# How can frequency analysis be applied to modern ciphers?

I am building a computer program that deciphers Caesar, Vigenere and monoalphabetic substitution ciphers. All of those are susceptible to frequency analysis. However, it does not seem to be real-world applicable considering the complexity of the modern-day ciphers. Where else can frequency analysis be applied in the modern day to make my project real-world applicable?

The only suggestion I have been given is an application to the Bitcoin encryption algorithm. I am not sure if this would be a viable option. Is it?

• Oct 3, 2019 at 21:46
• Outside the field of cryptography, frequency analysis is used in most data compression algorithms -- the ones that use either arithmetic coding or Huffman coding. Sep 11, 2022 at 0:31

## 3 Answers

Classical ciphers operate on letters. If we consider the frequency attack on classical ciphers it considers the frequency of the letters. Modern ciphers, if we consider only block ciphers, operates on blocks - 64, 128, or more bit blocks. Let see how you can perform a frequency attack on modern ciphers to infer some data.

## Example attack on Databases:

Let assume that a sensor network has deployed in a rural area to observe animals by researchers. When an animal's tag is detected, the nodes stores this information on a database table; here for simplicity only the type of the animal and date and time encrypted in ECB mode;

|  TYPE  | D&Time|
|-----------------
| 0x443  | 0x7FE |
| 0x122  | 0x1E0 |
| 0x443  | 0x651 |
| 0x443  | 0x6AA |
| 0x084  | 0x09C |
| 0x112  | 0xC9F |
| 0x443  | 0x18D |
| 0x112  | 0x76B |


Now, assume that the poachers cannot extract the encryption key from the nodes but they can read the information. What can they learn from this data?

In ECB mode $$E_k(x) = E_k(y)$$ iff $$x=y$$ and this can leak information. This mode is used for equality and count queries on encrypted databases, see the CryptDB paper by Popa et.al. An attacker can calualte the frequency of the ciphertext and infer information from them.

By the Kerckhoffs's principle, we assume that they know everything but the key. The poachers also know the region, so they know how frequently a deer or a wolf appears around this sensor node. Assuming that mostly deer than wolf then bears appears in this region.

If we look at the ciphertext frequencies the 0x443 appears most. So they deduce it actually represents a deer. And similarly 0x122 is a bear data.

Long story short result: Frequency analysis can break deterministic encryption. If you have pre-knowledge on the frequency of the data you can infer information without decryption. So a ninth-century attack of Arab philosopher and mathematician al-Kindi still lives.

The attack Naveed et.al is performed using electronic medical records. Some of the data is also publicly available like the frequency of the illnesses, drug sales, etc. So their experimental result shows that an alarming amount of sensitive information can be recovered

Mitigation: Don't use ECB mode. Then, you cannot have simple equality queries on the encrypted data. The CryptDB designed to execute queries of TPC-C ot measure the capability and performance.

The future may reside using equality under the FHE, however, it may not easily solve all of the problems.

Some further readings on modern frequency analysis;

• Your second link at the bottom of your answer doesn't work Nov 16, 2019 at 19:56

The Caesar cipher, Vigenere, monoalphabetic substitution, the autokey cipher, columnar transposition, the Playfair cipher, the Rail Fence cipher, disrupted transposition, the ADFGVX cipher, Quagmire III, etc., are all interesting and good to understand, but compared to modern cryptographic systems they are almost always utterly worthless for providing real-world confidentiality, and confidentiality is just one cryptographic service.

However, the program that you are building does have a real-world application that has interest and value: the frequency analysis of classical ciphers. Other such programs already exist, but perhaps you can make one that is better.

It is difficult to imagine a scenario in which one would want to use a classical cipher for a serious purpose (let's omit the one-time pad for a moment). One's level of assurance would be low, even employing the VIC cipher, which is a tougher nut to crack (because it can resist frequency analysis), even against an inefficient opponent. It is just not realistic. Again, the exception here is the one-time pad, sometimes called Vernam cipher.

The one-time pad has very limited applications in the world today, but those applications do exist and they are important. You could write your program with the one-time pad in mind. Again, this territory has already been gone over.

The problem you face is this: modern ciphers are made to be resistant to frequency analysis. If a modern cipher can be attacked in this way, like 3DES in ECB mode, it is proof of tremendous weakness.

As far as Bitcoin goes, there have been problems in Android's Java SecureRandom with reused R values, but I am not sure how this could be helped by standard frequency analysis, which seems most useful as a tool of classical cryptanalysis.

• This is wrong. al-Kindi's attack still in use on the modern cipher. And you can infer the plaintexts, see my answer and the articles. It was even on news could you provide any [reference] for your claim? Oct 4, 2019 at 7:01
• @kelalaka I am trying to understand what you said. I did not mention al-Kindi. I did not say that frequency analysis is absolutely worthless against every modern cipher in every application. I even showed an example where it is useful. You found a good example, from 2015, in which frequency analysis was used, albeit limited to a very specific example, but still real. I am sure some others may exist in the weaknesses of certain poorly designed applications. Oct 4, 2019 at 8:05
• The idea of frequency attack goes back to 9th century. By saying old articles date back to 2015, I can make an analogy; The padding oracle attack of Vaudany is back to 2002 and around 2012 find a realistic attack. Executing a real attack, however, is prohibited in many countries, so don't expect a real attack. The aim of encryption is using it. If you don't use delete it. There are many database breaches that the company just says hey we have encrypted everything, don't worry This is completely nonsensical. Oct 4, 2019 at 10:27
• Continue: one has to show the database schema because as you can see from the CryptDB paper you need the functionality on the encrypted data. They have tested their work on TPC-C. The simplest one is Select * from e_table WHERE TYPE="0x6e7" Dou you now any practical database work that prevents this? There is a pioneering work of kantarcioglu, however, it is not much practical. One can design with FHE, however, it is not practical. So, in short it is not limited in practice. If you look at the SO database questions, many ask how to avoid this in their schema. Oct 4, 2019 at 10:35
• @kelalaka I think you were right to identify that database weakness from 2015. So far I cannot think of other examples where frequency analysis would be useful. The M-209 was sometimes broken by frequency analysis, but that is not very useful. Oct 4, 2019 at 10:35

This answer applies to cryptographic algorithms in general, rather than specific cases where the plaintext data must have specific properties. For such a situation, see kelalaka's answer.

Frequency analysis is not a new attack, and as such, the encryption functions in use today are designed to resist frequency analysis.

Having said this, there are two aspects to a cryptographic algorithm's security: The encryption function (such as AES, DES, etc) and the Cipher Mode. This link gives a good explanation of the Cipher Modes.

The various cipher modes have different properties. The Electronic Code Book (ECB) cipher basically breaks the message into a series of blocks, and encrypts each block individually. So no block affects any other block's encryption. This means that although a secure encryption function may be used, this will only remove patterns in each of the blocks of the message. However, patterns which occur between multiple blocks will still be visible, as seen in the penguin images at the referenced Wikipedia article. This means that a sophisticated form of frequency analysis could be used to identify patterns between the blocks of an ECB-encrypted message. However, this is unlikely to retrieve the entire message.

ECB is safe to use for messages whose length is less than one message block, but is vulnerable when used for messages which are of greater length than a single block.

• This is wrong. al-Kindi's attack still in use on the modern cipher. And you can infer the plaintexts, see my answer and the articles. It was even on news could you provide any [reference] for your claim? Oct 4, 2019 at 7:01
• Thanks @kelalaka, I have clarified my answer, and removed the unfounded claim. Thank you for your example in your answer. Oct 4, 2019 at 21:56
• all cryptographic algorithms in use today are designed to resist frequency analysis. cryptographic algorithms. Block ciphers are prone to it, we prevent it by probabilistic encryption. Even if you use IV in CBC, some stupid person can use it with fixed IV. Oct 4, 2019 at 22:00
• Yes, I meant to say the encryption function. Oct 4, 2019 at 22:05
• ECB can be safe if you guarantee that there are no repeating blocks. ECB has no CPA security. Oct 4, 2019 at 22:21