Hey! I'm David, cofounder of zkSecurity and the author of the Real-World Cryptography book. I was previously a crypto architect at O(1) Labs (working on the Mina cryptocurrency), before that 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.
As I'm doing some research on Homomorphic Encryption, and trying to watch everything from Gentry (lots of videos available and he has a soothing voice), and possibly read everything from him (which I didn't do so don't take my word). Here's a vocab list I did for reference:
HE: Homomorphic Encryption
leveled: no bootstrapping, only compute circuits up to a degree (public key grows with the depth of the circuit for f )
modulus switching: (some messy operation that they got rid of in bra12)
bootstrapping: to make a bootstrappable somwehat homomorphic encryption scheme into a fully homomorphic encryption scheme (Gentry recommends not to use this, although it was the original way of building FHE)
IBE: Identity Based Encryption scheme (introduced by the famous Boneh paper)
IBFHE: Identity Based Fully Homomoprhic Encryption scheme (introduced by GSW)
Alex asked me if I knew a way of comparing two sets of data: two players want to compare their guesses on some game, without giving away their guesses. You could think of Zero-Knowledge protocols, but this is usually a one-way proof. This is actually the Socialist Millionaire Problem and it is solved by doing a multi-party computation of a function (a comparison function in our case) on two inputs (the two guesses in our case).
In cryptography, the socialist millionaire problem is one in which two millionaires want to determine if their wealth is equal without disclosing any information about their riches to each other. It is a variant of the Millionaire's Problem whereby two millionaires wish to compare their riches to determine who has the most wealth without disclosing any information about their riches to each other.
But how to make it fair? What if one party stops the protocol at one point, for example when he knows if the guesses are the same or not, so that the other party doesn't learn anything.
This seems like a difficult problem to solve, but an interesting problem that crypto should be able to solve.
Alex found this paper: A fair and efficient solution to the socialist millionaires’ problem, where they explain what they call a "fair" protocol. And the solution is quite elegant! I haven't read the whole thing but the idea is basically to compare bit by bit (I guess under the surface they must use garbled circuits) so that if one party stops the protocol early, he only has one bit of advantage over the other one.
Finally, the fairness of the fair version of our protocol is straightforward. Both Alice and Bob are unable to compute the result of the comparison before the beginning of step 4. Moreover, during the fourth step, Bob's advantage over Alice is at most one bit. So, if Bob decides to abort the protocol and tries to search the remaining bits by exhaustive research, Alice needs no more than twice as much time compared to Bob to compute the same result.
I'm reading stuff about HE (Homomorphic Encryption) and so why not share what I find? Hopefuly there will be more than one post on the subject, and they won't be too long, and they will make others learn something new
So what is Homomorphic Encryption?
Well let's start at the beginning shall we? It all began in 1977 when RSA was invented. If you recall, to encrypt with RSA you take your message/number \(m\) and raise it to the power of the public exponent \(e\) modulo a product of primes \(N\) like so:
\[ c = m^e \pmod{N} \]
An incredible property of textbook RSA is the fact that the scheme is malleable. If you have a ciphertext \(c\) but don't know what number \(m\) it's decrypting to, you can modify that ciphertext \(c\) so that it would decrypt to \(3m\) or \(9999m\) or more generally \(x \cdot m\). Just take your number \(x < N \) and encrypt it, then multiply that to your ciphertext like so:
\[ x^e \cdot c \pmod{N} \]
And notice that beneath the encryption, this is equal to
\[ x^e \cdot m^e \pmod{N} \]
That is the same as
\[ (x \cdot m)^e \pmod{N} \]
and will obviously decrypt as \( x \cdot m \). Tell me if I'm going too slow =)
This is called Homomorphic Encryption, which you might not want as a property when using RSA and that's why you should never use RSA without a padding system. The state of the art being OAEP.
This made Rivest and his friends raise the question, a year later in 1978, can this property be extended so that any kind of circuit could be computed on encrypted data? (R. Rivest, L. Adleman, and M. Dertouzos. On data banks and privacy homomorphisms. In Foundations of Secure Computation)
Fully Homomorphic Encryption
The applications thought were mostly "data manipulation" where you would want someone to manage/operate on your data without seeing it. Think banks, search engines, the cloud.
A bunch of other cryptosystem that provided homomorphic properties came to life after that. For example in 1999 the Paillier cryptosystem, which unlike RSA provides additive homomorphic encryption (RSA provides multiplicative homomorphic encryption).
The open problem was still out there. Could you create a cryptosystem that would provide enough homomorphic properties, that combined could compute any kind of circuits. A Fully Homomorphic Cryptosystem.
Any circuit can be simplified to these simple instructions: AND, OR and NOT, which would only need a cryptosystem to have addition, subtraction and multiplication to be able to emulate these instructions.
Modulo 2 would be enough, look, if \( x \in \mathbb{Z}_2 \) (the set of 0 and 1) then:
\( \text{AND}(x,y) = xy \)
\( \text{OR}(x,y) = 1 - (1-x)(1-y) \)
\( \text{NOT}(x) = 1 - x \)
With these properties combined you could then compute any circuit on encrypted data (think of a function, like AES() or select all my data that starts with the letter A, etc...).
So, just a recap so that we're on the same page. With a FHE Alice could send \( c = E(m) \) the encryption of her message \( m \) to Bob, and bob could compute the function \( f \) on the encrypted data so that it would decrypt to a function \(f\) on the plaintext:
\[ D(f(c)) = f(m) \]
Note that in reality, we can't really compute \( f \) on the ciphertext directly, what we do is that typically a Homomorphic Encryption scheme is defined with a function called Eval (for Evaluate) which we would use like that:
\[ D(\text{Eval}(f, c)) = f(m) \]
In 2009, Dan Boneh's doctorate student Craig Gentry finished his thesis, unveilling the first FHE (Fully Homomorphic Encryption):
In a presentation to my fellow Ph.D. admits four years ago, Dan highlighted fully homo- morphic encryption as an interesting open problem and guaranteed an immediate diploma to anyone who solved it. Perhaps I took him too literally.
A bit later Gentry et al simplified that scheme using only hard problems on integers instead of lattices, this was explained as well in another article for CACM here: Computing Arbitrary Functions of Encrypted Data
In this video I'm explaining what is that Galois Counter Mode that provides Authenticated Encryption with Associated Data (AEAD). You must have heard it combined with AES, and maybe used in TLS, ... This is just a small explanation, you can get more on the NIST specs.
My coworker Tomek found a vulnerability in Rack, a Ruby thingy that is used pretty much in every web framework (Rails, Ramaze, Sinatra...) to translate queries (POST and GET) into Ruby objects.
It all started when he read about how to_sym() used to work in Ruby versions prior to MRI 2.2. This function converts strings into symbols. A symbol is this thing: :symbol, that usually is tied to a value: :symbol => 'something' and that you must have seen written in a really nice way thanks to Ruby's lovely syntax sugar: symbol: 'something'. Developers use the to_sym() method a lot to transform the strings from GET/POST to symbols. Problem: the symbols created by to_sym() were stored in the memory and never freed. No garbage collection.
That is a problem if you let user input infinitely use that to_sym() method.
For example if you naively transformed the GET variables of a request to symbols through that function, then a malicious user could have queried that page with many ?stuff=something so that your code would have infinitely stocked symbols in memory until no memory was left.
Tomek was looking for the fastest method to fill up a server's memory. To do
this, he tested various forms of parameters to see what would be the most
effective. This included many small symbol names, large symbol names, and
nested parameters. The most efficient way ended up being a lot of concurrent requests with huge symbol names, the server would go down in less than 2 minutes.
And it was through the nested parameter testing that he found CVE-2015-3225
Potential Denial of Service Vulnerability in Rack
There is a potential denial of service vulnerability in Rack. This
vulnerability has been assigned the CVE identifier CVE-2015-3225.
Versions Affected: All.
Not affected: None.
Fixed Versions: 1.6.2, 1.5.4
Impact
------
Carefully crafted requests can cause a `SystemStackError` and potentially
cause a denial of service attack.
All users running an affected release should either upgrade or use one of the workarounds immediately.
In Ruby on Rails, one of the nice things you can do is use associative arrays (called hashes in Ruby) in your forms.
And that is thanks to Rack! But there is a small problem here, Rack parses these nested hashes by recursively calling itself:
def normalize_params(params, name, v = nil)
...
if params_hash_type?(params[k].last) && !params[k].last.key?(child_key)
normalize_params(params[k].last, child_key, v)
else
params[k] << normalize_params(params.class.new, child_key, v)
Here, Ruby seems to push a bunch of info to the stack before calling the new function. A bit like an assembly prolog.
The thing is, after a while the stack gets full and Ruby throws an exception. And then? Rack catch the exception? Tries again? We don't really know, but the program hangs there.
So Tomek found out that when you would send a POST request with a deep enough hash you would cause the program to hang. This is bad because of DoS attacks.
Also, the problem cannot happen in json because json has a nest limit, and doesn't happen in GET requests either for the same reason.
This evening I was at Braintree's office in Chicago for the new edition of the Owasp meeting. The offices were amazing and huge!
They had beer pong tables, ping pong tables, a mini-arcade and a bar! (all that right next to the workspace)
Plenty of nice conference rooms
Some had whiteboard tables!
And in the largest building of Chicago!
1 Trojaned Gems - You can’t tell you’re using one!
Brandon Myers, a security researcher at Trustwave, was the first one to talk. He found out that when you executed gem fetch or gem install, the ruby package manager, it would allow a Man In The Middle to do a DNS poisoning attack to redirect you to his servers. Even though everything happens over TLS! This is because gem doesn't check for the domain in the certificate of rubygems.org: it just checks that the server has a valid certificate and that's all. You would then download the gem on his server and... game over.
He said that the same thing was happening when developers were pushing their gems to rubygems.org ...
One way of mitigating the first MITM would be to just use curl or wget directly with https://www.rubygems.org and do whatever the gem fetch does to get the gems. Because curl or wget should have a correct implementation of TLS that dodge fake DNS responses (that's why DNSSEC is useless if you query a https webpage with a correctly signed certificate).
To mitigate the second attack Brandon talked about signed gems, and that it was far from being efficient since none of the top gems are signed.
Two other problem were that if a fix comes around, gem update --system is vulnerable to the attack (since gem is itself a gem) and not using the fully secure gem signing allows some dependencies of not being signed (and thus a MITM would be able to modify those).
2 Attacking and Defending DevOps
Patrick Thomas and Alec Gleason followed by explaining how much they pwned their client with heartbleed, passwords in clear and github hooks. They then explained how you could get more information out of a pwned machine if there was git, vagrant, chef, docker and other non-crypto stuff installed on the machine.
a software development method that emphasizes communication, collaboration (information sharing and web service usage), integration, automation, and measurement of cooperation between software developers and other IT professionals.
Basically, it's a way to get a quick x509 certificate for your server without knowing much about what is a x509 certificate:
You have a website. You want people to be able to log in on it from starbucks without the guy sitting at a near table reading the passwords in clear from the packets you're sending everyone around you. So you google a few random keywords like "secure website https" and you end up with a bunch of links and services (which may not be free) and you suddenly have to understand what are certificates, PKI, x509, public-key cryptography, RSA, FQDN, haaa.... Well, worry no more, Let's Encrypt was thought so that you wouldn't have to go through all that trouble and destroy your server on the way.
You download their program on your server that has the address www.example.com:
sudo apt-get install lets-encrypt
You run it as sudo telling it you want to get a certificate for your domain
lets-encrypt example.com
And voila.
Behing the curtains this is what happens:
lets-encrypt will generate a pair of RSA private key/public key and contact the CA with your public key.
The CA will register your public key
The program will then ask the CA to verify your domain.
The CA will answer with a set of challenges. These are some tasks you can complete to prove to the CA you own that domain. One of the common one is to upload a certain file at a certain address of that domain.
The program you installed will then do that for you and poll the CA for a confirmation
The CA will tell you "OK man, all is good".
The program will then generate another long term pair of private key/public key, generate a CSR (Certificate Signing Request) with the new public key and send that CSR to the CA.
The CA will extract the information, create a beautiful x509 certificate, sign it and send it back to you.
lets-encrypt will install the certificate on your server and set certain options (or not) to force https
By the way, the certificate you will get will be a DV certificate, meaning that they only verified that you owned the domain, nothing more. If you want an EV certificate this is what you will have to go through (according to wikipedia):
Establish the legal identity as well as the operational and physical presence of website owner;
Establish that the applicant is the domain name owner or has exclusive control over the domain name; and
Confirm the identity and authority of the individuals acting for the website owner, and that documents pertaining to legal obligations are signed by an authorised officer.
But how does it really work?
So! The lets-encrypt program you run on your server is open sourced here: https://github.com/letsencrypt/lets-encrypt-preview. It is called lets-encrypt-preview I guess because it isn't done yet. It's written in Python and has to be run in sudo so that it can do most of the work for you. Note that it will install the certificates on your server only if you are using Apache or Nginx. Also, the address of the CA is hardcoded in the program so be sure to use the official lets-encrypt.
The program installed on the CA is also open sourced! So that anyone can publicly review and audit the code. It's called Boulder and it's here: https://github.com/letsencrypt/boulder and written in Go.
Lets-encrypt and Boulder both use the protocol ACME for Automated Certificate Management Environment specified here as a draft: https://letsencrypt.github.io/acme-spec/
The whole protocol is happening over TLS. As a result the exchanges are encrypted, you know that you are talking to the CA you want to talk to (eventhough you might not use DNSSEC) and replay attacks should be avoided.
The whole thing is actually a RESTful API. The client, you, can do GET or POST queries to certains URI on the CA webserver.
Registration
The first thing you want to do is register. Registration is actually done by generating a new pair of RSA keys and sending them the public key (along with your info).
ACME specifies that you should use JWS for the transport of data. Json Web Signature. It's basically Json with authentication (so that you can sign your messages). It actually uses a variant of JWS called Jose that doesn't use a normal base64 encoding but that's all you should know for now. If you really want to know more there is an RFC for it.
So here what a request should look like with JWS (your information are sent unencrypted in the payload field (but don't worry, everything is encrypted anyway because the exchange happens over TLS)):
Boulder will check the signature with the public key you passed, verify the information you gave and eventually add you to its database.
Once you are registered, you can perform several actions:
Update your infos
Get one domain authorized (well actually as many as you'd like)
The server will authenticate you because you will now send your public key along AND you will sign your requests. This all runs on top of TLS by the way. I know I already said that.
Boulder's guts
Boulder is separated in multiple components, this makes the code clearer and ensure that every piece of code does what it is supposed to do and nothing more.
One of the components is called the Web-Front-End (WFE) and is the only one accepting queries from the Client. It parses them, verifies them and passes them to the Registration Authority (RA) that combines the other authorities together to produce a response. The response is then passed back to the WFE and to the client over the ACME protocol. Are you following?
TWe'll see what other authorities the RA has access to in the next queries the client can do. But just to talk about the previous query, the new registration query, the RA talks to the Storage Authority that deals with the database (Which is currently SQLlite) to register your new account.
All the components can be run from a single machine (for the exception of the CA that runs on another library), but they can also be run seperately from different machines that will communicate on the same network via AMQP.
New Authorization
Now that you are registered, you have to validate a domain before you can request a certificate.
You make a request to a certain URI.
Well to be exact you make a POST request to /new-authz, and the response will be 201 if it works. It will also give you some information about where you can go next to do stuff (/authz)
Here's the current list of API calls you can do and the relevant answers.
The server will pass the info to the Policy Authority (PA) and ask it if it is willing to accept such a domain. If so, it will then answer with a list of challenges you can complete to prove you own the domain, along with combinations of accepted challenges to complete. For now they only have two challenges and you can complete either one:
SimpleHTTPS
DVSNI
If you choose SimpleHTTPS the lets-encrypt client will generate a random value and upload something at the address formed by a random value the CA sent you and the random value you generated.
If you choose DVSNI, the client will create a TLS certificate containing some of the info of the challenge and with the public key associated to his account.
The client then needs to query the CA again and the CA's Validation Authority (VA) will either check that the file has been uploaded or will perform a handshake with the client's server and verify that specific fields of the certificates have been correctly filled. If everything works out the VA will tell the RA that will tell the WFE that will tell you...
After that you are all good, you can now make a Certificate Signing Request :)
New Certificate
The agent will now generate a new pair of private key/public key. And create a CSR with it. So that your long term key used in your certificate is not the same as your let's encrypt account.
a CSR example, containing the identifier and the public key
After reception of it, the Web-Front-End of Boulder will pass it to the Registration Authority (RA) which will pass it to the Certificate Authority (CA) that will do all the work and will eventually sign it and send it back to the chain.
1: Client ---new-cert--> WFE
2: WFE ---NewCertificate--> RA
3: RA ---IssueCertificate--> CA
4: CA --> CFSSL
5: CA <-- CFSSL
6: RA <------return--------- CA
7: WFE <------return------- RA
8: Client <------------- WFE
Oh and also. the CA is talking to a CFSSL server which is CloudFlare's PKI Toolkit, a nice go library that you can use for many things and that is used here to act as the CA. CFSSL has recently pushed code to be compatible with the use of HSM which is a hardware device that you HAVE to use to sign keys when you are a CA.
After reception the lets-encrypt client will install the fresh certificate along with the chain to the root on your server and voila!
You can now revoke a certificate in the same way, but interestingly you won't need to sign the request with your account key, but with the private key associated to your certificate's public key. So that even if you lose your agent's key you can still revoke the certificate.
Other stuff
There are a bunch of stuff that you will be able to do with the lets-encrypt client but that haven't been implemented yet:
Renew a certificate
Read the Terms of Service
Query OCSP requests (see if a certificate has been revoked)
...
Moar
This post is a simplification of the protocol. If you want to know more and don't want to dig in the ACME specs right now you can also take a look at Boulder's flow diagrams.
This is because when you are a CA you are suppose to keep the root certificate offline. So you sign a (few) certificate(s) with that root, you lock that root and you use the signed certificate(s) to sign other certificates. This is all very serious because if something goes wrong with the root certificate, you can't revoke anything and well... the internet goes wrong as a result (until vendors start removing these from their list of trusted roots).
So the keys for the certificate have to be generated during a "ceremony" where everything is filmed and everyone must authenticate oneself at least with two different documents, etc... Check Wikipedia's page on Key Ceremony it's "interesting".
Also, I received a note from Seth David Schoen and I thought that was an interesting anecdote to share :)
the name "Boulder" is a joke from the American children's
cartoon Looney Tunes: https://en.wikipedia.org/wiki/Wile_E._Coyote_and_The_Road_Runner
This is because the protocol that Boulder implements is called ACME,
which was also the name of the company that made the products
unsuccessfully used by the coyote to attempt to catch the roadrunner.
https://en.wikipedia.org/wiki/Acme_Corporation
Two of those products were anvils (the original name of the CA software)
and boulders.
The Hacking Week ended 2 weeks ago and EISTI got the victory.
I'm the proud creator of the crypto challenge number 4, still available here, which was solved 12 times.
I also wrote a Proof of Solvableness, reading it should teach you about a simple and elegant crypto attack on RSA: the Same Modulus Attack.
(Note that I wrote that back in January)
Let's start
We are presentend with 4 files:
alice.pub
irc.log
mykey.pem
secret
the irc.log reads like this:
Session Start: Thu Feb 05 20:49:04 2015
Session Ident: #mastercsi
[20:49] * Now talking in #mastercsi
[20:49] * Topic is 'http://www.math.u-bordeaux1.fr/CSI/ |||| http://www.youtube.com/watch?v=zuDtACzKGRs "das boot, ouh, ja" ||| http://www.koreus.com/video/chat-saut-balcon.html ||| http://blog.cryptographyengineering.com/ ||| http://www.youtube.com/watch?v=K1LZ60eMpiw ||| petit chat http://www.youtube.com/watch?v=eu2kVcWKvRo ||| sun : t'as le droit de boire quand même va'
[20:49] * Set by Jiss!~Jiss@2001:41d0:52:100::65d on Sat Nov 22 00:06:50
[20:49] <asdf> et donc j'ai chopé une vieille clé rsa qu'alice utilisait
[20:49] <qwer> alice la alice? tu te fous de moi ?
[20:49] <asdf> haha non
[20:49] <asdf> mais le truc est corrompu, ça a l'air de marcher pour chiffrer mais la moitié de la clé a disparu
[20:49] <qwer> attend j'ai sa clé publique qui traine quelque part, et même un fichier chiffré avec. me suis toujours demandé ce que c'était...
[20:50] <asdf> je t'ai envoyé le truc, mais ça m'étonnerait que ça soit la même clé non ?
[21:22] * Disconnected
Session Close: Thu Feb 05 21:22:11 2015
so alice.pub seems to be alice public rsa key. secret seems to be the file encrypted under this key and mykey.pem should be the partial key which was found.
It looks like prime1, prime2 and some other stuff are pretty short. I guess this is what he meant by "half the key" is messed up.
By the way this is what a RSA PrivateKey should look like:
> RSAPrivateKey ::= SEQUENCE {
version Version,
modulus INTEGER, -- n
publicExponent INTEGER, -- e
privateExponent INTEGER, -- d
prime1 INTEGER, -- p
prime2 INTEGER, -- q
exponent1 INTEGER, -- d mod (p-1)
exponent2 INTEGER, -- d mod (q-1)
coefficient INTEGER, -- (inverse of q) mod p
otherPrimeInfos OtherPrimeInfos OPTIONAL
}
So this is what exponent1, exponent2 and coefficient are. Just additional information so that computations are faster thanks to CRT.
the partial key and alice public key seems to share the same modulus. this is vulnerable. If our public/private exponents are not messed up, this means we can factor the modulus and thus inverse Alice's public key.
Let's retrieve all the info we have and put them in a file:
openssl rsa -pubin -in alice.pub -modulus -noout | sed 's/Modulus=//' | xclip -selection c
Here's the modulus. We know that our public key is 3, let's get the private key in the clipboard as well
here I parse mykey.pem with openssl. I select the lines I want with grep. It returns two results, the modulus and the private key. I select only the second line with tail. I select only the 7th column with awk. I remove the : with sed. And now I have a beautiful integer in my clipboard.
Okay so let's do a bit of Sage now:
# let's write the info we have
modulus = int(0xC6C83529A2388F146365C5F5FD4B0D888961B95DE10FFA8853A3C2CBED750E9959BD0FF872C2232F6BAD32624F356A82D062755E1E4FEDAE54E8CA2471FC8D13AC700EE25720D4D9089FD6FBD42F12E6A41E1C1DE81F578C32132AD08594E851841D0239CD410DEF11D1C15EE75B92F86A04F7C6C7F36B9046B8FB2FE29565B1)
public = 3
private = int(0x848578C66C25B4B84243D94EA8DCB3B05B967B93EB5FFC5AE26D2C87F3A35F10E67E0AA5A1D6C21F9D1E2196DF78F1AC8AEC4E3EBEDFF3C98DF086C2F6A85E0BEFC0CA19C5E2495549FEE52E513E7BE9F22207D24B847FBB0CB5BAB795C690053E652D11539A2D960FEADECB9B175487000F7812CEACF5DB83301606CC357DA3)
# now let's factor the modulus
k = (private * public - 1)//2
carre = 1
g = 2
while carre == 1 or carre == modulus - 1:
g += 1
carre = power_mod(g, k, modulus)
p = gcd(carre - 1, modulus)
print(p)
This does not work. This should work.
Let's re-do the maths:
We know that our private and public keys cancel out. This is RSA:
private * public = 1 mod phi(N)
so we have private * public - 1 = 0 mod phi(N)
So for any g in our ring, we should have g^(private * public - 1) = 1 mod N
This is how RSA works.
Let's write it like that: private * public - 1 = k with k a multiple of phi(n). And we know that phi(n) = (p-1)(q-1) is even. So it could be written like this: k = 2^t * r with r an odd number.
Now if we take a random g mod N and we do g^(k/2) it should be the square root of a 1.
The Chinese Remainder Theorem tells us that there are 4 square roots mod N:
1 mod p
-1 mod p
1 mod q
-1 mod q
and two of them should be 1 mod N and -1 mod N. The 2 others should be different from 1 and -1 mod N. That's what I was trying to find in my code.
Once we have found this x mod N which is a square root of 1 mod N, we know that it is either x = 1 mod p or x = -1 mod p.
If we are in the first case, we shoudl have x - 1 = 0 mod p which translates into x - 1 is a multiple of p.
Doing gcd(x - 1, N) should give us p the first prime. If you don't understand it maybe check Dan Boneh's explanation (proof end of page 3) which should be clearer than mine.
With p it's easy to get q the other prime.
But it doesn't work...
Ah! I forgot that g^(k/2) could equal 1 all the time if k/2 were to be a multiple of phi(n). So let's code a loop that divides k by 2 and tries any g^k until it is giving us something else than 1. Then we know how many times we have to divide k by 2 so it's not a multiple of phi(n) anymore.
It turns out we just have to do it 3 times. And then it magically works. A bit more of Sage gives us the primes:
# p and q our primes
p = gcd(carre - 1, modulus)
q = modulus // p
# now that we have factored N let's find alice decryption key
public = 65537
phi = (p - 1) * (q - 1)
private = inverse_mod(public, phi)
Now that we have Alice's private key there are two ways to decrypt our secret:
recreate a valid rsa key with those values and use openssl rsautl
figure out how openssl rsautl works to do it ourselves
Let's do the first one. We'll modify our mykey.pem for this:
first is coded the type 02 (integer), then the length (81 repeated twice because the value block is bigger than 127bits, so we set the first byte to 81 (10000001, the first bit means it is a long way of defining the length, the 7 following bits are the number of byte it will take to define the length, in our case only one and it will be the next one) and the second byte to the actual size), then there is our modulo in hexadecimal. Note that the most significant bit of our value has to be zero if it is a positive integer, that's why we use 41 instead of 40 and lead the payload with 00.
So let's take the time and break this apart:
3082 // some header
0122 // the length of everything that follows (in byte)
0201 // integer of size 1
00
028181 // integer of size 0x81 (our modulus)
00c6c83529a2388f146365c5f5fd4b0d888961b95de10ffa8853a3c2cbed750e9959bd0ff872c2232f6bad32624f356a82d062755e1e4fedae54e8ca2471fc8d13ac700ee25720d4d9089fd6fbd42f12e6a41e1c1de81f578c32132ad08594e851841d0239cd410def11d1c15ee75b92f86a04f7c6c7f36b9046b8fb2fe29565b1
0201 // integer of size 1 (our public key)
03
028181 // integer of size 0x81 (our private key)
00848578c66c25b4b84243d94ea8dcb3b05b967b93eb5ffc5ae26d2c87f3a35f10e67e
0aa5a1d6c21f9d1e2196df78f1ac8aec4e3ebedff3c98df086c2f6a85e0b
efc0ca19c5e2495549fee52e513e7be9f22207d24b847fbb0cb5bab795c6
90053e652d11539a2d960feadecb9b175487000f7812ceacf5db83301606
cc357da3
0202 // integer of size 2 (prime 1)
00f5
0202 // integer of size 2 (prime 2)
00cf
0202 // integer of size 2 (exponent 1)
00a3
0202 // integer of size 2 (exponent 2)
008a
0202 // integer of size 2 (coefficient)
00bd
Now let's remove everything which is after the modulus and let's refill the file with our own values. Let's go back in Sage to calculate them:
