Many commonly used cryptosystems could be quickly cracked by quick algorithms on quantum computers, requiring more creative approaches to digital security. In a recent study, a group of researchers created a stream cipher using three primitives for cryptography that were each based on a different mathematical model of chaos.
The resulting cryptographic method paves the way for secure digital communications in the post-quantum age and is resistant to attacks from powerful quantum computers. It can also be applied to low-cost machines.
While for most of us cryptographic systems are things that just run “under the hood,” they are an essential element in the world of digital communications.
However, the emergence of quantum computers in the near future may fundamentally alter the world of cryptography. Some of the most extensively used cryptosystems could be compromised by quick algorithms running on these devices, making them insecure. Cryptography researchers around the world are developing cutting-edge encryption techniques that can fend off attacks from quantum computers since they are well aware of this impending threat.
Chaos theory is actively being studied as a basis for post-quantum era cryptosystems. Chaos in mathematics is a characteristic of some dynamic systems that makes them very susceptible to the beginning conditions.
Although technically deterministic (non-random), these systems grow in such a complex manner that it is almost impossible to anticipate their long-term state with imperfect knowledge. Even slight rounding errors in the initial conditions result in diverging outputs.
This unique characteristic of chaotic systems can be leveraged to produce highly secure cryptographic systems, as a team of researchers from Ritsumeikan University, Japan, showed in a recent study published in IEEE Transactions on Circuits and Systems I.
Led by Professor Takaya Miyano, the team developed an unprecedented stream cipher consisting of three cryptographic primitives based on independent mathematical models of chaos.
The implementation and running costs of our cryptosystem are remarkably low compared with those of quantum cryptography. Our work thus provides a cryptographic approach that guarantees the privacy of daily communications between people all over the world in the post-quantum era.
Professor Takaya Miyano
The first primitive is a pseudorandom number generator based on the augmented Lorenz (AL) map. The key streams for encrypting/decrypting communications are created using the pseudorandom numbers generated using this method, which takes center stage in the second and possibly most impressive foundational novel mechanism for secret-key exchange.
Based on the synchronization of two chaotic Lorenz oscillators, which the two communicating users can independently and arbitrarily initialize, without either of them knowing the state of the other’s oscillator, this novel method for exchanging secret keys specifying the AL map uses random initialization.
To conceal the internal states of these oscillators, the communicating users (the sender and the receiver) mask the value of one of the variables of their oscillator by multiplying it with a locally generated random number. The masked value of the sender is then sent to the receiver and vice-versa.
The users can mask and exchange secret keys and then locally unmask them with straightforward calculations after a short period of time when these back-and-forth exchanges cause both oscillators to sync up almost perfectly to the same state. This is true even though the variables are randomly generated.
Finally, the third primitive is a hash function based on the logistic map (a chaotic equation of motion), which allows the sender to send a hash value and, in turn, allows the receiver to ensure that the received secret key is correct, i.e., the chaotic oscillators were synchronized properly.
The researchers showed that a stream cipher assembled using these three primitives is extremely secure and resistant to statistical attacks and eavesdropping since it is mathematically impossible to synchronize their own oscillator to either the sender’s or the receiver’s ones.
This is an unprecedented achievement, as Prof. Miyano states: “Most chaos-based cryptosystems can be broken by attacks using classical computers within a practically short time. In contrast, our methods, especially the one for secret-key exchange, appear to be robust against such attacks and, more importantly, even hard to break using quantum computers.”
In addition to its security, the proposed key exchange method is applicable to existing block ciphers, such as the widely used Advanced Encryption Standard (AES).
Moreover, the researchers could implement their chaos-based stream cipher on the Raspberry Pi 4, a small-scale computer, using Python 3.8. They even used it to securely transmit a famous painting by Johannes Vermeer between Kusatsu and Sendai, two places in Japan 600 km apart.
“The implementation and running costs of our cryptosystem are remarkably low compared with those of quantum cryptography,” highlights Prof. Miyano, “Our work thus provides a cryptographic approach that guarantees the privacy of daily communications between people all over the world in the post-quantum era.”
With the strength of chaos-based cryptography, the dangers of quantum computing may not be as great as we think.
