When will the internet be unsafe?

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Rudin
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16 Aug 2015, 6:37 am

eric76 wrote:

Rudin wrote:

I'll explain what encryption is based on, it is based on the factorization of large prime numbers. You see prime factorization cannot be solved "fast" by a computer, it takes many years when numbers are very large and that's what encryption is based on. The Riemann Hypothesis will find a efficient algorithm to factorize large numbers.

Some encryption, but not all encryption, is based on the use of prime numbers. More precisely, public key encryption is often based on the use of prime numbers, but even then, not all public key encryption. There is, after all, also elliptic curve encryption.

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If you are not following, perhaps a certain theorem will help. All non-prime positive integers can be expressed as the product of primes. 20=4x5=2x2x5 2 and 5 are both prime. 30=5x6=5x3x2 5,3,2 are all prime. It takes a computer a long time to find the prime factors of a large positive integer. So encryption is based on the prime factorization of large numbers which will take a computer many years to crack.

The key is that factoring enormously large numbers is considered to require a brute force algorithm that basically works by trying every possibility until it comes to an answer.

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The Riemann Hypothesis is a theorem, someone asked the question, it's not necessarily true.

No. It is a Hypothesis, not a Theorem. If and when it is proved to be true, it will then be a theorem.

I meant it was a hypothesis.

Prime-based encryption is used for Amazon, Ebay, ATMs, E-mail and PGP encryption. This explanation on StackOverflow was better than anything I could come up with:

"
Most basic and general explanation: cryptography is all about number theory, and all integer numbers (except 0 and 1) are made up of primes, so you deal with primes a lot in number theory.

More specifically, some important cryptographic algorithms such as RSA critically depend on the fact that prime factorization of large numbers takes a long time. Basically you have a "public key" consisting of a product of two large primes used to encrypt a message, and a "secret key" consisting of the primes used to decrypt the message. You can make the public key public, and everyone can use it to encrypt messages to you, but only you know the prime factors and can decrypt the messages. Everyone else would have to factor the number, which takes too long to be practical, given the current state of the art of number theory."

The consequences of the RH are often over-hyped. However for a brief period of time after the RH is proven (if it is proven) all RSA encrypted data will be accessible by hackers until the encryption system is switched to something else like you said, ECC. These hackers will be pulling of the credit card thievery of the century, there will be billions of dollars lost.

However, if someone simply finds a counter-example then there will be no proof which implies no algorithm.

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"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

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Rudin
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16 Aug 2015, 6:40 am

eric76 wrote:

Regarding P vs NP, it is not really about what can be done quickly and what cannot.

For example, an algorithm that has complexity O(n^1,000,000,000) would clearly be P, but it would grow really badly.

Spiderpig
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16 Aug 2015, 6:41 am

The Internet is unsafe enough if you know where to look for trouble.

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Kurgan
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16 Aug 2015, 7:31 pm

Rudin wrote:

Many of you are enjoying your internet. But the internet might not be safe anymore.

The internet is based on something called encryption, almost everything on the internet has an encryption component. This is an image explaining it:

If you don't understand how it works here is a tutorial that is optional to read, but I suggest you do:

----- BEGIN TUTORIAL-----

I've only worked with Linux and Windows but I'm sure there is a PGP client for Mac.

Step 1:

Go to http://www.gpg4usb.org and download the zip file.

Step 2:

Once the download is complete open the file location.

Step 3: Extract the zip file to a folder of your choice, it could be on a USB or somewhere in your hard drive.

Step 4: Click either Start Windows or right click under Start Linux 32/64-bit and click Run.

Step 5: Now we are going to generate a PGP public key. I've included pictures to help you guys out.

Your screen should look something like this,

Click Keys>Manage Keys>Keys>Generate Key. For E-Mail it doesn't have to be valid, something like user@WrongPlanet will work. Choose a name and make sure it doesn't expire. Also change it from 2048 bit to 4096 bit encryption. Then choose a good password. Generate it, it may take a while but don't kill the process.

Once your PGP key has generated you should have something similar to this:

So paste your key into the clipboard as shown by these pictures.

And,

Before I teach you how to import PGP keys I'll tell you what PGP key is used for, I'll put it in bold.

Bob is buying an item from Alice. They were communicated using a third-party email system that combs through all the E-mails. Bob wants to send his information to Alice, but doesn't want his security compromised. Solution=PGP. Bob imports Alice's PGP key and sends her his information but encrypts it and only she has the password.
So basically PGP is a P2P message system used by people who don't want their information compromised. Also, the NSA or any other security agency cannot break it. Encryption is also used by Amazon, ATMs, G-mail, Yahoo even WrongPlanet(possibly). opy

To import my PGP key just copy and paste it into the clipboard Import>Clipboard if you wanna send me an encrypted PM, feel free.

-----END TUTORIAL------

Anyway, there is a way that all PGP encryption can be broken. Encryption is based on the factorization of prime numbers and if the Riemann Hypothesis is solved all encryption will be breakable and nothing on the internet is safe, everything is open to hackers.

The Riemann Hypothesis is a very complex mathematical problem that has lot's of benefits but one of the consequences is the loss of the internet.

Both the police in several countries and hackers have tried to hack Countermail (OpenPGP based with a 4096 bit key) for years. No success so far. :mrgreen:

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Kurgan
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16 Aug 2015, 7:37 pm

Humanaut wrote:

If you've got nothing to hide, you've got nothing to encrypt.

A lot of people have (harmless, but illegal) stuff to hide. People in countries where porn is illegal still watch porn, people in theocratic countries still drink alcohol, 12% of all Norwegians in "clean" gyms do anabolic steroids (the number is 40-50% in bodybuilder gyms), lots of people everywhere download pirated software, etc. Privacy is a human right.

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eric76
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16 Aug 2015, 11:53 pm

Rudin wrote:

More specifically, some important cryptographic algorithms such as RSA critically depend on the fact that prime factorization of large numbers takes a long time. Basically you have a "public key" consisting of a product of two large primes used to encrypt a message, and a "secret key" consisting of the primes used to decrypt the message. You can make the public key public, and everyone can use it to encrypt messages to you, but only you know the prime factors and can decrypt the messages. Everyone else would have to factor the number, which takes too long to be practical, given the current state of the art of number theory."

The consequences of the RH are often over-hyped. However for a brief period of time after the RH is proven (if it is proven) all RSA encrypted data will be accessible by hackers until the encryption system is switched to something else like you said, ECC. These hackers will be pulling of the credit card thievery of the century, there will be billions of dollars lost.

However, if someone simply finds a counter-example then there will be no proof which implies no algorithm.

A proof of the Riemann Hypothesis by itself might not necessarily lead to an algorithm to factor primes. The bigger danger is that the advances in theory that would be needed to prove the hypothesis might possibly lead to better factoring algorithms. Of they might not.

The real future danger could result from the development of a quantum computer. And that wouldn't need the Riemann Hypothesis to be proven.

---

What is currently a real problem is that of using weak keys. There are plenty of them around.

If you get a chance, take a look at this paper: https://factorable.net/weakkeys12.extended.pdf

---

By the way, have you ever wondered where the prime factors come from in generating a cryptographic key that relies on using two very large prime numbers and their product? It's not like the computer proves that the two factors are actually prime. And the computer definitely does not have a list of known prime numbers that large. From what I recollect, the algorithms to generate the keys do tests on the candidate factors that would lead to a reasonable amount of confidence that the factors are actually prime by using tests that give a good indication about whether the number is probably prime.

Humanaut
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17 Aug 2015, 3:25 am

Kurgan wrote:

Humanaut wrote:

If you've got nothing to hide, you've got nothing to encrypt.

A lot of people have (harmless, but illegal) stuff to hide.

True.

Quote:

People in countries where porn is illegal still watch porn, people in theocratic countries still drink alcohol, 12% of all Norwegians in "clean" gyms do anabolic steroids (the number is 40-50% in bodybuilder gyms), lots of people everywhere download pirated software, etc.

Also true.

Quote:

Privacy is a human right.

Another truth.

However, and I am sure you agree with the fact that these truths do not stack into a coherent argument where "lots of people everywhere" can be said to have a right to keep pirated software private.

Rudin
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17 Aug 2015, 7:07 am

Quote:

By the way, have you ever wondered where the prime factors come from in generating a cryptographic key that relies on using two very large prime numbers and their product? It's not like the computer proves that the two factors are actually prime. And the computer definitely does not have a list of known prime numbers that large. From what I recollect, the algorithms to generate the keys do tests on the candidate factors that would lead to a reasonable amount of confidence that the factors are actually prime by using tests that give a good indication about whether the number is probably prime.

No I haven't, but you make a good point.

They probably use sieves to help them find primes. Also there are algorithms to determine if a number is divisible by another number which they probably use.

Also, weak keys aren't use in any of the ciphers I use so I'm good. The only places I use encryption is for my laptop, PGP to prevent sensitive information being read by a third party and to encrypt sensitive data on the computer. I believe PGP is RSA and LUKS is 256-bit AES which is not secure enough so I am going to try to encrypt it with Veracrypt or Truecrypt today.

I'll read that article later today.

Kurgan
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17 Aug 2015, 9:31 am

Humanaut wrote:

However, and I am sure you agree with the fact that these truths do not stack into a coherent argument where "lots of people everywhere" can be said to have a right to keep pirated software private.

If someone chooses to crack and upload his licensed version of something on the internet, people should be allowed to download it. Even if you're against piracy, you can't deny the fact that privacy is more important than finding out who uses pirated software.

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Rudin
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18 Aug 2015, 11:05 am

eric76 wrote:

Regarding P vs NP, it is not really about what can be done quickly and what cannot.

For example, an algorithm that was O(n^1,000,000,000) would clearly be P, but it would grow really badly.

An algorithm that has complexity O(n^(10^9)) is by definition, P (polynomial time) anything that doesn't run in O(n^c) is not a P-class algorithm. Something like O(n log n) is NP.

I'm glad you brought up P vs NP it is a really interesting open problem.

Primality tests in particular are interesting. Primality tests have only been discovered to run in NP.

Fermat's theorem has complexity O(k log^3(n)) and is not accurate. There are infinitely many Carmicheal numbers (composite numbers that satisfy Fermat's theorem).

A algorithm made by undergraduate students is proven to be the most efficient. It is known as AKS and has complexity Õ(log^12(n)) which is NP. I didn't know what Õ meant so I searched it up https://en.wikipedia.org/wiki/%C3%95#Mathematical_use.

Rudin
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18 Aug 2015, 11:17 am

I am actually currently reading a book called Visions of Infinity by Ian Stewart. It's fairly good and talks about interesting mathematical problems along with the history of them.

I know some of the stuff he talks about and it's geared more toward people at a lower mathematical level than myself but it is still pretty interesting. I am pondering ordering Matt Parker's Things to Make and Do in the Fourth Dimension I might order it today if I have time. It seems like a good book, however the title is misleading at best, he doesn't talk about the fourth dimension for his entire book. It actually has a fairly small part. He talks about a lot of the stuff he talks about on Numberphile which I am subscribed to happily.

I am also going to read Prime Obsession by John Derbyshire because I am simply fascinated with the Riemann Hypothesis and the Riemann character. The book is divided into two parts history and mathematics. I am also planning on reading about about Asperger's my mother gave me.

It's weird how this thread was hijacked and turned into a thread for discussions about mathematics and cryptography.

Humanaut
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18 Aug 2015, 11:24 am

Kurgan wrote:

Humanaut wrote:

However, and I am sure you agree with the fact that these truths do not stack into a coherent argument where "lots of people everywhere" can be said to have a right to keep pirated software private.

You might not like it, but it is not uncommon that the right to privacy ends where a violation of property rights begins, and distributing pirated software could ultimately compromise ones right to freedom too.

Humanoid2436
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18 Aug 2015, 11:55 am

first of all, must say this is a really nice topic :roll:
Now I must finish some work and unfortunately cannot read all your interesting posts but I will do it tomorrow
internet has never been a safe place.

In fact the whole world and Universe seem to me as very dangerous place and no matter what you do and how smart you are :skull: :skull: you must inevitably die. :skull: :skull:
So living itself is the worst of all games or things which you can practice. :evil:
The Universe is really odd and living on Earth or doing anything on internet is really dangerous. :evil:
:roll: :roll: And its even worse, because you are living it means (very probably or in fact surely) that someone else cannot live or your life causes death of other innocent people and much worse you are the cause of death of many innocent animals. :!: :!: :roll: :roll:
:idea:
I am depressed the whole day and maybe thoughts on death make me happy because when you die you sucesfully solve this issue. :roll: :roll: :idea: :idea:

Rudin
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18 Aug 2015, 12:09 pm

Humanoid2436 wrote:

first of all, must say this is a really nice topic :roll:

Now I must finish some work and unfortunately cannot read all your interesting posts but I will do it tomorrow
internet has never been a safe place.

In fact the whole world and Universe seem to me as very dangerous place and no matter what you do and how smart you are :skull:

you must inevitably die. :skull:

So living itself is the worst of all games or things which you can practice. :evil:

The Universe is really odd and living on Earth or doing anything on internet is really dangerous. :evil:

And its even worse, because you are living it means (very probably or in fact surely) that someone else cannot live or your life causes death of other innocent people and much worse you are the cause of death of many innocent animals. :!:

I am depressed the whole day and maybe thoughts on death make me happy because when you die you sucesfully solve this issue. :roll:

Why thank you, I do my best to make my posts interesting.

You are right, death is inevitable and the internet is never truly safe, if it were there would be no such thing as hackers. I meant that anyone who is fairly competent in security can use the internet with little fear of being hacked i.e. make purchases, check E-mail stuff like that.

Ichinin
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19 Aug 2015, 3:52 am

Sweetleaf wrote:

What the hell, why did someone go and come up with a math problem that would cause 'loss of the internet'

Not true, if RSA fails, there is ECC, DH and Knapsack to take over.

People have tried breaking RSA and similar algorithms for a long time. It is true that there are attacks that circumvent the crypto, but that attack is against the crypto protocol, not the math problem itself.

There is a lot of quacks and attentionseekers trying to claim that this and that is possible, i know of one that claimed that she had an algorithm that was able to find insanely large primes in a second. She also said that she had an unbreakable encryption - that got analysed and disproved and broken in the discussion under the article.

What is really scary is that people like that has the tendency to be able to get millions in funding for their craptography.

Rudin
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19 Aug 2015, 7:02 am

Ichinin wrote:

Sweetleaf wrote:

What the hell, why did someone go and come up with a math problem that would cause 'loss of the internet'

Government funding, it's scary.

The government, the NSA in particular, wants to be able to break RSA encryption because many criminals and activists use it to communicate online.

What you said about attention-seekers reminds me of Kazakhstani mathematician that claimed to have solved the Navier-Stokes equations, the Riemann Hypothesis and Yangs-Mill. He also claimed to have found a proof that the Euler-Mascheroni constant is irrational. These proofs were quickly disproved.

BTW:

The Euler-Mascheroni constant denoted by the Greek letter, γ (gamma) can be represented as follows:

γ=lim_{N -> infinity} [1+1/2+1/3+...+1/N-log N]

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