How can you explain the discrete logarithm problem?
The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to the base g of h in the group G. Discrete logarithm problem is not always hard. The hardness of finding discrete logarithms depends on the groups.
Is it possible to solve the discrete logarithm problem?
The discrete logarithm problem is considered to be computationally intractable. That is, no efficient classical algorithm is known for computing discrete logarithms in general. A general algorithm for computing logb a in finite groups G is to raise b to larger and larger powers k until the desired a is found.
What is discrete logarithm attack?
Discrete Logarithm Attacks. Many cryptosystems could be broken if we can compute discrete log- arithms quickly, that is, if we could solve the equation ax = b in a large finite field.
How do you find the discrete log?
A discrete logarithm is just the inverse operation. For example, take the equation 3k ≡ 12 (mod 23) for k. As shown previously, k = 4 is a solution, but it is not the only solution. Because 322 ≡ 1 (mod 23), it also follows that if n is an integer, 34 + 22n ≡ 12 × 1n ≡ 12 (mod 23).
What are discrete logarithms explain how are they used in public key cryptography?
All the systems use the properties of the multiplicative group modulo p , denoted Z∗p, for a prime p. Their security ultimately depends on the intractability of solving the Discrete Logarithm Problem (DLP): namely, if you are given g∈Z∗p and gnmodp then find n.
What is the importance of discrete logarithms?
Aside from the intrinsic interest that the problem of computing discrete logarithms has, it is of considerable importance in cryptography. An efficient algorithm for discrete logarithms would make several authentication and key-exchange systems insecure.
Is RSA a discrete logarithm?
RSA labs makes a similar statement: The discrete logarithm problem bears the same relation to these systems as factoring does to the RSA system: the security of these systems rests on the assumption that discrete logarithms are difficult to compute.
Is the discrete log problem NP complete?
NP-complete. The subset of NP to which all problems in NP can be reduced, i.e. the subset of NP that is NP-hard. All problems which are in NP but neither in P nor in NPC.
What is weak key cryptography?
In cryptography, a weak key is a key, which, used with a specific cipher, makes the cipher behave in some undesirable way. Nevertheless, it is considered desirable for a cipher to have no weak keys. A cipher with no weak keys is said to have a flat, or linear, key space.
What is factorization problem cryptography?
Factoring is the act of splitting an integer into a set of smaller integers (factors) which, when multiplied together, form the original integer. Factoring is the underlying, presumably hard problem upon which several public-key cryptosystems are based, including the RSA algorithm.
How does key stretching work?
Key stretching is the practice of converting a password to a longer and more random key for cryptographic purposes such as encryption. This is generally recognized as making encryption stronger as it ensures that the encryption itself is reasonably hard.
What are the disadvantages of DES?
The main disadvantage to DES is that it is broken using brute-force search. However, using 3DES mitigates this issue at the cost of increasing execution time. DES is also vulnerable to attacks using linear cryptanalysis. However, it takes 247 known plaintexts to break DES in this manner.
What is discrete logarithm used for In cryptography?
The discrete logarithm problem is used in cryptography. Given values for a, b, and n (where n is a prime number), the function x = (a^b) mod n is easy to compute. For example, if a = 3, b = 4, and n = 17, then x = (3^4) mod 17 = 81 mod 17 = 81 mod 17 = 13.
Is it possible to UN-compute the discrete logarithm problem?
For example, if a = 3 and n = 17, then: In addition to the discrete logarithm problem, two other problems that are easy to compute but hard to “un-compute” are the integer factorization problem and the elliptic-curve problem.
Is there a discrete log problem for elliptic curves?
The known methods of attack on the elliptic curve (EC) discrete log problem that work for all curves are slow, making encryption based on this problem practical. However, several eﬃ cient methods for solving the EC discrete log problem for speciﬁc types of elliptic curves are known.
How do you find the kth power of a discrete log?
Let g be the generator of Z p*; then the discrete logarithm problem reduces to computing a, given (g, p, g a mod p) for a randomly chosen a< (p−1). If we want to find the kth power of one of the numbers in this group, we can do so by finding its kth power as an integer and then finding the remainder after division by p.