknowns (5 unknowns in our example))
We then have a discrete logarithm to compute, but a small one, that we can compute efficiently thanks to Shanks' method (baby-step giant-step) or Pollard's rho algorithm.
Then we have multiple ways of writing our
x (modulo the factors of
p - 1) and we find out what is
x with CRT (I explained this step in my airbus write-up here).
Proofs + Video
You can read through the Algorithm (or watch the video bellow), but I don't really like to do that since I'm not really good at memorizing something if I don't understand the nut and bolts of it. So here's a really good paper written by D. R. Stinson that demonstrate where the equations of the algorithm come from. And here's an explanation + example of the algorithm:
Interactive Protocols are basically a discussion between a Prover and a Verifier where the Prover has some piece of information he wants to prove, without giving out the information.
It is often illustrated with Peggy and Victor and their super tunnel.
Usualy it takes 3 steps:
- Prover sends a fixed value.
- Verifier sends a challenge.
- Prover answers the challenge.
The Verifier can then verify the answer based on the fixed value. If the answer is correct, the Verifier can assume the Prover knows what he's trying to prove. Sometimes you have to repeat the protocols multiple time to be sure, and not all problems have an Interactive Proof.
Classic examples of such proofs can be found on the Discrete Logarithm problem (we'll talk about that later) and the Hamiltonian Cycle problem.
Interactive Protocols are verified if they are :
- Complete: a Prover can successfully answer the challenge if he is honest.
- Sound : a dishonest Prover cannot convince the Verifier he knows the secret.
In the real definitions we use probabilities (an honest prover still has a small chance of making a mistake, a dishonest prover still has a small chance of convincing the Verifier).
We also often want a 3rd condition on our Interactive Protocols: we want it to be Zero-knowledge, no information about our secret should be leaked in this interaction.
Here are how you prove each one of them:
- Completeness: Can the Prover answer correctly thanks to his secret?
- Soundness: From the point of view of the Verifier. If the Prover can correctly answer two different challenges for the same fixed value (however he crafted the answers and the fixed value), does it mean that he must know the secret then?
- Zero-Knowledgeness: If you see a transcript of a recorded instance of this interaction, will you learn anything about the secret? (See if you can create fake transcripts)
There are also notions of weak Zero-knowledge, strong Zero-knowledge, dishonnest verifiers, etc...
But let's talk about something else.
Since we said that a recorded transcript of a past interaction has no value (if it is zero-knowledge), then we could assume that there is no way of proving something by showing an old transcript, or by showing a transcript with yourself.
Don't fool yourself! Yes we can. We do this by using hash functions that we deem random enough.
The idea is that, by replacing the Verifier by a random oracle, we cannot predict the challenges and we thus cannot craft a fake transcript (we like to use random oracles instead of hashes, to prove some scheme is secure).
a random oracle is an oracle (a theoretical black box) that responds to every unique query with a (truly) random response chosen uniformly from its output domain. If a query is repeated it responds the same way every time that query is submitted.
What is interesting is that this protocol was used in a Signature Scheme.
Interactive Proof of a Discrete Logarithm
The most famous academic example of Interactive Protocol is done using the Discrete Logarithm problem.
<g> = G, with
g of order
q. The Prover wants to show he knows
g^x = y.
- the Prover sends
t = g^e
- the Verifier sends a challenge
- the Prover sends
d = e + cx
The Verifier can then compute
y^c * t = g^(e + cx) and see if it equals
g^d = g^(e + cx)
A transcript would look like this:
(t, c, d)
Non-Interactive Proof of a Discrete Logarithm
Doing this with a non-interactive protocol, it would look like this:
(t, h(t), d) with
h a hash function.
This is what Schnorr's Signature is doing:
t = g^e
c = H(m || t)
d = e - x*c
he would then send
(c, d) as the signature, along the message
m. Which is basically a hash of the message with a proof that he knows the secret
To verify the signature you would use the public key
y = g^x to compute
y^c * g^d = t and then you would compute the hash. It would give you the proof that the signer knows
x (authentication, non-repudiation) and that the message hasn't been tampered (integrity).
So this is one way of using Non-interactive proofs!
A great FAQ, written by Marc Joye (Thomson R&D), on Whitebox Cryptography.
Thansk Tancrède for the link!
Q1: What is white-box cryptography?
A major issue when dealing with security programs is the protection of "sensitive" (secret, confidential or private) data embedded in the code. The usual solution consists in encrypting the data but the legitimate user needs to get access to the decryption key, which also needs to be protected. This is even more challenging in a software-only solution, running on a non-trusted host.
White-box cryptography is aimed at protecting secret keys from being disclosed in a software implementation. In such a context, it is assumed that the attacker (usually a "legitimate" user or malicious software) may also control the execution environment. This is in contrast with the more traditional security model where the attacker is only given a black-box access (i.e., inputs/outputs) to the cryptographic algorithm under consideration.
Q2: What is the difference with code obfuscation?
Related and complementary techniques for protecting software implementations but with different security goals include code obfuscation and software tamper-resistance. Code obfuscation is aimed at protecting against the reverse engineering of a (cryptographic) algorithm while
software tamper-resistance is aimed at protecting against modifications of the code.
All these techniques have however in common that the resulting implementation must remain directly executable.
Or as Francis Gabriel writes here
Code obfuscation means code protection. A piece of code which is obfuscated is modified in order to be harder to understand. As example, it is often used in DRM (Digital Rights Management) software to protect multimedia content by hiding secrets informations like algorithms and encryption keys.