Store Now, Decrypt Later (SNDL)

• Nation states and individual actors intercept and store encrypted data like passwords and bank details.

• They cannot open the files but believe that within 10-20 years, they will have access to a quantum computer capable of breaking the encryption in minutes.

• This procedure is known as Store Now, Decrypt Later or SNDL.

• Information around today will still be valuable in a decade.

Quantum Computing Threat

• Nation states and individual actors are aware of the quantum computing threat.

• The National Security Administration says that a sufficiently large quantum computer if built, would be capable of undermining all widely deployed public key algorithms.

• Quantum computing will break encryption as we know it today.

• The US Congress passed legislation mandating all agencies to start transitioning to new methods of cryptography that can't be broken by quantum computers.

History of Encryption

• Before the 1970s, private information was exchanged by meeting up in person and sharing a secret key.

• Modern cryptography uses public key algorithms like RSA.

• RSA works by every person having two large prime numbers that they keep secret and multiply together to get a bigger number.

• The bigger number is made public for everyone to see.

• To send a private message, the sender uses the recipient's big public number to garble the message.

• The message is garbled in such a way that it is impossible to ungarble without knowing the two prime factors that made the number.

• This is an asymmetric key system, where different keys are used to encrypt and decrypt the message.

• Modern cryptography uses prime numbers that are around 313 digits long.

• Factoring a product of two primes this big, even with a supercomputer, would take around 16 million years.

Quantum Computing Basics

• Quantum computers consist of qubits which can be in an arbitrary combination of states, a superposition of zero and one.

• If we have two qubits, they can exist simultaneously in a superposition of 0, 1, 2, and 3.

• With three qubits, we can represent eight states.

• With 20 qubits, we can represent over a million different states, meaning we can simultaneously compute over a million different answers.

• With 300 qubits, we can represent more states than there are particles in the observable universe.

• All of the answers to computation are embedded in a superposition of states but you can't simply read out this superposition.

• To harness the power of a quantum computer, you need a smart way to convert the superposition of states into one that contains only the information you want.

Shor's Algorithm

• Peter Shor and Don Coppersmith figured out how to take a quantum Fourier transform.

• It works just like a normal Fourier transforms, apply it to some periodic signal, and it returns the frequencies that are in the signal.

• The quantum Fourier transform allows us to extract frequency information from a periodic superposition.

• Shor's algorithm uses the quantum Fourier transform to extract the prime factors of a number in superposition.

• It is exponentially faster than the best-known classical algorithm, the General Number Field Sieve.

Shor's Algorithm Example

• Let's say we have a number N, which is the product of two primes, p and q.

• For the sake of this example, let's set N equal to 77.

• To find the prime factors of 77 with Shor's algorithm, we need to create a superposition of all the possible factors.

• We use a quantum Fourier transform to extract the period of the superposition, which gives us the prime factors of N.

• A quantum computer can execute this method even for a very large number in a short period.

Quantum Key Distribution

• Quantum key distribution (QKD) is a way of sharing a secret key over an insecure channel.

• QKD works by using the principles of quantum mechanics to detect eavesdropping.

• Alice sends Bob a stream of photons encoded with random polarizations.

• Bob measures the polarizations and tells Alice which ones he received.

• They use this information to create a shared secret key.

• Any attempt to intercept the photons will alter their polarizations, alerting Alice and Bob to eavesdropping.

Limitations of Quantum Computing

• Most applications of quantum computing are useless because of the difficulty of converting the superposition of states into one that contains only the information you want.

• Quantum computers are good at solving specific problems like factoring large numbers and simulating quantum systems.

• Quantum computers will not replace classical computers because they are not good at solving general problems like word processing and web browsing.

Recap of the previous video

• The previous video explained how RSA encryption works using prime numbers.

• It showed how difficult it is to factorize a number that is a product of two large prime numbers.

• Factoring such a number is the basis of cracking RSA encryption.

Finding the factors of large numbers

• To factorize a large number, it is necessary to find its prime factors.

• With a product of really big primes, it is difficult to guess the prime factors.

• A fact about numbers can be used to find the factors.

Using a fact about numbers to find factors

• Pick a number g that doesn't share any factors with N.

• If you multiply g by itself over and over, you will always eventually reach a multiple of N plus one.

• In other words, you can always find some exponent r, such that g to the power of r, is a multiple N plus one.

Example of using the fact to find factors

• Pick any number that is smaller than 77.

• Multiply it by itself once, twice, or three times, and then divide each of these numbers by 77.

• The remainder will eventually be one.

• The exponent used to get the remaining one can be used to find two new numbers that probably share factors with N.

• Use Euclid's algorithm to find the shared factors between those numbers and N, which should give p and q.

Recap of the process to find factors

• To find the prime factors p and q of a number N, make a bad guess, g.

• Find out how many times r you have to multiply g by itself to reach one more than a multiple of N.

• Use that exponent to calculate two new numbers that probably do share factors with N.

• Use Euclid's algorithm to find the shared factors between those numbers and N, which should give you p and q.

How a quantum computer speeds up the process

• The key process that a quantum computer speeds up is step two, finding the exponent you raise G2 to equal one more than a multiple of N.

Understanding the cycle of remainders

• The remainders cycle and they will just keep cycling.

• The exponent that yields a remainder of one is 20, which is 10 more than the first exponent that yielded a remainder of one.

• So, we know that 8 to the 30 and 8 to the 40, and 8 raised to any power divisible by 10 will also be one more than a multiple of 77.

• If you pick any remainder, the next time you find that same remainder, the exponent will have increased by 10.

• The exponent R that gets us to one more than a multiple of n can be found by looking at the spacing of any remainder, not just one.

Using a quantum computer to factorize a large number

• Quantum computers can factorize any large product of two primes.

• First, split up the qubits into two sets.

• The first set is prepared in a superposition of zero and one, and all the numbers up to 10 power of 1,234.

• The other set contains a similar number of qubits all left in the zero state for now.

• Make a guess G, which most likely doesn't share factors with N.

• Raise G to the power of the first set of qubits and then divide by N.

Quantum Factorization

• A superposition of all numbers and the remainder of raising G to power is created using qubits.

• Two sets of qubits are entangled through this operation, but the superposition cannot be measured.

• Measuring only the remaining part of the superposition produces a random remainder that occurs multiple times.

• The exponents in the superposition that give the same remainder are separated by the same amount r.

Finding the Number

• Since the remainder is now the same for all states, it can be put aside to get a periodic superposition.

• Applying the quantum Fourier transform to this superposition produces states containing one over R.

• By performing a measurement and inverting it, R can be found.

Breaking RSA Encryption

• If R turns out to be even, then it can be used to turn a bad guess G into two numbers that likely share factors with N.

• The factors of N can be found using Euclid's algorithm and the encryption can be broken.

• Several thousand perfect qubits would be needed, but current qubits are imperfect, so additional redundant qubits are required.

• Breaking the encryption would require around 20 million physical qubits, which is still beyond our current capabilities.

Alternatives to RSA Encryption

• Scientists have been looking for new ways to encrypt data that can withstand attacks from both normal and quantum computers.

• The National Institute of Standards and Technology (NIST) launched a competition to find new encryption algorithms that aren't vulnerable to quantum computers.

• Cryptographers from all over the world submitted 82 different proposals, which were rigorously tested.

• NIST selected four of the algorithms to be part of their post-quantum cryptographic standard.

• Three of the algorithms are based on the mathematics of lattices.

Lattice-Based Encryption

• Lattice-based encryption is one of the post-quantum cryptographic standards proposed by NIST.

• Lattice-based encryption involves finding the closest lattice point to a target point using vectors.

• The vectors that make up the lattice are provided, and the number of lattice points grows exponentially with the number of dimensions.

• Solving the closest vector problem becomes extremely hard for computers in a high number of dimensions, making it difficult for hackers to break the encryption.

Lattice Cryptography

• Lattice cryptography is a method of encryption that uses a set of vectors to create a lattice.

• The vectors are kept secret, but the lattice is shared publicly.

• To send a message, a point is picked on the lattice and some random noise is added to it.

• The recipient decodes the message by figuring out which lattice point is closest to the message point.

• This method is extremely difficult to solve for both normal and quantum computers.

Importance of Researchers and Cryptographers

• There is an army of researchers, mathematicians, and cryptographers working behind the scenes to keep our secret data safe.

• They are the unsung heroes that will keep us safe moving forward and avoid mass surveillance by governments.

• They are also responsible for keeping critical infrastructure protected and allowing us to live as if quantum computers were never invented in the first place.