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.

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# Schnorr's Signature and non-interactive Protocols posted December 2014

## Interactive Protocols

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:

1. Prover sends a fixed value.
2. Verifier sends a 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 :

1. Complete: a Prover can successfully answer the challenge if he is honest.
2. 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:

1. Completeness: Can the Prover answer correctly thanks to his secret?
2. 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?
3. 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.

## Non-interactive Protocols

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.

we have <g> = G, with g of order q. The Prover wants to show he knows x in g^x = y.

1. the Prover sends t = g^e
2. the Verifier sends a challenge c
3. 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.

## Schnorr's Signature

This is what Schnorr's Signature is doing:

1. t = g^e
2. c = H(m || t)
3. 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 x.

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!

# FAQ on Whitebox Cryptography posted December 2014

A great FAQ, written by Marc Joye (Thomson R&D), on Whitebox Cryptography.

http://joye.site88.net/papers/Joy08whitebox.pdf

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.

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# Last exam of my life posted December 2014

And I just passed the last exam of this semester, which should be the last exam of my life =) Now is time to take a few days to relax and eat nice food because it will soon be christmas ^^ (or holidays, as I heard some american say to avoid saying christmas).

A few interesting things I had to do during my exams these last few days:

• Simple Power Analysis (SPA). Guess what algorithm is used from smartcards' traces and calculate the exponent if it's a binary exponentiation

In the picture you can see two patterns, "1" is represented by two operations in the algorithm, and one of them is squaring which happens also when you have a "0" in your exponent's binary representation. So following the computations revealed by the power trace you can guess the binary representation of the exponent.

I had to read this article explaining two malloc implementations and their vulnerabilities. GNU Lib C (used in Linux) and System V AT&T (used in Solaris, IRIX). I knew the double chained list system but System V uses a different approach: binary tree and also a realfree function that completes the free function.

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# Airbus crypto challenge write-up posted December 2014

Airbus made a "private" challenge called « Trust the future » and accessible only by some selected schools (epitech, insa, and others). I wasn't invited to participate but there was a "crypto" challenge I thought was interesting. Since the challenge just finished I'm posting the write up.

## Crypto challenge #1

We have 4 certificates and a challenge1 file that seems to be a s/mime file of a pkcs#7 enveloped-data object.

2.4.3 EnvelopedData Content Type
This content type is used to apply privacy protection to a message. A sender needs to have access to a public key for each intended message recipient to use this service. This content type does not provide authentication.

and

3.2 The application/pkcs7-mime Type
The application/pkcs7-mime type is used to carry CMS objects of several types including envelopedData and signedData. The details of constructing these entities is described in subsequent sections. This section describes the general characteristics of the application/pkcs7-mime type.

rfc2633

## Certificates

We dump the info of each certificates in human readable format, openssl has commands for that (I think certtool does as well, but I'm on windows using cmder and openssl is the one included).

openssl x509 -in alice.crt -text -noout -out alice.crt.txt

We see that alice, bob and charly use the same rsa exponent (3).

## Reminder: RSA

If you're familiar with RSA (and it's highly probable you are if you read this blog) you can skip this section.

RSA is an asymmetric encryption scheme (also used as a signature). It works by generating a set of private key/public key, the private key is of course kept private and the public key is publicly disclosed. If someone wants to send us a private message he can encrypt it with our public key and we will be able to decrypt it with the private key. The public key is the pair of number (n, e) where n is called the modulus and e is called the exponent. If we want to encrypt a message m with the public key we "basically" do c = m^e modulo n and send c. To decrypt it we use our private key d like this: m = c^d modulo n.

The math behind this is that n is generated from two secret primes p and q (big enough) n = p x q and d = e^-1 modulo (p-1)(q-1), (p-1)(q-1) being phi(n) being the order of the multiplicative group Z/nZ. The security comes from the fact that it's Computationally Hard to find the inverse of e if we don't know p and q. By the way, Heartbleed (a recent attack on openssl) led to finding one of the prime, thus the entire decomposition of n.

## Textbook RSA vs real life RSA

This is all theory. And in practice we have to go through several steps to encrypt an ascii message, make sure it is of length lesser than the modulus, make sure the modulus is big enough, etc...

Textbook RSA is also deterministic and thus not semantically secure (see my previous post) + it is malleable: imagine you intercept c, and of course you know (n, e) (the public key). You could compute c' = 2^e * c = 2^e * m^e = (2m)^e modulo n, this would correctly decrypt as 2m.

Thus, to counter those in practice, RSA Encrytion uses padding (usually OAEP) to make it probabilist and not malleable.

## Let's go back to our challenge

We open our challenge1 file:

MIME-Version: 1.0
Content-Disposition: attachment; filename="smime.p7m"
Content-Type: application/x-pkcs7-mime; smime-type=enveloped-data; name="smime.p7m"
Content-Transfer-Encoding: base64

BAYTAkZSMQ4wDAYDVQQHEwVQYXJpczEXMBUGA1UEAxQOY2FAZXhhbXBsZS5jb20C
EGOE4rIYS8v1jszxDKemVjwwDQYJKoZIhvcNAQEBBQAEggEAweI1fG/FPxzF4Odu
sSJL6PJOiDklHPlUqYCQxFSfG6+3vLEAbdKpgtVsHS0+a0IhItAfeNoAmXdreJFi
6M6U7j0ee4iqgXXbuG8vSsZTYbyUmzuQRgdByu5vGr3FvWxSlvvI8tr/d/cRDqMt

To read that we need to extract the pkcs7 object and parse it. Openssl allows us to do this:

openssl smime -in challenge1 -pk7out -out challenge1.p7m
openssl asn1parse -text -in challenge1.p7m

We get an annoying dump of info to read. With three of those things:

 95:d=6  hl=2 l=  16 prim: INTEGER           :6384E2B2184BCBF58ECCF10CA7A6563C
113:d=5  hl=2 l=  13 cons: SEQUENCE
115:d=6  hl=2 l=   9 prim: OBJECT            :rsaEncryption
126:d=6  hl=2 l=   0 prim: NULL
128:d=5  hl=4 l= 256 prim: OCTET STRING      [HEX DUMP]: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

Which means the same message was sent to three recipients, identified by their serial number which we recognize as being our alice, bob and charly.

We also get this at the end:

 1110:d=4  hl=2 l=   9 prim: OBJECT            :pkcs7-data
1121:d=4  hl=2 l=  20 cons: SEQUENCE
1123:d=5  hl=2 l=   8 prim: OBJECT            :des-ede3-cbc
1133:d=5  hl=2 l=   8 prim: OCTET STRING      [HEX DUMP]:01D4CE3AF4D17ABB

Which means that the data sent (after this dump) is encrypted by 3DES version 3 (three different keys) in CBC mode with an IV 01D4CE3AF4D17ABB.

### Reminder: DES-EDE3-CBC

I like to put reminders like this so you don't have to switch to Wikipedia if you don't remember what are those letters.

DES (Data Encryption Standard) is the famous no-longer-used block cipher (because it was broken ages ago). EDE3 short for the third version of the Triple DES block cipher (that is still considered secure today, it was a response to DES no longer being secure) which uses 3 different keys. Encrypting is done like this:

• we encrypt with key1
• then we decrypt with key2
• then we encrypt again with key3
E(k3, D(k2, E(k1, M)))

Hence the triple DES.

CBC is a mode of operation. A block cipher can only encrypt/decrypt blocks of a certain size (64bits with DES). If you want to do more (or less) you have to use a mode of operation (and a padding system).

### Chinese Remainder Theorem

Here the interesting thing is that the same message was sent to three different recipients, encrypted with the same exponent (3). Let's write down the informations we have:

c1 = m^3 modulo n1
c2 = m^3 modulo n2
c3 = m^3 modulo n3

c1 being the encrypted message sent to Alice, n1 being Alice's modulus, and so on...

We have a system with one unknown: the message. The Chinese Remainder Theorem works in a similar fashion to Lagrange Interpolation (anecdote time: it is used in Shamir's Secret Sharing).

So that we have:

m^3 = c1 * n2 * n3 * ((n2 * n3)^-1 [n1]) +
c2 * n1 * n3 * ((n1 * n3)^-1 [n2]) +
c3 * n1 * n2 * ((n1 * n2)^-1 [n3])
modulo n1 * n2 * n3

A brief explanation: We have c1 = m^3 modulo n1, to place it in a formula modulo n1 * n2 * n3 we have to cancel it when it's modulo n2 or modulo n3. How to make something congruent to zero when its modulo n2 or n3 ? Make it a multiple of n2 or n3. So we multiply c1 with n2 and n3. But then when it will be modulo n1 we will have the value c1 * n2 * n3 which is not correct (c1 = m^3 modulo n1 !). So let's cancel the n2 and n3 with their inverse modulo n1. We then have c1 * n2 * n3 * ((n2 * n3)^-1 [n1]). We do this with all the equations to find the bigger equation. This is the Chinese Remainder Theorem. Simple no?

And this result is even more useful since we know that:

m < n1
m < n2
m < n3
=>
m^3 < n1*n2*n3

Of course if m was greater than one of the modulus then it would decrypt incorrectly. So what we have is:

m^3 = something modulo n1*n2*n3
=>
m^3 = something

That's right, we can get rid of the modulo. We then do a normal cubic root and we find m.

Here's the quick python code I hacked together for this:

(by the way we can quickly get the modulus of each recipients with openssl: openssl x509 -in alice.crt -modulus)

## 6384E2B2184BCBF58ECCF10CA7A6563C (Alice)
c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

## 9F9D51BC70EF21CA5C14F307980A29D8 (Bob)

## A6D4EF4DD38B1BB016D250C16A680470 (Charly)

## base16 -> base10
c1 = int(c1, 16)
c2 = int(c2, 16)
c3 = int(c3, 16)
n1 = int(n1, 16)
n2 = int(n2, 16)
n3 = int(n3, 16)

## extended euclide algorithm
def xgcd(a,b):
"""Extended GCD:
Returns (gcd, x, y) where gcd is the greatest common divisor of a and b
with the sign of b if b is nonzero, and with the sign of a if b is 0.
The numbers x,y are such that gcd = ax+by."""
prevx, x = 1, 0;  prevy, y = 0, 1
while b:
q, r = divmod(a,b)
x, prevx = prevx - q*x, x
y, prevy = prevy - q*y, y
a, b = b, r
return a, prevx, prevy

## chinese remainder formula
n2n3 = n2 * n3
n1n3 = n1 * n3
n1n2 = n1 * n2

n2n3_ = xgcd(n2n3, n1)[1]
n1n3_ = xgcd(n1n3, n2)[1]
n1n2_ = xgcd(n1n2, n3)[1]

m3 = c1 * n2n3 * n2n3_ + c2 * n1n3 * n1n3_ + c3 * n1n2 * n1n2_

m3 = m3 % (n1n2 * n3)

print(m3)

from decimal import *

getcontext().prec = len(str(m3))
x = Decimal(m3)
power = Decimal(1)/Decimal(3)

if diff == Decimal(0):
print("x is the cubic number of", ranswer)
else:
print("x has a cubic root of ", answer)

Note:

• The xgcd function is included in sage but here I use Python so I included it in the code.
• We need to use the decimal package to calculate the cubic root because our number is too big.

We then get this big ass number that we convert to hexadecimal (hex(number) in python). This yields:

0001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff004f8957408f0ea202c785b95e206b3ba8da3dba7aea08dca1

We refer once more to the RFCs

8.1 Encryption-block formatting

A block type BT, a padding string PS, and the data D shall be formatted into an octet string EB, the encryption block.

          EB = 00 || BT || PS || 00 || D .           (1)

The block type BT shall be a single octet indicating the structure of the encryption block. For this version of the document it shall have value 00, 01, or 02. For a private- key operation, the block type shall be 00 or 01. For a public-key operation, it shall be 02.

The padding string PS shall consist of k-3-||D|| octets. For block type 00, the octets shall have value 00; for block type 01, they shall have value FF; and for block type 02, they shall be pseudorandomly generated and nonzero. This makes the length of the encryption block EB equal to k.

rfc 2315

We have our 3DES key: 4f8957408f0ea202c785b95e206b3ba8da3dba7aea08dca1 to use.

Let's get the hexdump the end of the file (you can use commandline utilities like base64, hexdump, dd and xdd):

openssl smime -in challenge1 -pk7out > b64file
base64 -d b64file > hexfile
hexdump -s 1135 hexfile
dd
xdd

And finally decrypt our encrypted file with openssl since it provides a command for that:

openssl des-ede3-cbc -d -iv 01D4CE3AF4D17ABB -K 4f8957408f0ea202c785b95e206b3ba8da3dba7aea08dca1 -in encrypted

Voila ! That was really fun :)

# Is there any particular reason to use Diffie-Hellman over RSA for key exchange? posted December 2014

I was wondering why RSA was used in the SSL handshake, and why Diffie-Hellman was used instead in a Perfect Forward Secrecy scheme.

http://security.stackexchange.com/questions/35471/is-there-any-particular-reason-to-use-diffie-hellman-over-rsa-for-key-exchange

There is, however, an advantage of DH over RSA for generating ephemeral keys: producing a new DH key pair is extremely fast (provided that some "DH parameters", i.e. the group into which DH is computed, are reused, which does not entail extra risks, as far as we know). This is not a really strong issue for big servers, because a very busy SSL server could generate a new "ephemeral" RSA key pair every ten seconds for a very small fraction of his computing power, and keep it in RAM only, and for only ten seconds, which would be PFSish enough.

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# Real Life side-channels attacks posted December 2014

Some funny slides from Vitaly Shmatikov on side channels attacks: http://www.cs.utexas.edu/~shmat/courses/cs361s/sidechannels.pdf

So you can tell what someone is typing just by analyzing the sound of the fingers on the keyboard, from a certain distance.

If you observe someone typing at his computer from an outside window, you can analyze the reflections in many objects (glass teapots, plastic bottles, spoons!!! and even eyes).

Like we weren't worried enough.

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# Network/Software Security posted December 2014

I was trying to find some info about the Heap and malloc (for the level 14 of microcorruption) when I ran into some very good videos from the Infosec Institute. I cannot find the name of the speaker but damn he's so good I just lost 2 hours of my life just watching his videos about nmap, pentesting, metaspoilt, and so on...

Here is his video on Heap Overflow:

And his talks are on several different youtube channels. I don't know how legit this is, and if someone can find the name of that guy I would love to know more about him. More about him: Advanced Recon, Advanced Exploitation, and so on...

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# Pandoc : Markdown -> LaTeX posted December 2014

I just turned in my cryptanalysis project: A Linear Cryptanalysis of A5/2 with Sage. Had to write a rapport in LaTeX.

Now I have to finish the challenges over at Microcorruption and produce a write up in LaTeX as well.

Well LaTeX is awful as a writing syntax. I'd rather focus on writing with John Gruber's excellent markdown syntax and then later convert it to LaTeX. And Pandoc does just that! It's magic. Now that I have all my .md files I concat them with a quick python script

output = ""
for ii in range(1,15):
markdown = open(str(ii) + ".md", "r")
output += "\n\n"
markdown.close()

output_file = open("rapport.md", "w")
output_file.write(output)
output_file.close()

A bit of Pandoc magic and voilà ! I have a beautiful .tex

Now let's finish Microcorruption (or at least try :D it's getting pretty hard).

PS: I use Markdown for everything. This blog is written in Markdown and then converted to HTML. I'm also writing a book in Markdown. Well Markdown is awesome.

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