Quantum computers, based on the laws of quantum mechanics, represent one of the most promising technologies of the future. However, they can seriously threaten the security of current digital systems, including internet communications, banking systems and cryptocurrencies.

DIFFERENCES WITH CLASSICAL COMPUTERS AND PROPERTIES
Unlike traditional computers, which process information through bits that assume binary states (0 or 1), quantum computers use qubits. They can assume values ​​of 0 and 1 at the same time. Thanks to phenomena such as superposition and entanglement, qubits can exist in multiple states at the same time, allowing parallel calculations and unthinkable computational power.

However, qubits must be kept at temperatures close to absolute zero (-273°C) to minimize quantum decoherence (this phenomenon, due to the loss of coherence of the wave function, would prevent the superposition of states). Furthermore, current quantum computers suffer from intrinsic errors that require correction techniques, increasing the complexity of scaling such machines. For this reason, it will take at least another 6-7 years to become fully active and inexpensive.

CRYPTOGRAPHY SYSTEMS
Many digital technologies today are based on cryptographic systems that guarantee privacy and security. Among these, asymmetric key cryptography is particularly vulnerable to quantum computers. In particular, Shor's algorithm allows to factorize large numbers and solve the discrete logarithm problem in reasonable times, effectively attacking algorithms such as:

1) ECDSA (Elliptic Curve Digital Signature Algorithm), which is based on elliptic curves (used by cryptocurrencies).

2) RSA, based on the factorization of prime numbers.

3) Diffie-Hellman, used for secure key exchange.

In elliptic curve cryptography, security is based on the difficulty of solving the discrete logarithm problem R=P+Q: given a point P on the elliptic curve, which derives from R, it is extremely difficult (today impossible) to trace back to the point Q.
Shor's algorithm uses superposition to simultaneously generate all possible solutions. The wrong solutions cancel each other out through quantum interference, leaving only the correct one.

R=P+Q.
R: public key
P: is a known point on the elliptic curve (generator point).
Q: is the secret value that we want to keep safe (private key).

In a classical system, knowing the public key (R) and the generator point (P) makes it practically impossible to trace the private key (Q). But with a quantum computer, Shor's algorithm would exploit the power of parallel computing to quickly determine Q.
Shor's algorithm is revolutionary for its ability to solve problems that seem impossible for classical computers, such as factoring large prime numbers (used in RSA) or discrete logarithms (used in elliptic curves).

Regarding Bitcoin or Ethereum, digital signatures use this scheme to ensure the integrity of transactions:

-Generation of the private key: the user chooses a random number Q as the private key.
-Calculation of the public key R=P⋅Q. Here, multiplication represents a geometric operation defined on the elliptic curve (scalar multiplication). The public address is derived from the public key. R is visible on the blockchain.

A quantum computer could use Shor's algorithm to quickly solve the discrete logarithm R, determining the private key Q. The attacker could:
1) Sign fake transactions.
2) Steal funds associated with known public addresses.

In short, the quantum computer uses:
1) Quantum superposition: it generates a superposition of all possible Q values ​​that could solve the problem.

2) Constructive and destructive interference: using the quantum Fourier transform, the calculations eliminate the wrong solutions through destructive interference, leaving only the correct ones.

3) The final result gives Q, thus solving the problem in a logarithmically increasing time, much faster than classical algorithms.

Grover's algorithm is less effective but can accelerate brute force attacks against symmetric cryptography.

VULNERABLE SECTORS
1) Bitcoin uses the ECDSA algorithm for digital signatures. A quantum attack could:
-Expose private keys: addresses with already known public keys (address), such as those of Satoshi Nakamoto, could be compromised. A quantum computer would be able to calculate the corresponding private key and steal the funds. The funds could be transferred to never-used addresses (and therefore with unknown public key) to mitigate the risk.

2) Ethereum's architecture introduces additional risks: by signing a transaction, the public key is exposed, so the private key can be forced. Again, the temporary solution could be to move the funds to an empty address that has never been used.

3) Banking systems: banks use cryptographic systems for secure transactions and access. An attack could expose all customer data.

4) Internet and private communications: encryption protocols such as TLS, which protect web communications, are based on RSA or Diffie-Hellman. A quantum attack could intercept security and private communications, as well as sensitive information (passwords), and decrypt archived messages.

WHICH ALGORITHMS CAN REPLACE ELLIPTICAL CURVES?
Post-quantum cryptography focuses on algorithms that are resistant to quantum attacks. Candidates include:

1) Code-based cryptography: based on problems related to encoding and error correction.

2) Lattice-based cryptography: uses mathematical problems related to multidimensional grids.

3) Multivariate polynomial cryptography: is based on multivariate polynomial equations.

4) Hash-based signatures: secure against quantum attacks, but less efficient for multiple signatures.

Therefore, to mitigate the risks that will affect today's technologies in 5-10 years, it is essential to adopt post-quantum cryptography algorithms and begin a gradual transition towards resistant systems.

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