Have you watched The Imitation Game (2014)? Set against the backdrop of World War II, it tells the story of Alan Turing, who led a team of codebreakers beating the Nazi cryptography machine. In the process, Turing designs an electromechanical codebreaking machine, the modern computer’s precursor. Here's the trailer:
The most prolific method of modern encryption is RSA, which first went public in 1977. Modern RSA encryption uses two prime numbers that are multiplied and mathematically manipulated to give the user a public and a private key. The public key is distributed openly and is used to encrypt messages. Only the person with the private key can decrypt these encrypted messages. This video explains how it works:
Theoretically, cracking RSA is easy - simply reverse the mathematical manipulation to get the two prime numbers. In practice though, the time taken for prime number factorisation on a computer  increases exponentially with the size of the numbers. The security of RSA systems relies on using extremely large numbers - it’s recommended your primes should multiply to give a number at least 2048 binary digits long. Although computing power has come a long way since 1977, that looks uncrackable for now. It took the equivalent of 2000 years of computer time to factor a 768-bit number in 2009 (see details).
But beware the quantum computer. Where classical computers crunch data stored as ones or zeros, quantum computers can work with superpositions of ones and zeroes. This enables the quantum computer to factorise numbers much, much faster than a classical one. In 1994, MIT professor Peter Shor devised a quantum algorithm that solves the factoring problem faced by classical computers - four years before the first two-qubit quantum computer was built!
In early March this year, MIT professor Isaac Chuang and his team used Shor’s algorithm to factor the number 15. That may not sound impressive (you could work out in your head that 15 is 5 times 3). It’s not even the biggest number factored by a quantum computer, but it was important because the computer design used in this demonstration is scalable. 
Until recently, there was no clear path for quantum computers to be made big enough to make them threatening. Now the US National Security Agency is warning businesses that they need to go to quantum-safe encryption. The security of their data - and your emails and bank transactions - depends on it. Here's a news report from earlier this year: https://www.technologyreview.com/s/600715/nsa-says-it-must-act-now-again...
Thinking of making a film about cracking RSA? You might want to check out Sneakers (1992), a comedy about a box capable of breaking into nearly any computer system.
Researchers are looking into new classical methods of cryptography like lattice-, hash- and code-based cryptography that could be quantum-safe. Read about some of the ongoing research here: https://www.quantamagazine.org/20150908-quantum-safe-encryption/
Ironically, quantum physics could also be the solution to the problem it made. Quantum Key Distribution was invented by scientists including contest judge Artur Ekert. QKD for short, it uses quantum mechanical properties like entanglement and the no-cloning theorem to ensure secure encryption of data. 
Check out this video for a splendid explanation of quantum cryptography: