Google Patentsopen access

Quantum resistant blockchain with multi-dimensional quantum key distribution — Nxgen Partners Ip, Llc (US11621836B2)

ABSTRACT

Abstract

A system for generating a blockchain including an input for receiving a plurality of groups of data. Blockchain processing circuitry generates the blockchain for the plurality of groups of data. The blockchain processing circuitry generates the blockchain by performing a first hash using the first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution and orbital angular momentum. The blockchain processing circuitry establishes the first block of the blockchain using the first group of data, the first nonce and the first digital signature. The blockchain processing circuitry performs a second hash using the second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for the second block, wherein the hash function uses encryption based on the quantum key distribution and the orbital angular momentum. The circuitry establishes the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 16/673,447, filed Nov. 4, 2019, entitled QUANTUM RESISTANT BLOCKCHAIN WITH MULTI-DIMENSIONAL QUANTUM KEY DISTRIBUTION, now U.S. Pat. No. 10,708,046, issued on Jul. 7, 2020, which claims the benefit of U.S. Patent Application No. 62/757,477, filed Nov. 8, 2018, entitled QUANTUM RESISTANT BLOCKCHAIN WITH MULTI-DIMENSIONAL QKD, the specifications of which are incorporated by reference herein in their entirety.

TECHNICAL FIELD

The present invention relates to blockchains, and more particularly, to a quantum resistant blockchain using quantum key distribution.

BACKGROUND

Quantum computers process information according to the laws of quantum mechanics. This means an increase in computational processing for specific problems (i.e. function inversion with Grover's algorithm, and factoring large numbers into prime factors with Shor's algorithm).

Blockchain offers an open, public, distributed ledger that has many applications, including digital currencies. The security of this ledger depends on the difficulty of solving certain cryptographic problems which are threatened by the potential of quantum computation. Specifically, hashes as used in signing the blocks of the ledger can be compromised.

The principal threat is Grover's algorithm, which can dramatically speed up function inversion. This allows the generation of a modified pre-image from a given hash (a hash collision) allowing a signed data block to be modified. This destroys authenticity of the ledger entries undermining the entire blockchain.

The second threat is Shor's algorithm, which applies to any part of blockchain that relies on asymmetric key cryptography. The main problem is that of breaking RSA encryption. RSA relies on the ease of multiplying prime numbers in contrast to the difficulty of factoring large numbers into prime factors. Shor's algorithm speeds-up this process exponentially, effectively breaking RSA encryption. Variants of Shor's algorithm do the same for other asymmetric key cryptosystems.

To counter these threats few quantum-resistant cryptographic tools have been developed. Currently, the National Institute of Standards and Technology is responsible for navigating this threat. Congress has tasked NIST with R&D in cryptographic standards and tools to counter the threat of quantum computation. No standards are developed yet. Therefore, a quantum version of Blockchain is needed that is resistant to Quantum attacks.

SUMMARY

The present invention, as disclosed and described herein, in one aspect thereof comprises a system for generating a blockchain including an input for receiving a plurality of groups of data. Blockchain processing circuitry generates the blockchain for the plurality of groups of data. The blockchain processing circuitry generates the blockchain by performing a first hash using the first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution and orbital angular momentum. The blockchain processing circuitry establishes the first block of the blockchain using the first group of data, the first nonce and the first digital signature. The blockchain processing circuitry performs a second hash using the second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for the second block, wherein the hash function uses encryption based on the quantum key distribution and the orbital angular momentum. The circuitry establishes the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates an information-theocratic protocol using a combination of quantum key distribution and orbital angular momentum;

FIG. 2 illustrates a blockchain structure;

FIG. 3 illustrates various types of quantum algorithms;

FIG. 4 illustrates different Quantum Blockchain issues;

FIG. 5 illustrates uses of Grover's algorithms;

FIG. 6 illustrates different types of Cryptographic Systems;

FIG. 7 illustrates a multilayer blockchain protocol;

FIG. 8 illustrates a flow diagram of a process for aggregating transactions;

FIG. 9 illustrates the components of a quantum resistant blockchain protocol;

FIG. 10 is a functional block diagram of a system for generating orbital angular momentum within a communication system;

FIG. 11 is a functional block diagram of the orbital angular momentum signal processing block of FIG. 10 ;

FIG. 12 is a functional block diagram illustrating the manner for removing orbital angular momentum from a received signal including a plurality of data streams;

FIG. 13 illustrates a single wavelength having two quanti-spin polarizations providing an infinite number of signals having various orbital angular momentums associated therewith;

FIG. 14 A illustrates a plane wave having only variations in the spin angular momentum;

FIG. 14 B illustrates a signal having both spin and orbital angular momentum applied thereto;

FIGS. 15 A- 15 C illustrate various signals having different orbital angular momentum applied thereto;

FIG. 15 D illustrates a propagation of Poynting vectors for various Eigen modes;

FIG. 15 E illustrates a spiral phase plate;

FIG. 16 illustrates a block diagram of an OAM processing system utilizing quantum key distribution;

FIG. 17 illustrates a basic quantum key distribution system;

FIG. 18 illustrates the manner in which two separate states are combined into a single conjugate pair within quantum key distribution;

FIG. 19 illustrates one manner in which 0 and 1 bits may be transmitted using different basis within a quantum key distribution system;

FIG. 20 is a flow diagram illustrating the process for a transmitter transmitting a quantum key;

FIG. 21 illustrates the manner in which the receiver may receive and determine a shared quantum key;

FIG. 22 more particularly illustrates the manner in which a transmitter and receiver may determine a shared quantum key;

FIG. 23 is a flow diagram illustrating the process for determining whether to keep or abort a determined key;

FIG. 24 illustrates a functional block diagram of a transmitter and receiver utilizing a free-space quantum key distribution system;

FIG. 25 illustrates a network cloud-based quantum key distribution system;

FIG. 26 illustrates a high-speed single photon detector in communication with a plurality of users;

FIG. 27 illustrates a nodal quantum key distribution network;

FIG. 28 is a flow diagram of a process for creating a blockchain using quantum key distribution with orbital angular momentum; and

FIG. 29 is a functional block diagram of a system for performing the process of FIG. 28 .

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of a quantum resistant blockchain with multi-dimensional quantum key distribution system are illustrated and described, and other possible embodiments are described. The figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments.

Referring now to FIG. 1 , a new information-theoretic secure protocol is introduced that is robust for current and future quantum attacks. It is a quantum resistant blockchain protocol 102 that uses multi-dimensional QKD 104 using Orbital Angular Momentum (OAM) states 106 of photons. Photons are quantas of electromagnetic signals and therefore suitable for this protocol as the 2-dimensional QKD has already been demonstrated both in fiber optics as well as satellite communications. This multi-dimensional QKD protocol 104 can also be extended for development of a global QKD network and “quantum Internet” and extend quantum-safe blockchain platforms to a global scale.

Classical and Quantum Computation

The physical laws relevant to the information processing system are important to understanding the limitations of computation. In general, Quantum Mechanics adds features that do not exist in classical mechanics. Physical quantities are “quantized,” i.e. cannot be subdivided. Quantum mechanics further requires physical states to evolve in such a way that cloning a state into an independent copy is not possible. This is used in quantum cryptography to prevent information copying. Quantum mechanics also describes systems in terms of superposition that allow multiple inputs to be processed simultaneously, though only one can be observed at the end of processing, and the outcome is probabilistic in nature. Finally, quantum mechanics allows for entanglement that is not possible in classical physics.

Many computational algorithms and data structures have been developed for use on classical computers. Many of these algorithms have parallels on quantum computers but due to the quantum mechanical nature of the information processing could have far greater power. The simplest example of this is called Deutsch's Problem, which demonstrates that quantum computation can be significantly faster than classical computation.

RSA encryption relies on the fact that multiplication of large primes is easy and thus fast but factoring large composite numbers into two prime factors is very difficult and thus slow. Hash functions have the important property of being easy to calculate but difficult to invert. They provide a unique fingerprint precisely because it is very difficult to take a given hash value and find a chosen pre-image that yields that hash.

The threat of quantum computation is that such algorithms become useless because the premise of asymmetric effort of computation is invalidated. Quantum computing provides potential attacks on many cryptographic systems and algorithms. As of now, no quantum computer exists to perform such computations, though there is no doubt as to the usefulness of the algorithms themselves and of their threat to cryptographic systems. Such a quantum computer would need to have at least as many qubits as the output of the computations, e.g. 256 logical qubits to encode a hashing function with a 256-bit output. Each logical Qbit will likely need to be composed of some unknown large number of physical qubits, and the current state of the art quantum computer have small number of physical qubits.

Blockchain

The first blockchain structure was initially developed in the context of the digital currency Bitcoin to solve the problem of multiple spending.

The core component implements an open, distributed, cryptographically signed digital ledger that is secure against modification and verifiable by anyone. To prevent bulk rewriting of an entire sequence of blocks from some point in the past as well as attacks to deny service or grow the chain faster than legitimate sources can, a work requirement is added to make rewriting long chains prohibitive. For our purposes here, the relevant structure of blockchain amounts to the following description as illustrated in FIG. 2 .

The blockchain 202 consists of a sequence of blocks 204 that are stored on and copied between publicly accessible servers 205 . Each block 204 consists of four fundamental elements including the <figure-callout id="206" label="f

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 16/673,447, filed Nov. 4, 2019, entitled QUANTUM RESISTANT BLOCKCHAIN WITH MULTI-DIMENSIONAL QUANTUM KEY DISTRIBUTION, now U.S. Pat. No. 10,708,046, issued on Jul. 7, 2020, which claims the benefit of U.S. Patent Application No. 62/757,477, filed Nov. 8, 2018, entitled QUANTUM RESISTANT BLOCKCHAIN WITH MULTI-DIMENSIONAL QKD, the specifications of which are incorporated by reference herein in their entirety.

TECHNICAL FIELD

The present invention relates to blockchains, and more particularly, to a quantum resistant blockchain using quantum key distribution.

BACKGROUND

Quantum computers process information according to the laws of quantum mechanics. This means an increase in computational processing for specific problems (i.e. function inversion with Grover&#39;s algorithm, and factoring large numbers into prime factors with Shor&#39;s algorithm).

Blockchain offers an open, public, distributed ledger that has many applications, including digital currencies. The security of this ledger depends on the difficulty of solving certain cryptographic problems which are threatened by the potential of quantum computation. Specifically, hashes as used in signing the blocks of the ledger can be compromised.

The principal threat is Grover&#39;s algorithm, which can dramatically speed up function inversion. This allows the generation of a modified pre-image from a given hash (a hash collision) allowing a signed data block to be modified. This destroys authenticity of the ledger entries undermining the entire blockchain.

The second threat is Shor&#39;s algorithm, which applies to any part of blockchain that relies on asymmetric key cryptography. The main problem is that of breaking RSA encryption. RSA relies on the ease of multiplying prime numbers in contrast to the difficulty of factoring large numbers into prime factors. Shor&#39;s algorithm speeds-up this process exponentially, effectively breaking RSA encryption. Variants of Shor&#39;s algorithm do the same for other asymmetric key cryptosystems.

To counter these threats few quantum-resistant cryptographic tools have been developed. Currently, the National Institute of Standards and Technology is responsible for navigating this threat. Congress has tasked NIST with R&amp;D in cryptographic standards and tools to counter the threat of quantum computation. No standards are developed yet. Therefore, a quantum version of Blockchain is needed that is resistant to Quantum attacks.

SUMMARY

The present invention, as disclosed and described herein, in one aspect thereof comprises a system for generating a blockchain including an input for receiving a plurality of groups of data. Blockchain processing circuitry generates the blockchain for the plurality of groups of data. The blockchain processing circuitry generates the blockchain by performing a first hash using the first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution and orbital angular momentum. The blockchain processing circuitry establishes the first block of the blockchain using the first group of data, the first nonce and the first digital signature. The blockchain processing circuitry performs a second hash using the second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for the second block, wherein the hash function uses encryption based on the quantum key distribution and the orbital angular momentum. The circuitry establishes the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates an information-theocratic protocol using a combination of quantum key distribution and orbital angular momentum;

FIG. 2 illustrates a blockchain structure;

FIG. 3 illustrates various types of quantum algorithms;

FIG. 4 illustrates different Quantum Blockchain issues;

FIG. 5 illustrates uses of Grover&#39;s algorithms;

FIG. 6 illustrates different types of Cryptographic Systems;

FIG. 7 illustrates a multilayer blockchain protocol;

FIG. 8 illustrates a flow diagram of a process for aggregating transactions;

FIG. 9 illustrates the components of a quantum resistant blockchain protocol;

FIG. 10 is a functional block diagram of a system for generating orbital angular momentum within a communication system;

FIG. 11 is a functional block diagram of the orbital angular momentum signal processing block of FIG. 10 ;

FIG. 12 is a functional block diagram illustrating the manner for removing orbital angular momentum from a received signal including a plurality of data streams;

FIG. 13 illustrates a single wavelength having two quanti-spin polarizations providing an infinite number of signals having various orbital angular momentums associated therewith;

FIG. 14 A illustrates a plane wave having only variations in the spin angular momentum;

FIG. 14 B illustrates a signal having both spin and orbital angular momentum applied thereto;

FIGS. 15 A- 15 C illustrate various signals having different orbital angular momentum applied thereto;

FIG. 15 D illustrates a propagation of Poynting vectors for various Eigen modes;

FIG. 15 E illustrates a spiral phase plate;

FIG. 16 illustrates a block diagram of an OAM processing system utilizing quantum key distribution;

FIG. 17 illustrates a basic quantum key distribution system;

FIG. 18 illustrates the manner in which two separate states are combined into a single conjugate pair within quantum key distribution;

FIG. 19 illustrates one manner in which 0 and 1 bits may be transmitted using different basis within a quantum key distribution system;

FIG. 20 is a flow diagram illustrating the process for a transmitter transmitting a quantum key;

FIG. 21 illustrates the manner in which the receiver may receive and determine a shared quantum key;

FIG. 22 more particularly illustrates the manner in which a transmitter and receiver may determine a shared quantum key;

FIG. 23 is a flow diagram illustrating the process for determining whether to keep or abort a determined key;

FIG. 24 illustrates a functional block diagram of a transmitter and receiver utilizing a free-space quantum key distribution system;

FIG. 25 illustrates a network cloud-based quantum key distribution system;

FIG. 26 illustrates a high-speed single photon detector in communication with a plurality of users;

FIG. 27 illustrates a nodal quantum key distribution network;

FIG. 28 is a flow diagram of a process for creating a blockchain using quantum key distribution with orbital angular momentum; and

FIG. 29 is a functional block diagram of a system for performing the process of FIG. 28 .

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of a quantum resistant blockchain with multi-dimensional quantum key distribution system are illustrated and described, and other possible embodiments are described. The figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments.

Referring now to FIG. 1 , a new information-theoretic secure protocol is introduced that is robust for current and future quantum attacks. It is a quantum resistant blockchain protocol 102 that uses multi-dimensional QKD 104 using Orbital Angular Momentum (OAM) states 106 of photons. Photons are quantas of electromagnetic signals and therefore suitable for this protocol as the 2-dimensional QKD has already been demonstrated both in fiber optics as well as satellite communications. This multi-dimensional QKD protocol 104 can also be extended for development of a global QKD network and “quantum Internet” and extend quantum-safe blockchain platforms to a global scale.

Classical and Quantum Computation

The physical laws relevant to the information processing system are important to understanding the limitations of computation. In general, Quantum Mechanics adds features that do not exist in classical mechanics. Physical quantities are “quantized,” i.e. cannot be subdivided. Quantum mechanics further requires physical states to evolve in such a way that cloning a state into an independent copy is not possible. This is used in quantum cryptography to prevent information copying. Quantum mechanics also describes systems in terms of superposition that allow multiple inputs to be processed simultaneously, though only one can be observed at the end of processing, and the outcome is probabilistic in nature. Finally, quantum mechanics allows for entanglement that is not possible in classical physics.

Many computational algorithms and data structures have been developed for use on classical computers. Many of these algorithms have parallels on quantum computers but due to the quantum mechanical nature of the information processing could have far greater power. The simplest example of this is called Deutsch&#39;s Problem, which demonstrates that quantum computation can be significantly faster than classical computation.

RSA encryption relies on the fact that multiplication of large primes is easy and thus fast but factoring large composite numbers into two prime factors is very difficult and thus slow. Hash functions have the important property of being easy to calculate but difficult to invert. They provide a unique fingerprint precisely because it is very difficult to take a given hash value and find a chosen pre-image that yields that hash.

The threat of quantum computation is that such algorithms become useless because the premise of asymmetric effort of computation is invalidated. Quantum computing provides potential attacks on many cryptographic systems and algorithms. As of now, no quantum computer exists to perform such computations, though there is no doubt as to the usefulness of the algorithms themselves and of their threat to cryptographic systems. Such a quantum computer would need to have at least as many qubits as the output of the computations, e.g. 256 logical qubits to encode a hashing function with a 256-bit output. Each logical Qbit will likely need to be composed of some unknown large number of physical qubits, and the current state of the art quantum computer have small number of physical qubits.

Blockchain

The first blockchain structure was initially developed in the context of the digital currency Bitcoin to solve the problem of multiple spending.

The core component implements an open, distributed, cryptographically signed digital ledger that is secure against modification and verifiable by anyone. To prevent bulk rewriting of an entire sequence of blocks from some point in the past as well as attacks to deny service or grow the chain faster than legitimate sources can, a work requirement is added to make rewriting long chains prohibitive. For our purposes here, the relevant structure of blockchain amounts to the following description as illustrated in FIG. 2 .

The blockchain 202 consists of a sequence of blocks 204 that are stored on and copied between publicly accessible servers 205 . Each block 204 consists of four fundamental elements including the first hash 206 of the preceding block; the data content of the block (i.e. the ledger entries) 208 ; the nonce 210 that is used to give a form to the hash; and the second hash 212 of the block.

By including the hash 206 of the preceding block 204 , each successive block strengthens the authenticity claim for the preceding block. Blocks 204 early in the chain 202 cannot be modified without modifying all subsequent blocks or the modification will appear as an inconsistency in the hashes. Also, adding the data 208 to the hash makes the data unmodifiable without breaking the consistency of the block sequence. Adding a nonce 210 that is used to impose a signature structure to the hash requires significant work to be performed to generate a new block 204 . This implements the work requirement, thereby preventing the recreation of a long chain of blocks 204 to supersede the existing chain 202 with modified data.

Quantum Computation Algorithms

Referring now to FIG. 3 , to understand blockchain in the context of quantum computing and quantum attacks, you must understand two fundamental quantum algorithms 302 : Grover&#39;s Algorithm 304 and Shor&#39;s Algorithm 306 . Grover&#39;s 304 is a search algorithm to find a unique input to a black box function which operates faster than a brute force search, thus compromising hash functions of deficient length. Shor&#39;s 306 provides an exponential speed increase in factoring integers and, can be applied to the hidden subgroup and discrete logarithm problems. These problems are at the heart of breaking many known asymmetric ciphers, and thus are relevant to breaking things like public key cryptography and digital signatures. Taken together, the two quantum algorithms 302 present a significant danger to systems implementing blockchain.

Grover&#39;s Algorithm

Blockchain relies on the computation of hashes to provide security against modification of the past blocks. The chain is secure against extended revision by both its distributed nature and the computational effort required to re-compute a chain of blocks. Modification of a single block is secured by the difficulty of finding a hash collision with the existing hash, which amounts to the problem of inverting the hash function.

Grover&#39;s algorithm 304 is specifically a solution to the problem of finding a pre-image of a value of a function that is difficult to invert. If we are given a signature that is the hash value of some data s=H(d), and the function H(d) can be implemented on a quantum computer, then Grover&#39;s algorithm 304 allows us to find d for a given sin time of order O(√n) where n is the size of the space of valid hashes. In other words, it allows us to generate hash collisions more efficiently than brute force search, which would be (n).

For a hash of length k bits this means that we have a significant speedup by a factor of 2 k/2 . This can be very large even for small values of k.

Shor&#39;s Algorithm

Shor&#39;s Algorithm 306 provides a significant improvement in the efficiency of factoring large numbers. Thus, Shor&#39;s algorithm 306 can be used to attack RSA encryption and related problems. The complexity of the general number field is super-polynomial (run time longer than any polynomial in the input length) but sub-exponential (shorter than exponential in the input length). Shor&#39;s algorithm 306 on the other hand is polynomial in the input length, making the gain in speed roughly exponential. In practical terms, this makes RSA keys of 4096 bits in practice unbreakable with classical computation, but breakable with quantum computation. The consequence is that any aspect of a blockchain implementation that relies on RSA or similar algorithms would be vulnerable to quantum attack.

The first target of Shor&#39;s algorithm 306 was the factoring of large composite integers consisting of a product of two large primes. However, factoring is a specific case of the more general hidden subgroup problem, and modifications of Shor&#39;s algorithm 306 can solve all such problems. This allows solution of problems such as the discrete logarithm problem, which in turn makes such cryptographic algorithms as ElGamal encryption, Diffie-Helman key exchange, the Digital Signature Algorithm, and elliptic curve cryptography insecure. The existence of Shor&#39;s algorithm 306 demonstrates that a quantum computer opens vulnerabilities beyond that of just hash collision generation or function inversion by Grover&#39;s algorithm 304 .

Threat to Blockchain

As shown in FIG. 4 , in quantum computing, there are two issues invalidating the promises of blockchain. First, the inversion of hashes 404 is assumed to be computationally difficult. If this can be simplified by a quantum computer, the authenticity of the blockchain can no longer be guaranteed and the authenticity of entries in the blockchain is compromised. Grover&#39;s algorithm can do so significantly faster than the classical brute force search.

Referring now to FIG. 5 , Grover&#39;s algorithm can be used in two ways to attack the blockchain. The first is that it can be used to search for hash collisions 504 which can be used to replace blocks without disturbing the integrity of the blockchain. The second is that it can speed up the generation of nonces 506 , to the point that entire chains of records can be recreated with consistent modified hashes quickly enough to weaken the integrity of the chain. In both cases the algorithm is used to find the pre-image of a given value under a difficult to invert function.

As a secondary threat, in any part of a blockchain implementation that uses public/private key cryptography 406 , whether it is in an information exchange between parties or in digital signatures, a quantum computer may be able to break the security of the encryption.

Grover&#39;s Algorithm Attack (Full Collision)

If full collisions of hash values can be generated, it is possible to take a modified block content and a given hash and add trivial data to the content to make the given hash consistent with the block content. In general, this problem is computationally difficult. The general case assumes that it requires a brute force search through the possible source data with enough additional bits to finish the hash space until a case is found that matches the known hash value. For an ideal hash, this requires linear time in the size of the hash space. Weaknesses in the hash function can reduce this time, but generally the reduction is not large. The expected classical run time is of order (n) for this classical attack.

Grover&#39;s algorithm runs in time (√n), and so would give a speedup of (√n) compared to classical collision search algorithm. This makes it possible to insert a modified block into the chain without compromising the sequential consistency of the blocks. This speed increase is equivalent to finding a hash collision by brute force with half as many bits in the hash. Since this attack is only moderately fast, one could consider increasing (doubling) the hash length, but the computational effort to calculate the nonce with longer hashes would limit the ability to generate the chain and would make the blockchain not viable.

The worst scenario (asymmetric case) is the attacker has a quantum computer and the defender has only a classical computer. A slightly better scenario is when both parties have the same computational capability (symmetric case) because then there is hope that the balance of time to generate hashes and to invert hashes remains like the classical case. If this is true, the operational consequence is that whoever gets quantum computational capacity first has an advantage, but only until the defending parties develop the capacity themselves. At that point, we expect that either the system is again viable, or the system is broken beyond repair and must be discarded.

Grover&#39;s Algorithm Attack (Mining Time)

The mining step of the blockchain growth has another problem: the calculation of the nonce. This calculation adds computational cost to re-writing the chain and amounts to finding a pre-image to a partially defined hash. Grover&#39;s algorithm could speed up the generation of nonces, making the reconstruction of the blockchain from a modified block forward much faster, thereby opening the attack of regenerating the chain by undermining the computational effort of extension.

It becomes feasible for a party with a quantum computer to rapidly outperform competitors, who have only classical computing capacity, in generating additional blocks on the chain. In crypto-currency applications this means that the mining step becomes much shorter and thus allows individuals to obtain more currency than others by mining faster. In the case of a consensus blockchain for other ledger applications, the fastest miners will dominate the generation of new blocks and thus can take control of the content of the blockchain.

Of course, if the generation of nonces is even faster, there is nothing to prevent a wholesale re-creation of an entire blockchain in short time, and then substituting that history by growing faster than others can grow the true chain. Since the longest chain is conventionally chosen as the accepted truth, the faster growing chain will come to dominate the blockchain, basically re-writing history.

Threats Outside of Hashes

Hashes are susceptible to Grover&#39;s algorithm for finding function pre-images. Shor&#39;s algorithm, on the other hand, is highly effective at factoring integers and solving the hidden subgroup problem. Any part of blockchain that uses public/private key algorithms is susceptible to attack with Shor&#39;s algorithm. The algorithm serves to find the two prime factors of a composite integer used as a public key in an algorithm like RSA. Being able to factor the integer, which is computationally challenging on classical computers, gives the attacker the private key of the public/private pair. That makes it possible for the attacker to forge messages, signatures, etc. While this is not a threat to the blockchain structure of linked hashes, nor to the generation or re-generation of nonces, it means that, for example, any content that is signed may be forged by an intermediary in the process, passing the forged content on to the blockchain where it gets incorporated and thereby gains the validity of being part of the publicly readable and verifiable record.

Also, any encrypted communications used in blockchain infrastructure is vulnerable to an attacker who can break the cryptographic security of the communications. While this is removed from the core features of a blockchain, it is important.

Quantum-Resistant Cryptography

The advances in quantum computers will have a major impact on algorithms used for cryptographic applications. According to the Information Assurance Directorate (IAD) of the NSA, algorithms used in national security systems require twenty years for full deployment and should be designed to protect information for at least thirty years. One cannot predict if or when a large-scale quantum computer will ever be manufactured, however many anticipate such a system within these timescales. Therefore, the development of cryptographic algorithms which are “quantum resistant” has been determined to be a national priority.

Quantum resistant cryptography, also known as post-quantum cryptography, is a field that includes potential attacks using a quantum computer as part of the analysis of (classical) cryptographic algorithms. Although there are some insights in this area, as mentioned above, it is still a very new area with uncertainty and no accepted standards. To remedy this problem Congress has tasked the National Institute of Standards and Technology (NIST) to “research and identify, or if necessary, develop cryptography standards and guidelines for future cybersecurity needs, including quantum-resistant cryptography standards.” NIST has already initiated this process, and public updates to this process are posted to &lt;http://www.nist.gov/pqcrypto&gt;.

Post-Quantum Cryptography for Blockchain with Hash Functions

Even though standards are still being developed for quantum resistant cryptography, parts that will be important for designing systems that involve blockchain based technologies can be determined. The first part is related to the hashing function itself. As described, Grover&#39;s algorithm provides a quadratic speedup over classical algorithms for evaluating hash functions. Since this speedup is not an exponential speedup like Shor&#39;s algorithm, this means that the computational complexity of a function that is needed for secure applications can be restored by increasing the number of bits used in the calculation. At most one needs only twice the number of bits due to the quadratic speedup of the algorithm.

As previously described, there are two parts in which hash functions are used to protect a blockchain. The primary method based on inverting a hash or finding a collision is computationally difficult. The difficulty in finding a different data block with the same hash grows with the length of the hash. The complexity is of order (n) classically, but (√n) with Grover&#39;s algorithm for a hash space of size n. So, if a certain level of difficulty is required for security, a quantum-resistant standard will require twice the hash length of a similar requirement that considers only classical algorithms.

The second way in which blockchain may utilize hash functions for security is by signing a block. This is done for example by finding a nonce such that the first m bits of the block&#39;s hash are zero. This is equivalent to computing a partial collision of the hash function and is computationally difficult. This difficulty is precisely the ‘proof of work’ that a signature is designed to require. Just as the hash length k can be increased in order to maintain a level of protection against a quantum attack, so also the length m required for signing the block can be increased to ensure a minimal ‘proof of work’. However, this comes at the expense of making the required work computationally twice as hard per additional bit, or equivalently take twice as long, for the classical devices that are used to sign a data block. Therefore, there will be an inherent trade-off between the system requirements necessary for implementing any blockchain protocol that uses hash-based block signatures and protecting against a spoofing attack from a quantum machine.

Post-Quantum Cryptography Outside of Blockchain Hash Function

If the blockchain ledger needs to be distributed, then encryption schemes will be required. Other protocols are also needed to define what entities are (allowed to expand the blockchain), where identity verifications or digital signatures might be utilized. In many of these cases, current standard cryptographic algorithms are generally insufficient to protect against the threat of quantum computing.

A key issue with many current cryptographic algorithms is that security relies on the difficulty of a mathematical problem. The asymmetric key encryption scheme of RSA relies on the difficulty of prime factorization of large numbers; the Digital Signature Algorithm (DSA), the standard for digital signatures, is based on the problem of computing discrete logarithms; and the Elliptic Curve Digital Signature Algorithm (ECDSA) is a variant of DSA and example of elliptic curve cryptography (ECC). All three types of problems of factorization, discrete logarithms, and ECC can easily be solved by Shor&#39;s algorithm on a quantum computer.

Although NIST has yet to define quantum-resistant cryptographic standards, there are several classes of cryptographic systems 602 , as shown in FIG. 6 , that are relatively robust to attacks from either classical or quantum devices. Some of them are:

Hash-based cryptography 604 . The classic example is Merkle&#39;s hash-tree public-key signature system, building upon a one-message-signature idea of Lamport and Diffie. Code-based cryptography 606 . The classic example is McEliece&#39;s hidden Goppa-code public-key encryption system. Lattice-based cryptography 608 . The example that has perhaps attracted the most interest, not the first example historically, is the Hoffstein-Pipher-Silverman “NTRU” public-key-encryption system. Multivariate-quadratic- equations cryptography 610 . One of many interesting examples is Patarin&#39;s Hidden Field Equations public-key-signature system, generalizing a proposal by Matsumoto and Imai.

Quantum Cryptography

Another strategy for future crypto-systems involves leveraging quantum features in new technology. This field of quantum cryptography is distinct from post-quantum cryptography which relies purely on classical methods and present-day technologies to protect against potential future quantum attacks. Instead, quantum cryptography is itself part of quantum information science and looks for how quantum effects can create fundamentally new ways of doing cryptography.

The primary and most mature technology that has come out of quantum cryptography is quantum key distribution (QKD) as is more fully described herein below. QKD is a protocol by which a random bitstream can be generated between parties. Once established, this random bitstream message is used as a one-time pad (OTP) to encrypt a secret message. This method of distributing a secret shared key is not secured by mathematical complexity like normal methods of distributing cryptographic keys (e.g. Diffie-Hellman), but instead is based on the laws of quantum physics itself. This security specifically comes from the Quantum No-cloning theorem, a consequence of the Heisenberg uncertainty principle which states a signal made of individual quantum particles cannot be copied without introducing observable errors, preventing any eavesdropper from avoiding detection. Once a random key has been established between two parties with a QKD protocol, the encrypted message is considered unconditionally secure.

QKD is the most mature technology within the field of quantum information science. Commercial companies exist that will sell transmitters and receivers, and such systems have been used in both the private and public sectors. The technology currently requires private networks (e.g. dark fibers), cannot be repeated or routed and is currently limited to city scale networks. Although there are limitations, the technology is still developing fast and so will likely become more extensive in the near-future.

In addition to QKD, there are other ideas that are being researched that could make a significant impact to blockchain based systems. For instance, information can be encoded and transmitted directly into a quantum stream (rather than just using a quantum channel to distribute a key). There has also been a proposal for a “Quantum Bitcoin,” which uses a classical blockchain ledger but uses quantum methods to mine and verify a block. There are also protocols to encode and store information such as a ledger in a quantum system making the information tamper-proof. There are also quantum bit commitment protocols which may be a type of alternative to digital signature schemes. Many of these ideas may have promise, however these technologies are currently at early stages with many of the technologies being at least as difficult to implement as quantum computing itself.

As indicated, current blockchain relies on two one-way computational technologies: cryptographic hash functions and digital signatures. Most blockchain platforms rely on the elliptic curve public-key cryptography (ECDSA) or the large integer factorization problem (RSA) to generate a digital signature. The security of these algorithms assumes computational complexity of certain mathematical problems.

As described, a universal quantum computer would enable efficient solving of these problems, thereby making corresponding digital signature algorithms, including those used in blockchains, insecure. As described, Shor&#39;s quantum algorithm solves factorization of large integers and discrete logarithms in polynomial time and Grover&#39;s search algorithm allows a quadratic speedup in calculating the inverse hash function. This will enable a so-called 51-percent attack, in which a syndicate of malicious parties controlling most of the network&#39;s computing power would monopolize the mining of new blocks. Such an attack would allow the perpetrators to sabotage other parties&#39; transactions or prevent their own spending transactions from being recorded in the blockchain. Other attacks with quantum computing on blockchain technology as well as possible roles of quantum algorithms in the mining process are considered in recent work.

The security of blockchains can be enhanced by using post-quantum digital signature schemes for signing transactions. Such schemes are robust against attacks with quantum computers. However, this robustness relies on unproven assumptions. Also, post-quantum digital signatures are computationally intensive and are not helpful against attacks that utilize the quantum computer to dominate the network&#39;s mining.

In addition to the blockchains based on mining principles there are other approaches to distributed ledgers maintenance, e.g. Byzantine fault tolerance (BFT) replication and practical BFT replication. All the proposed approaches either require use of digital signatures, and hence are vulnerable to quantum computer attacks, or pairwise authenticated channels at least. The pairwise authentic channel ensures that each message was not tampered while passing but does not solve the transferability issue.

The way to guarantee authentication in the quantum era is to use quantum key distribution (QKD), which guarantees information-theoretic (unconditional) security based on the laws of quantum physics. QKD can generate a secret key between two parties connected by a quantum channel (for transmitting quantum states) and a public classical channel (for post-processing procedures). The technology enabling QKD networks have been demonstrated in many experiments and is now publicly available through some suppliers.

Kiktenko and Pozhar described a blockchain platform that combines (i) the original BFT state-machine replication without use of digital signatures (“broadcast protocol”), (ii) A 2-dimensional QKD for providing authentication, and implemented an experiment demonstrating its capability in an urban QKD network. This 2-dimensional scheme is robust against presently known capabilities of the quantum computer, but it is not robust against future quantum attacks.

The utility of QKD for blockchains may appear counterintuitive, as QKD networks rely on trust among nodes, whereas the earmark of many blockchains is the absence of such trust. More specifically, one may argue that QKD cannot be used for authentication because it itself requires an authenticated classical channel for operation. However, each QKD communication session generates a large amount of shared secret data, part of which can be used for authentication in subsequent sessions. Therefore, a small amount of “seed” secret key that the parties share before their first QKD session confirms their secure authentication for all future communication. In this way, QKD can be used in lieu of classical digital signatures.

Referring now to FIG. 7 , consider a blockchain protocol 702 within a two-layer network with n nodes. The first layer is a QKD network 704 with pairwise communication channels that permit establishing information-theoretic (unconditionally) secure private key for each pair of nodes. The second (classical) layer 706 is used for transmitting messages with authentication tags based on information-theoretic secure hashing that are created using the private keys procured in the first layer. For example, a blockchain with a digital currency can be considered. The operation of the blockchain is based on two procedures: (i) creation of transactions and (ii) construction of blocks that aggregate new transactions. As shown in FIG. 8 , new transactions are created at step 802 by those nodes who wish to transfer their funds to another node. Each individual new transaction record is constructed like those in Bitcoin, i.e. contains the information about the sender, receiver, time of creation, amount to be transferred, and a list of reference transactions that justifies that the sender has enough funds for the operation. This record is sent at step 804 via authenticated channels to all other n−1 nodes, thereby entering the pool of unconfirmed transactions. Each node checks these entries with respect to their local copy of the database and each other, in order to verify that each transaction has sufficient funds and forms an opinion regarding the transaction&#39;s acceptability at step 806 . At this stage, the community does not attempt to exclude double-spending events. Next, the unconfirmed transactions are aggregated into a block at step 808 . The classical blockchain practice of having the blocks proposed by individual “miners” is eliminated, because it is vulnerable to quantum computer attacks in at least two ways. First, transactions are not arranged with digital signatures. This means that a miner has complete freedom to fabricate arbitrary transactions and include them in the block. Second, a node with a quantum computer is able to mine new blocks significantly faster than any classical node. This opens a possibility for attacks such as the 51-percent attack described above.

Instead, it is better to create blocks in a decentralized fashion. To that end, one can use the broadcast protocol proposed in the classic paper by Shostak and Lamport. This secure protocol allows achieving a Byzantine agreement in any network with pairwise authenticated communication provided that the number of dishonest parties is less than n/3. At a certain moment in time (e.g. every ten minutes), the network applies the protocol to each unconfirmed transaction, arriving at a consensus regarding the correct version of that transaction (thereby eliminating double-spending) and whether the transaction is acceptable. Each node then forms a block out of all acceptable transactions, sorted according to their time stamps. The block is added to the database. In this way, the same block will be formed by all honest parties, thereby removing the possibility of a “fork” which is the situation in which several different versions of a block are created simultaneously by different miners.

Because the broadcast protocol is relatively forgiving to the presence of dishonest or faulty nodes, this blockchain setup has significant tolerance to some of the nodes or communication channels not operating properly during its implementation. While the broadcast protocol is relatively data intensive, the data need not be transmitted through quantum channels. Quantum channels are only required to generate private keys.

While this protocol seems to be efficient against quantum attacks on the distribution of transactions and formation of blocks, the database is still somewhat vulnerable while it is stored. A possible attack scenario is as follows: a malicious party with a quantum computer works offline to forge the database. It changes one of the past transaction records to its benefit and performs a Grover search for a variant of other transactions within the same block such that its hash remains the same, to make the forged version appear valid. Once the search is successful, it hacks into all or some of the network nodes and substitutes the legitimate database by its forged version. However, the potential of this attack to cause significant damage is low, because the attacker would need to simultaneously hack at least one-third of the nodes to alter the consensus. Also, because the Grover algorithm offers only a quadratic speed-up with respect to classical search algorithms, this scenario can be prevented by increasing the convention on the length of the block hash to about a square of its safe classical value.

A new information-theoretic secure protocol that is robust for current and future quantum attacks is shown generally in FIG. 9 . It is a quantum resistant blockchain protocol 908 that uses multi-dimensional QKD 902 using Orbital Angular Momentum (OAM) states of photons with blockchain 906 . Photons are quantas of electromagnetic signals and therefore suitable for this protocol as the 2-dimensional QKD has already been demonstrated both in fiber optics as well as satellite communications. This multi-dimensional protocol can also be extended for development of a global QKD network and “quantum Internet” and extend quantum-safe blockchain platforms to a global scale.

Orbital Angular Momentum (OAM)

The orbital angular momentum (OAM) component will now be more fully described. Referring now more particularly to FIG. 10 , there is illustrated a functional block diagram of a system for generating the orbital angular momentum “twist” within a communication system, such as that illustrated with respect to FIG. 3 , to provide a data stream that may be combined with multiple other data streams for transmission upon a same wavelength or frequency. Multiple data streams 1002 are provided to the transmission processing circuitry 1000 . Each of the data streams 1002 comprises, for example, an end to end link connection carrying a voice call or a packet connection transmitting non-circuit switch packed data over a data connection. The multiple data streams 1002 are processed by modulator/ demodulator circuitry 1004 . The modulator/ demodulator circuitry 1004 modulates the received data stream 1002 onto a wavelength or frequency channel using a multiple level overlay modulation technique, as will be more fully described herein below. The communications link may comprise an optical fiber link, free-space optics link, RF microwave link, RF satellite link, wired link (without the twist), etc.

The modulated data stream is provided to the orbital angular momentum (OAM) signal processing block 1006 . Each of the modulated data streams from the modulator/ demodulator 1004 are provided a different orbital angular momentum by the orbital angular momentum electromagnetic block 1006 such that each of the modulated data streams have a unique and different orbital angular momentum associated therewith. Each of the modulated signals having an associated orbital angular momentum are provided to an optical transmitter

CLAIMS

Claims ( 26 )

What is claimed is:

1. A method for generating a blockchain, comprising:

performing a first hash using a first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution using N-state qudits, where N is greater than 2;

establishing the first block of the blockchain using the first group of data, the first nonce and the first digital signature;

performing a second hash using a second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for a second block, wherein the hash function uses encryption based on the quantum key distribution using N-state qudits, where N is greater than 2 ; and

establishing the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

2. The method of claim 1 , wherein the hash function further uses quantum key distribution using orbital angular momentum to perform the first hash and the second hash.

3. The method of claim 1 , wherein the hash function further uses quantum key distribution using an orthogonal function to perform the first hash and the second hash.

4. The method of claim 1 , wherein the step of performing the first hash further comprises:

receiving the first group of data and the first nonce;

generating a first secret key using a quantum key generation process;

generating the first digital signature using the generated first secret key.

5. The method of claim 4 , wherein the step of generating the first secret key further comprises:

selecting a series of random bits;

assigning a random basis to each of the selected series of random bits;

generating a first photon polarization state for each of the selected series of random bits responsive to the selected series of random bits and the assigned random basis for the selected series of random bits; and

determining the first secret key responsive to matching portions of the first photon polarization states and second photon polarization states.

6. The method of claim 4 further including processing the first group of data and the first nonce to associate an orthogonal function with the first group of data and the first nonce.

7. The method of claim 6 , wherein the orthogonal function comprises at least one of a modified Hermite polynomials, Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials and Laguerre functions.

8. A system for generating a blockchain, comprising:

an input for receiving a plurality of groups of data;

blockchain processing circuitry for generating the blockchain for the plurality of groups of data, wherein the blockchain processing circuitry generates the blockchain by:

performing a first hash using a first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution using N-state qudits, where N is greater than 2;

establishing the first block of the blockchain using the first group of data, the first nonce and the first digital signature;

performing a second hash using a second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for a second block, wherein the hash function uses encryption based on the quantum key distribution using N-state qudits, where N is greater than 2; and

establishing the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

9. The system of claim 8 , wherein the hash function further uses quantum key distribution using orbital angular momentum to perform the first hash and the second hash.

10. The system of claim 8 , wherein the hash function further uses quantum key distribution using an orthogonal function to perform the first hash and the second hash.

11. The system of claim 8 , wherein the performing the first hash by the blockchain processing circuitry further comprises:

receiving the first group of data and the first nonce;

generating a first secret key using a quantum key generation process;

generating the first digital signature using the generated first secret key.

12. The system of claim 11 , wherein the generating the first secret key by the blockchain processing circuitry further comprises:

selecting a series of random bits;

assigning a random basis to each of the selected series of random bits;

generating a first photon polarization state for each of the selected series of random bits responsive to the selected series of random bits and the assigned random basis for the selected series of random bits; and

determining the first secret key responsive to matching portions of the first photon polarization states and second photon polarization states.

13. The system of claim 11 , wherein the blockchain processing circuity further generates the blockchain by processing the first group of data and the first nonce to associate an orthogonal function with the first group of data and the first nonce.

14. The system of claim 13 , wherein the orthogonal function comprises at least one of a modified Hermite polynomials, Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials and Laguerre functions.

15. A method for generating a blockchain, comprising:

performing a first hash using a first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution combined with orbital angular momentum to provide greater than 2-state qudits;

establishing the first block of the blockchain using the first group of data, the first nonce and the first digital signature;

performing a second hash using a second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for a second block, wherein the hash function uses encryption based on the quantum key distribution combined with the orbital angular momentum to provide greater than 2-state qudits; and

establishing the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

16. The method of claim 15 , wherein the step of performing the first hash further comprises:

receiving the first group of data and the first nonce;

generating a first secret key using a quantum key generation process;

generating the first digital signature using the generated first secret key.

17. The method of claim 16 , wherein the step of generating the first secret key further comprises:

selecting a series of random bits;

assigning a random basis to each of the selected series of random bits;

generating a first photon polarization state for each of the selected series of random bits responsive to the selected series of random bits and the assigned random basis for the selected series of random bits; and

determining the first secret key responsive to matching portions of the first photon polarization states and second photon polarization states.

18. The method of claim 15 further including processing the first group of data and the first nonce to associate an orthogonal function with the first group of data and the first nonce.

19. The method of claim 18 , wherein the orthogonal function comprises at least one of a modified Hermite polynomials, Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials and Laguerre functions.

20. The method of claim 15 , wherein the hash function uses encryption based on quantum key distribution using N-state qudits, where N is greater than 2.

21. A system for generating a blockchain, comprising:

an input for receiving a plurality of groups of data;

blockchain processing circuitry for generating the blockchain for the plurality of groups of data, wherein the blockchain processing circuitry generates the blockchain by:

performing a first hash using a first group of data and a first nonce as an input to a hash function to generate a first digital signature for a first block, wherein the hash function uses encryption based on quantum key distribution combined with orbital angular momentum to provide greater than 2-state qudits;

establishing the first block of the blockchain using the first group of data, the first nonce and the first digital signature;

performing a second hash using a second group of data, a second nonce and the first digital signature as an input to the hash function to generate a second digital signature for a second block, wherein the hash function uses encryption based on the quantum key distribution combined with the orbital angular momentum to provide greater than 2-state qudits; and

establishing the second block of the blockchain using the second group of data, the second nonce, the first digital signature and the second digital signature.

22. The system of claim 21 , wherein the performing the first hash by the blockchain processing circuitry further comprises:

receiving the first group of data and the first nonce;

generating a first secret key using a quantum key generation process;

generating the first digital signature using the generated first secret key.

23. The system of claim 22 , wherein the step of generating the first secret key by the blockchain processing circuitry further comprises:

selecting a series of random bits;

assigning a random basis to each of the selected series of random bits;

generating a first photon polarization state for each of the selected series of random bits responsive to the selected series of random bits and the assigned random basis for the selected series of random bits; and

determining the first secret key responsive to matching portions of the first photon polarization states and second photon polarization states.

24. The system of claim 21 further including processing the first group of data and the first nonce to associate with the first group of data and the first nonce an orthogonal function.

25. The system of claim 24 , wherein the orthogonal function comprises at least one of a modified Hermite polynomials, Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials and Laguerre functions.

26. The system of claim 21 , wherein the hash function uses encryption based on quantum key distribution using N-state qudits, where N is greater than 2.

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