So, if you remember from when we talked about Diffie-Hellman, this is a key exchange protocol that relies on the Discrete Logarithm problem and the Diffie-Hellman assumption.
Usually this is done over a finite field GF p , but now we have just defined a group based on Elliptic Curves which we can use as well. In this case, Alice has a private key and a public key , where G is the base point. Similarly, Bob has and. Alice and Bob exchange public keys, and then each of them can compute a common point. This protocol relies on the assumption that the DL problem is infeasible in the elliptic curve which requires a base point G of high order and the Diffie-Hellman assumption.
Each algorithm has its own details, but the important problem used as a foundation for each of them is the Discrete Logarithm problem over Elliptic Curves as we have defined it here. Beware, however, that similarly to other algorithms, ECC algorithms rely also on other conditions. For example, for ECDSA and DSA there is a secret parameter that must be unique, and two signatures with the same value for this parameter will reveal your secret key.
As usual, if you implement cryptography. Somewhere before the weekend I was discussing about Padding Oracles with a friend and somehow it came up that there was no public tool using timing information for this kind of attacks. Also, some weeks ago I added the CBC-R encryption part to my scripts, in order to be able to encrypt arbitrary information as long as we are able to control the IV. So in this post I'm going to write about these two things: CBC-R encryption and a web based padding oracle attack script using timing information. I wanted to share with you guys the little challenge I prepared for the Campus Party Europe.
The wargame was organized by SecurityByDefault and took place during the last couple of days.
I was asked to prepare a cryptography challenge for it, and I delivered a little problem that became the level 4 challenge in the crypto category. The idea was to give some real crypto instead of the typical break-classic-crypto or find-the-needle-in-the-haystack challenges. You have to find the trick ;-. I'll leave the challenge here, and the solution will be published in SecurityByDefault in some time.
If you have questions or want to share ideas with me you can use the comments, but please do not spoil the solution for other readers! In one of our missions we have intercepted an email containing a file encrypted with AES in ECB mode with a bit key. Together with the file there was what we suspect is the AES key encrypted with a RSA key, which we found to be as follows:. Although it was a tough mission, our Operations team did a great job and was able to provide the following information on the target:.
In addition, the Operations team modified the hardware used by our target and was able to collect a pair of RSA signatures over the same data. One of these signatures contains a fault injected thanks to our hardware modification, while the other one is the correct signature. These are the signature values:.
S1: 3aec8ecfb47c42cb0ecd2a3bdccdd7ff0afb8ca2cdcaedd4dd50aafea8afdadf69c1c6cfda3b3aed01dadcfdd6e9f63a4cc39e1acbfd04dea72bf07cd5fecced6d38a7baced9d3cab27cbc5abaa8d2b5f53f66aeddaedd4a2beeafcadbba9fc28c4a19fddff90f8ce96df8e6eec48ce0cbdae6aa19a01ec87e7bed8efc6dbd6e8cbcafd8f3b27fe4b98fe9aeb4cdedb1bb0fe0eb8fbde30a6ee. S2: eeaddacade5af3a1db70c1e64ef9ac0a3c7e8fdfeadca44fea6b1cbee13cafdaddb5a79fa19ffdf3a61d32fc75edc7a0bedca66faaa7efeada75dda2ec53f3cafaaa3eadcaaa8ba0ff20f1c53deeafdc2ad1a0fefffa7f72edf1f8ce66c1e5fcdfecae46cea3d7d7a5f1c2a0efeb5bc1b8a4af4fb21fce1daeceb3b1e6ba31e7c4bc01bc2ebae4dcd1d3df8b9bcafcbf.
It is critical for the mission to obtain the contents of the encrypted file. Your task is to obtain the contents of the AES file. This means no padding, just modular exponentiation. For keys smaller than the modulus, the padding is null i. After seeing how the ElGamal system works, today we are going to take a look at the RSA public key cryptosystem. The RSA algorithm was first published by Rivest, Shamir and Adleman in and is probably the most used crypto algorithm today.
Despite this fact, the algorithm seems to have been invented by Clifford Cocks, a british mathematician who worked for a UK intelligence agency.
Since this work was never published due to the top-secret classification, the algorithm received its name from Rivest, Shamir and Adleman who were the first to discuss it publicly. A document declassified in revealed the fact that Clifford Cocks had actually described an equivalent system in If you want more accurate and complete descriptions, take a crypto book such as the Handbook of Applied Cryptography I've linked in most of my posts :. The RSA algorithm is based on the assumption that integer factorization is a difficult problem.
This means that given a large value n , it is difficult to find the prime factors that make up n. Based on this assumption, when Alice and Bob want to use RSA for their communications, each of them generates a big number n which is the product of two primes p,q with approximately the same length. Next, they choose their public exponent e , modulo n.
Typical values for e include 3 which is not recommended! From e , they compute their private exponent d so that:. Where is the Euler's totient of n. This is a mathematical function which is equal to the number of numbers smaller than n which are comprimes with n, i. If n is a prime p , then its totient is p-1 since all numbers below p are comprimes with p. In the case of the RSA setup, n is the product of two primes. In that case, the resulting value is lcm p-1 q-1 because only the multiples of p and q are not comprimes with n. Once our two parties have their respective public and private exponents, they can share the public exponents and the modulus they computed.
Once the public key i. When this message is received, it can be decrypted using the private key and a modular exponentiation as well:. In the commands above, I first create two random primes below and compute n. Then I create a IntegerModRing object to compute things modulo lcm p-1,q-1 and perform the computation of the private exponent as the inverse of the public exponent on that ring.
Next, I create a new ring modulo N. Then I can use the public exponent to encrypt a message m and use the private exponent to decipher the cryptotext c We have seen it works with our previous example, but that doesn't prove that it really works always. I could have chosen the numbers carefully for my example and make them work.
Euler's theorem tells us that given a number n and another number a which does not divide n the following is true:. Therefore, and since , for any message m that does not divide n the encryption and decryption process will work fine.
However, for values of m that divide n we need to use more advanced maths to prove the correctness. Another way to prove it is to use Fermat's little theorem and the Chinese Remainder Theorem. I will explain these theorems in my next post and then I will provide a complete proof based on them. In the case of RSA, digital signatures can be easily computed by just using d instead of e. So, for an RSA signature one would take message m and compute its hash H m. Then, one would compute the signature s as:.
For verifying the signature, the receiving end would have to compute the message hash H m and compare it to the hash contained in the signature:.
Therefore, if the hash computed over the received message matches the one computed from the signature, the message has not been altered and comes from the claimed sender. In order to completely break RSA, one would have to factor n into it's two prime factors, p and q. Otherwise, computing d from e would be hard because p-1 and q-1 are not known and n is a large number which means that computing its totient is also difficult.
In a few posts I will show an algorithm to solve the factorization problem. However, another way to break RSA encrypted messages would be to solve a discrete logarithm. Indeed, since , if one solves the discrete logarithm of c modulo n , the message would be recovered. Luckily, we already know that discrete logs are not easy to compute. And in this case, solving one does not break the whole system but just one message. Let's go a little further in our way to understand the way the DNIe works. In my previous post I talked about the device authentication procedure and today I'll talk about what happens next, how Secure Messaging protects all the subsequent communication.
Start a Wiki. RC4 is a widely used stream cipher; see Category:Stream ciphers. In practice today, since the relaxation in US export restrictions, and because almost every personal computer connected to the Internet , everywhere in the world, includes US-sourced web browsers such as Mozilla Firefox or Microsoft Internet Explorer , almost every Internet user worldwide has access to quality cryptography i. Similar ebooks. Modern cryptography intersects the disciplines of mathematics , computer science , and engineering. Until modern times cryptography referred almost exclusively to encryption , which is the process of converting ordinary information plaintext into unintelligible gibberish i. Introduction to Modern Cryptography provides a rigorous yet accessible treatment of this fascinating subject.
So basically at the end of this process, both ends share a pair of keys that can be used for protecting the confidentiality and the integrity of subsequent messages. In our last post we learnt about the Discrete Lograithm problem, why it is a difficult problem and how we can attempt to solve it if the numbers are manageable.
Of course, in a real setting we wouldn't use 16 bit numbers as in my example, but at least bit numbers nowadays and most likely even bigger numbers. We will look at how ElGamal uses the DL problem to provide public key encryption and digital signatures. Skip to search form Skip to main content.
With the rise of new network architectures and services, the field encompasses much more than traditional communication where each side is of a single user. It also covers emerging communication where at least one side is of multiple users.
New Directions of Modern Cryptography presents general principles and application paradigms critical to the future of this field. View via Publisher. Alternate Sources. Save to Library.
Summary. Modern cryptography has evolved dramatically since the s. With the rise of new network architectures and services, the field encompasses much. Buy New Directions of Modern Cryptography on ykoketomel.ml ✓ FREE SHIPPING on qualified orders.
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