An elliptic curve is a graph of the equation y2 = x3 + ax + b. The Bitcoin network uses the variant with a = 0 and b = 7
Bitcoin critics say that the system is based on the mathematical foundation of ECDSA elliptic cryptography, which could theoretically be cracked after the invention of quantum computers. For example, the BTC price drop in 2019 was explained as panic due to the news that Google had achieved quantum supremacy.
Now scientists have calculated exactly how many qubits are required to break the ECDSA elliptic curve encryption, which is used to encode keys in the Bitcoin network.
The Bitcoin protocol uses the following constants:
Elliptic curve equation: y2 = x3 + 7
Simple modulus = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1 = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
Reference point = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 F81D08B 9C47D08B
Order = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141
This set of options is known as secp256k1.
Scientists from the Sussex Center for Quantum Technology (University of Sussex, UK) and Universal Quantum have calculated the number of physical qubits needed to break the 256-bit encryption of keys in the Bitcoin network using an elliptic curve within an available time frame in which it makes sense.
As the calculation showed, it would take 317 × 106 physical qubits to break the encryption within an hour using the surface code. It would take 13×106 physical qubits to break a cipher in a day.
By comparison, the world's largest quantum computer, IBM, is now running at 127 qubits.
To achieve “quantum supremacy”, Google tried to use a Bristlecone 72 qubit computer, but failed, so they limited themselves to the 53 qubit Sycamore version, on which this achievement was recorded. The system was designed in such a way as to solve one specific mathematical problem not several orders of magnitude faster than a supercomputer on a traditional architecture. Google's claim to achieve quantum supremacy drew criticism from colleagues, but their paper was accepted by Nature.
The calculation of the number of qubits for breaking encryption on elliptic curves was published on January 25, 2022 in the journal AVS Quantum Science (doi: 10.1116/5.0073075).
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