Why is asymmetric encryption weaker than symmetric encryption?

Question

Cryptography

Here is a YouTube video where the author claims (11:30):

With the same length of keys, symmetric and asymmetric, the cryptographic strength of the algorithms is different. And the cryptographic strength of symmetrical ones is higher. asymmetrical below. ...

How to explain it?

Answer 1

With the same length of keys, symmetric and asymmetric, the cryptographic strength of the algorithms is different. And the cryptographic strength of symmetrical ones is higher. asymmetrical below. ...
How to explain it?

This is because symmetric and asymmetric algorithms have a different nature.
Symmetric ones (like AES) use a key to generate a transformation from the input block to the output block. The number of bits in the key directly determines the size of the transformation space -- a 128-bit key gives ^2,128 possible output unit values ​​for each input.
Asymmetric uses the key differently, so you need to consider a specific algorithm. RSA uses key bits to store the product of two prime numbers. A 128-bit key gives 64-bit primes. Factorization of a 128-bit number does not require a search of ^{2,128 options and takes} about a second on ordinary modern hardware .

Answer 2

Elliptic cryptography is asymmetric and with the same key lengths as symmetric cryptography.

Answer 3

I really do not understand the essence of the question.
There is encryption resistance to hacking. It depends on the algorithm and on the key length.
Achieving a given strength - for different algorithms requires different key lengths. It's kind of obvious.
Asymmetric algorithms are such that, in addition to cryptographic strength, they also require asymmetry (well, it’s quite an obvious phrase; however, here you need to understand what “asymmetry” is). And for the sake of an additional requirement - almost all ... ; in this case, "the strength drops with the same key length" or "you need to increase the key for the same strength."
(If nothing needs to be sacrificed to improve, then this improvement is made, and the element "A" in the picture by reference is thrown into the trash.)

Answer 4

This is due to the fact that
(1) For modern symmetric algorithms, there are no known attacks that work faster than exhaustive search of keys. For example, if the key length (say for AES) is 128 bits, then the cipher can be cracked in only 2 ^ 128 operations. It can't be faster.
(2) For most asymmetric algorithms, this statement is not true, since for them there are cracking algorithms that work faster than exhaustive search. For ease of calculation, I will not give examples of the fastest hacking algorithms, but I will consider one algorithm that demonstrates this situation.
The stability of many asymmetric systems is based on the complexity of solving the discrete logarithm problem. And it can be solved faster than by exhaustive search, for example, by the Shanks method, which has the complexity of sqrt (key length). Therefore, a 128-bit key provides not 2^128 but only 2^64 cracking difficulty. Therefore, to achieve a cracking difficulty of 2^128, the key length must be 256 bits.
So it turns out that to achieve the complexity of breaking 2 ^ 128, 128 bits are enough for symmetric circuits, and 256 for asymmetric circuits. But here I will make a reservation again that there are more efficient algorithms than the Shanks method, so the keys of asymmetric circuits are usually even longer.