Discrete Log Problem - Applied Cryptography

Summary

The script discusses the discrete log problem, a cornerstone of Diffie-Hellman's security, which involves finding the exponent x in a^x mod n = b, where n is a large prime and a is a generator. It's a hard problem with no known polynomial-time solution, making it a foundation for cryptographic security. The script also explores the concept of a generator and its ability to produce a permutation of numbers in a group, demonstrating with a code example using a small prime and a generator. The difficulty of the discrete log problem is highlighted, as it remains intractable for large primes, underpinning the security of cryptographic systems.

Takeaways

• 🔒 The Diffie-Hellman security property is closely related to the discrete log problem, which is considered a hard problem in cryptography.
• 📚 Discrete logs are analogous to continuous logs but operate over a discrete set of numbers, making them more complex to solve.
• 🔍 In the discrete log problem, the goal is to find 'x' in the equation a^x ≡ b (mod n), which is difficult when 'n' is a large prime number.
• 🔑 A number 'a' is a generator if it can produce all possible numbers in the group Zn when raised to different powers.
• 🤔 The discrete log problem is believed to be hard because the fastest known solutions are exponential, requiring an impractical amount of computation for large 'n'.
• 💡 The script demonstrates a code example that shows the permutation of numbers generated by a specific generator and modulus.
• 👀 The demonstration uses a small prime number and a generator to visually show the sequence of numbers produced, emphasizing the difficulty in predicting the position of a number in the sequence.
• 🚫 There is no known polynomial-time solution to the discrete log problem, suggesting that it remains computationally infeasible for large primes.
• 💡 The script implies that the security of the Diffie-Hellman key exchange relies on the assumption that the discrete log problem is hard to solve.
• 🤖 The hypothetical scenario in the quiz assumes the existence of an adversary with access to a powerful computer and a fast discrete log procedure, questioning the security of the Diffie-Hellman key.
• 🔑 The quiz highlights the importance of the discrete log problem in the context of cryptographic security and the potential implications if a fast solution were found.

Q & A

What is the discrete log problem?

The discrete log problem involves solving for x in the equation a^x = b modulo n, where a and b are known values, and n is a large prime number.

How does the discrete log problem differ from continuous logarithms?

Continuous logarithms involve solving a^x = b in a continuous group with well-known efficient methods, whereas discrete logs deal with discrete groups and are much harder to solve, especially when n is a large prime number.

Why is the discrete log problem considered hard?

The discrete log problem is considered hard because no efficient (polynomial-time) solution has been found, and the best-known solutions are exponential in the size of n.

What is a generator in the context of discrete logarithms?

A generator g in the context of discrete logarithms is an element that, when raised to various powers, produces all the elements of the group Zn in a permutation.

Why is using a large prime number n important in the discrete log problem?

Using a large prime number n is important because it ensures the existence of the discrete log and increases the problem's computational difficulty, making it intractable for large n.

What is the significance of the generator permutation demonstration in the script?

The generator permutation demonstration shows how raising a generator to various powers covers all elements in the group Zn, illustrating the concept of a generator and the sequence involved in discrete logs.

How does the exponential search for solving the discrete log problem work?

The exponential search involves trying all possible powers of the generator until the correct one is found, which is computationally intensive and impractical for large n.

What are the assumptions about the passive attacker in the quiz scenario?

The passive attacker can eavesdrop on messages and has access to powerful computational resources, including a fast procedure for computing discrete logs and modular exponentiation.

Can the passive attacker break a Diffie-Hellman key with the given resources?

Yes, if the attacker has a fast procedure for computing discrete logs on any inputs, they can use it to break the Diffie-Hellman key by solving the discrete log problem with the eavesdropped messages.

Why is it important to choose a large n for secure cryptographic systems?

Choosing a large n ensures that solving the discrete log problem remains computationally infeasible, providing security against attacks that rely on solving this problem.

Outlines

00:00

🔍 Understanding the Discrete Log Problem

The discrete log problem is closely related to the Diffie-Hellman security property. Unlike continuous logarithms, discrete logs operate within a discrete group. The problem involves solving for x in the equation a^x ≡ b (mod n), which is considered difficult when n is a large prime number. Discrete logs may not always exist, but if n is a large prime and a is a generator, they do. This difficulty forms the basis for many cryptographic systems.

🔢 Demonstrating Generator Permutations

To illustrate the concept of a generator, a code example shows the permutation produced by raising a generator to every power from 1 to n-1. Using 277 as the prime number and 5 as the generator, the output demonstrates that these operations result in a permutation of all numbers from 1 to 276. This demonstrates the difficulty in predicting the sequence, which is a key aspect of the discrete log problem.

🧩 Solving the Discrete Log Problem

The challenge of the discrete log problem lies in finding the power needed to raise a generator to produce a given number. This problem is believed to be hard because no polynomial-time solutions exist; the fastest known methods are exponential. Solving the problem requires trying all possible powers, making it computationally infeasible for large n, which grows exponentially with the number of bits in n.

💡 Implications for Cryptographic Security

The intractability of the discrete log problem underpins the security of cryptographic systems like Diffie-Hellman. Assuming no efficient solution exists, large n ensures security. In this context, even with powerful computational resources, solving the problem remains impractical, thus safeguarding cryptographic keys against passive attackers who can only eavesdrop on messages.

🔐 Exploring Diffie-Hellman Key Security

The quiz question explores whether a passive attacker, equipped with a fast procedure for computing discrete logs and modular exponentiation, can break a Diffie-Hellman key. The attacker can eavesdrop on messages between Alice and Bob, gaining access to all transmitted values. The possible answers consider whether such an attacker can break the key with these resources.

Keywords

💡Diffie-Hellman

Diffie-Hellman is a cryptographic protocol that enables two parties to establish a shared secret key over an insecure channel. In the video, the security of this protocol is closely tied to the difficulty of the discrete log problem. The script mentions that the security property of Diffie-Hellman is based on the assumption that it's hard to compute the discrete logarithm, which is a key aspect of the video's theme on cryptographic hardness.

💡Discrete Log Problem

The discrete log problem is a mathematical problem related to the security of the Diffie-Hellman key exchange. It involves finding the exponent x in the equation a^x ≡ b (mod n), where a and b are known, and n is a large prime number. The video explains that this problem is believed to be computationally hard, which is why it forms the basis of the security in Diffie-Hellman.

💡Continuous Log

A continuous log, also known as a logarithm in the usual sense, is the inverse operation to exponentiation where you find the power to which a number must be raised to obtain another number. In the script, it's contrasted with the discrete log to illustrate the difference in solving for x in the context of discrete groups as opposed to real numbers.

💡Discrete Group

A discrete group in the context of the video refers to a set of discrete elements that follow specific group properties, such as closure, associativity, identity, and invertibility. The script uses the example of a group formed by integers modulo n, where the discrete log problem is considered.

💡Generator

In the context of the video, a generator is an element of a group that can produce all other elements of the group when raised to successive powers. The script explains that if 'a' is a generator, then a^x (mod n) will produce a permutation of the group Zn, which is crucial for the discrete log problem.

💡Permutation

A permutation in the video refers to a rearrangement of elements in a set such that each element is uniquely mapped to another element. The script demonstrates that raising a generator to different powers modulo n results in a permutation of the numbers in the group Zn.

💡Modulo Operation

The modulo operation is a mathematical operation that finds the remainder of division of one number by another. In the script, it's used in the context of the discrete log problem where a^x ≡ b (mod n) to denote that a raised to the power x is congruent to b modulo n.

💡Cryptographic Hardness

Cryptographic hardness refers to the difficulty of solving a mathematical problem that underpins the security of a cryptographic system. The video discusses the hardness of the discrete log problem as it relates to the security of the Diffie-Hellman protocol.

💡Exponential Time

Exponential time in the context of the video refers to the computational complexity of an algorithm, where the time required to solve a problem grows exponentially with the size of the input. The script states that the fastest known solutions to the discrete log problem are exponential, which makes it computationally infeasible for large inputs.

💡Passive Attacker

A passive attacker in the script is an entity that eavesdrops on the communication between two parties without altering the messages. The video discusses the capabilities of such an attacker, including access to a powerful computer and a hypothetical fast procedure for computing discrete logs, in the context of breaking the Diffie-Hellman key.

💡Modular Exponentiation

Modular exponentiation is a mathematical operation where an exponentiation is performed and then reduced modulo another number. In the script, it's mentioned as a fast procedure that outputs the result of a base raised to a power modulo a module, which is a fundamental operation in the Diffie-Hellman key exchange.

Highlights

The discrete log problem is closely related to the Diffie-Hellman security property.

Discrete logs are analogous to continuous logs but operate over a discrete group.

Efficient methods exist to compute continuous logarithms, unlike the discrete log problem.

The discrete log problem involves solving for x in a^x ≡ b (mod n), where n is a large prime number.

The existence of a discrete log is guaranteed when n is a large prime and a is a generator.

A generator g produces a permutation of numbers in the group Zn when raised to successive powers.

A demonstration code is provided to show the permutation produced by a generator and modulus.

Using 277 as a prime number and 5 as a generator demonstrates the permutation sequence.

The discrete log problem asks if one can determine the position of a number in the permutation sequence or the power needed to generate it.

There is no known polynomial-time solution for the discrete log problem; all known solutions are exponential.

The difficulty of solving the discrete log problem is attributed to the lack of efficient algorithms discovered by experts.

The exponential nature of the problem makes it intractable for large n, regardless of computational resources.

The size of n grows exponentially with the number of bits needed, impacting the feasibility of solving the discrete log.

The assumption that the discrete log problem is hard underpins the security of cryptographic systems.

In the quiz scenario, an adversary with access to a powerful computer and a fast discrete log procedure is considered.

The question posed is whether the adversary can break a Diffie-Hellman key with their resources and procedures.

Transcripts

The hard problem that is closely related

to the Diffie-Hellman security property

is the discrete log problem.

Discrete logs are like continuous logs

but over a discrete group.

So continuous log if we have a to the x equals b,

and we know a and b, we can solve for x.

That's the log base a of b,

and they're well know efficient ways to compute these logarithms.

One of the earliest use of computers

was to compute these tables of logarithms.

With discrete numbers, this gets much more interesting.

So now we have a to the x equals b,

modulo sum value n,

and our goal is to solve for x,

which is the discrete log base a of b,

and this turns out to be, as far as everyone can tell,

a very hard problem when n is a large prime number.

It's not clear that discrete log always exists,

and for certain choices of a, b, and n, it would not exists,

but if we choose n as a large

prime number and a as a generator,

well then by definition, it must exist.

What it means for our number to be a generator

is that if we raise g to each power,

what we get is the permutation of the numbers in the group Zn.

So as a little demonstration, certainly not a proof,

here's a code that produces the permutation

for given some generator and some modulus,

raises the generator to every power

between 1 and the modulus minus 1.

So we can try that with a fairly

small prime number so you can see the results.

We'll use 277 as our prime number

and 5 as a generator for 277.

One could check that in a root force way

just to show that it all produces all the numbers,

and we'll see that in the output for generator permutation.

These are the results and

you can see the first 1 is 5, that's 5 to the 1;

25 is 5 to the 2; 125 is 5 to the 3.

The next one is 71 because 5 to the 4 mod to 77 is 71,

and if we look at all the numbers here,

it would be a permutation on the numbers from 1 to 276.

Other than the early ones,

it would be fairly hard to predict where

a number is in this sequence.

You could certainly compute the whole sequence to find it.

The question the discrete log is asking

is given a number, can you figure out

where it would be in this sequence

or can you figure out the power that you need

to raise the generator to find it,

and the claim is that that's hard to do.

Showing this sequence really is not enough

to convince you that that's hard to do,

and there's no proof that it's hard.

The reason people believe it's hard

is that many smart people have tried to find

good ways of doing this, and none of the

solutions rendered polynomial time

that the fastest known solutions are exponential.

That means essentially that the only way to solve

this is to try all possible powers

until you find the one that works.

You can do a little better than that by trying

powers in a clever way, and you can

exclude some of the powers more quickly,

but you can't do any better than doing this exponential search,

which is exponential in the size of n

so this is something we have to be careful about when we

talk about hard problems.

When we say it's exponential, well it's not exponential in the value of n.

It's linear in the value of n.

We just need to try n operations,

but the magnitude of n

grows as 2 to the number of bits needed to break down n.

So as long as that's the best solution to discrete log,

then for very large n,

it is intractable no matter how many computer resources you have,

you can't do this exponential search.

You can't find the value of x that's the

discrete log of b, base a, mod n.

So as long as no one can find a

fast way to solve the discrete log problem,

as long as n is large and is an

arbitrary instance of this problem,

we think that it should be hard to

compute x given a and b and the modulus.

So for this quiz, we will assume that we have

and advisory that's passive

so all it can do is ease drop on the messages,

but they also have access to a powerful computer resource,

they have a procedure dlog that is

a fast procedure for computing discrete

logs that works on any inputs,

and they have modular exponentiation,

a fast procedure that outputs

base to the power mod modules.

And now the question is can they break a Diffie-Hellman key?

So we're assuming that they're passive attackers,

so they've eased dropped on all the

messages between Alice and Bob,

so they have all these values that were sent over the secure channel,

and the possible answers are no

that it's impossible with no

more resources or information,

or yes there is a way to do it,

and here's the way that she would compute that.