RSA, first described in 1977, is the most famous public-key cryptosystem. It has two main use-cases:
Although RSA's security is based on the difficulty of factoring large composite numbers, in recent years the cryptosystem has received criticism for how easy it is to implement incorrectly. Major flaws have been found in common deployments, the most notorious of these being the ROCA vulnerability which led to Estonia suspending 760,000 national ID cards.
These challenges introduce you to the many footguns of RSA, and soon see you performing attacks which have caused millions of dollars of damage in the real world.
All operations in RSA involve modular exponentiation.
Modular exponentiation is an operation that is used extensively in cryptography and is normally written like: 2^{10} mod 17
You can think of this as raising some number to a certain power (2^{10} = 1024
), and then taking the remainder of the division by some other number (1024 mod 17 = 4
). In Python there's a built-in operator for performing this operation: pow(base, exponent, modulus)
In RSA, modular exponentiation, together with the problem of prime factorisation, helps us to build a "trapdoor function". This is a function that is easy to compute in one direction, but hard to do in reverse unless you have the right information. It allows us to encrypt a message, and only the person with the key can perform the inverse operation to decrypt it.
Find the solution to 101^{17} mod 22663
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RSA encryption is modular exponentiation of a message with an exponent e
and a modulus N
which is normally a product of two primes: N = p * q
.
Together the exponent and modulus form an RSA "public key" (N, e)
. The most common value for e
is 0x10001
or 65537
.
"Encrypt" the number 12
using the exponent e = 65537
and the primes p = 17
and q = 23
. What number do you get as the ciphertext?
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RSA relies on the difficulty of the factorisation of the modulus N
. If the primes can be found then we can calculate the Euler totient of N
and thus decrypt the ciphertext.
Given N = p*q
and two primes:p = 857504083339712752489993810777
q = 1029224947942998075080348647219
What is the totient of N
?
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The private key d
is used to decrypt ciphertexts created with the corresponding public key (it's also used to "sign" a message but we'll get to that later).
The private key is the secret piece of information or "trapdoor" which allows us to quickly invert the encryption function. If RSA is implemented well, if you do not have the private key the fastest way to decrypt the ciphertext is to first factorise the modulus.
In RSA the private key is the modular multiplicative inverse of the exponent e
modulo the totient of N
.
Given the two primes:p = 857504083339712752489993810777
q = 1029224947942998075080348647219
and the exponent:e = 65537
What is the private key d
?
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I've encrypted a secret number for your eyes only using your public key parameters:N = 882564595536224140639625987659416029426239230804614613279163
e = 65537
Use the private key that you found for these parameters in the previous challenge to decrypt this ciphertext:c = 77578995801157823671636298847186723593814843845525223303932
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How can you ensure that the person receiving your message knows that you wrote it?
You've been asked out on a date, and you want to send a message telling them that you'd love to go, however a jealous lover isn't so happy about this.
When you send your message saying yes, your jealous lover intercepts the message and corrupts it so it now says no!
We can protect against these attacks by signing the message.
Imagine you write a message M
. You encrypt this message with your friend's public key: C = M^{e0} mod N_{0}
.
To sign this message, you calculate the hash of the message: H(M)
and "encrypt" this with your private key: S = H(M)^{d1} mod N_{1}
.
In real cryptosystems, it's best practice to use separate keys for encrypting and signing messages.
Your friend can decrypt the message using their private key: m = C^{d0} mod N_{0}
. Using your public key they calculate s = S^{e1} mod N_{1}
.
Now by computing H(m)
and comparing it to s
: assert H(m) == s
, they can ensure that the message you sent them, is the message that they received!
Sign the flag crypto{Immut4ble_m3ssag1ng}
using your private key and the SHA256 hash function.
Challenge files:
- private.key
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So far we've been using the product of small primes for the modulus, but small primes aren't much good for RSA as they can be factorised using modern methods.
What is a "small prime"? There was an RSA Factoring Challenge with cash prizes given to teams who could factorise RSA moduli. This gave insight to the public into how long various key sizes would remain safe. Computers get faster, algorithms get better, so in cryptography it's always prudent to err on the side of caution.
These days, using primes that are at least 1024 bits long is recommended—multiplying two such 1024 primes gives you a modulus that is 2048 bits large. RSA with a 2048-bit modulus is called RSA-2048.
Some say that to really remain future-proof you should use RSA-4096 or even RSA-8192. However, there is a tradeoff here; it takes longer to generate large prime numbers, plus modular exponentiations are predictably slower with a large modulus.
Factorise the 150-bit number 510143758735509025530880200653196460532653147
into its two constituent primes. Give the smaller one as your answer.
Resources:
- How big an RSA key is considered secure today?
- primefac-fork
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Here is my super-strong RSA implementation, because it's 1600 bits strong it should be unbreakable... at least I think so!
Challenge files:
- inferius.py
- output.txt
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Why is everyone so obsessed with multiplying two primes for RSA. Why not just use one?
Challenge files:
- output.txt
Resources:
- Why do we need in RSA the modulus to be product of 2 primes?
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It was taking forever to get a 2048 bit prime, so I just generated one and used it twice.
If you're stuck, look again at the formula for Euler's totient.
Challenge files:
- output.txt
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Using one prime factor was definitely a bad idea so I'll try using over 30 instead.
If it's taking forever to factorise, read up on factorisation algorithms and make sure you're using one that's optimised for this scenario.
Challenge files:
- output.txt
Resources:
- The Elliptic Curve Factorization Method
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Smallest exponent should be fastest, right?
Challenge files:
- salty.py
- output.txt
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My primes should be more than large enough now!
Challenge files:
- modulus_inutilis.py
- output.txt
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We have a supercomputer at work, so I've made sure my encryption is secure by picking massive numbers!
Challenge files:
- source.py
- output.txt
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I asked my friends to encrypt our secret flag before sending it to me, but instead of using my key, they've all used their own! Can you help?
Challenge files:
- source.py
- output.txt
Resources:
- RSA: how to factorize N given d
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Okay so I got a bit carefree with my last script, but this time I've protected myself while keeping everything really big. Nothing will stop me and my supercomputer now!
Challenge files:
- source.py
- output.txt
Resources:
- Twenty Years of Attacks on the RSA Cryptosystem
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Poor Johan has been answering emails all day and the students are all asking the same questions. Can you read his messages?
Challenge files:
- johan.py
- output.txt
Resources:
- Twenty Years of Attacks on the RSA Cryptosystem
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Finding large primes is slow, so I've devised an optimisation.
Challenge files:
- descent.py
- output.txt
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I've found a super fast way to generate primes from my secret list.
Challenge files:
- marin.py
- output.txt
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I need to produce millions of RSA keys quickly and the standard way just doesn't cut it. Here's yet another fast way to generate primes which has actually resisted years of review.
Challenge files:
- fast_primes.py
- key.pem
- ciphertext.txt
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Here's a bunch of RSA public keys I gathered from people on the net together with messages that they sent.
As excerpt.py shows, everyone was using PKCS#1 OAEP to encrypt their own messages. It shouldn't be possible to decrypt them, but perhaps there are issues with some of the keys?
Challenge files:
- excerpt.py
- keys_and_messages.zip
Resources:
- The recent difficulties with RSA
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It seems like my method to generate fast primes was not completely secure. I came up with a new approach to improve security, including a factorization backdoor in case I ever lose my private key. You'll definitely need some complex techniques to break this!
You may need to tweak the recursion limit (sys.setrecursionlimit(n)
in Python) in your programming language to get your solution working.
Challenge files:
- complex_primes.py
- output.txt
Challenge contributed by joachim
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Can custom padding save one from some of the mistakes we already covered?
Challenge files:
- pad_encrypt.py
- output.txt
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Let's Decrypt was too easy, let's do it again!
Challenge contributed by Robin_Jadoul and Thunderlord
Connect at nc socket.cryptohack.org 13394
Challenge files:
- 13394.py
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