The security level of a cryptographic primitive, like a cipher or hash function, is a measure of its strength. The security level is typically represented as a number of "bits of security," also known as security strength, where n-bit security means that the attacker would need to perform 2^{n} operations to break it.

Interesting right?

In this article, we'll discuss the Bit Security of Discrete Logarithms in detail.

Let us start:

Understanding Bit Security

Bit Security of any algorithm means the adversary needs to perform 2^{n-bit} operations to break the message.

For example, How strong is 256-bit Encryption?

When you encrypt something, you take the plaintext or unencrypted data and apply an algorithmic operation to produce an encrypted ciphertext. The algorithm youâ€™re using is considered a key.

For example, imagine Jack and Diane, and Jack wants to tell Diane, "Hi Diane, Are you a great Coder?".

Jack will take his message and scramble it into ciphertext using an algorithm or cipher and the encryption key. He will now give Diane the message and the key, which can be used to decrypt it and make it readable.

As long as nobody else gets the "key," the ciphertext (encrypted text) is worthless because it canâ€™t be read.

A bit is a fundamental unit of information. In this context of today's discussion, the bits of security, let's call them (n), refer to the number of operations an attacker would need to perform to guess the value of the private key. The larger the key, the harder it is to guess or crack. Because this key is in 1s and 0s, each bit has two possible values. To break or guess the key, the attacker must perform 2^{n} operations.

That may be too abstract, so here's an example: Assume you have a 2-bit key. That means it will have 2^{2 }(4) different values.

That would be easy to crack for a computer. Still, as critical sizes increase, it becomes extensively tricky for a modern computer to accurately interpret the value of the private key in any reasonable amount of time.

So, how strong is 256-bit encryption?

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Bit Security of Discrete Logarithms

Again, this depends on the algorithm, and it varies from asymmetric to symmetric encryption. Discrete Logarithm follows Asymmetric encryption in which the problem is easy to solve in one direction and challenging to solve in the reverse direction.

So, Asymmetric encryption is based on mathematical problems that are simple to solve in one direction (encryption) but extremely difficult to reverse (decryption).

Let us take an example:

To make Coffee, we need Milk + Coffee powder.

When we have the right items, we can prepare whatever we want to. Right? We have just learned how to prepare a coffee. Isnâ€™t it so easy? đź«ˇ

So, it is easy in one direction.

Now, think of the reverse situation in which you are given a coffee and asked to estimate the exact quantity of milk and coffee powder used to prepare that Coffee. This is undoubtedly a problematic or impractical task. Hence, it takes work in the reverse direction.

To conclude the example, preparing the coffee in one direction is easy but in the reverse direction is hard or impractical.

As a result, attacks on public key, asymmetric cryptosystems are typically much faster than brute-force key space searches that plague private key, symmetric encryption schemes.

Therefore, when we discuss the security level of public key cryptography, we are referring to a calculation of the computational hardness of the implementation in comparison to the best and most well-known attack at the time. Symmetric encryption strength is little easier to calculate as compared to Asymmetric encryptions.

A 256-bit key can have 2^{256} possible combinations. As we discussed earlier, a 2-bit key would have 4 possible combinations (and be easily crackable by the attacker).

Weâ€™re dealing in exponentiation here, so each time we raise the exponent, n, we increase the number of possible combinations. So, 2^{256} is nothing but 2 x 2, x 2, x 2â€¦ 256 times.

As we've discussed, the best method for breaking an encryption key is "brute-forcing," which is essentially just trial and error.

Therefore, if the key is 256 bits long, there are 2^{256} possible combinations, and a hacker must try the majority of these 2^{256} combinations before drawing a conclusion.

Since the success rate for guessing the key is typically around 50%, it won't take all of them, but the amount of time required to perform so would be far longer than a person's lifetime.

So, The total number of combinations for a 256-bit private key is 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936 (that's 78 digits).

No Supercomputer on the planet could solve that in a reasonable amount of time.

The world's fastest supercomputer, Tianhe-2 (MilkyWay-2), cannot break 256-bit AES encryption because it would take billions of years to do so.

Cryptography allows you to store sensitive information or send it through insecure networks (such as the Internet).

How is cryptography used in cryptocurrency?

With the help of cryptography, cryptocurrency transactions are anonymous, secure, and "trustless," meaning there is no need for a bank, credit card company, government, or another intermediary to be present. Instead, you can conduct business with anyone without knowing anything about them.

What is the role of Cryptography in Information Security?

Cryptography safeguards data availability, ensures data integrity, ensures the right sender or receiver, and protects information confidentiality.

Specify the Key differences between Symmetric Key and Asymmetric Key encryption.

Symmetric key encryption is quicker, more efficient and simple, having a single private key and is employed when a huge volume of data needs to be sent. Asymmetric key encryption is slower and more complex. Both private and public keys are used. It's used to transmit and receive a small amount of data.

Conclusion

To conclude the discussion, we have extensively discussed the Bit-Security of Discrete Logarithms in Cryptography. We have also gone through the one-way function and talked about Asymmetric and Symmetric encryption. At last, we took the example of 256-bit encryption as an example to understand the bit-security of Discrete Logarithms.

If you like this article and want to learn more about Cryptography, please refer to these articles and enhance your knowledge.