Quantum Cryptography Quotes

if a message protected by quantum cryptography were ever to be deciphered, it would mean that quantum theory is flawed,

if N is large enough, it is virtually impossible to deduce p and q from N, and this is perhaps the most beautiful and elegant aspect of the RSA asymmetric cipher.

The value we provide at Mayflower-Plymouth exists at the convergence of various technologies and studies including Blockchain, cryptography, quantum computing, permaculture design principles, artificial intelligence, stigmergy, forestry, economics, additive manufacturing, big data, advanced logistics and more.

The value we provide at Mayflower exists at the convergence of various new technologies and studies including Blockchain, cryptography, quantum computing, artificial intelligence, stigmergy, additive manufacturing, big data, advanced logistics and more.

the enemy of security: repetition leads to patterns, and cryptanalysts thrive on patterns.

The NSA employs more mathematicians, buys more computer hardware, and intercepts more messages than any other organization in the world.

Enigma was considered invulnerable, until the Poles revealed its weaknesses.

This apparently innocuous observation would lead to the first great breakthrough in cryptanalysis.

like Turing and the cryptanalysts at Bletchley Park, the Navajo were ignored for decades. Eventually, in 1968, the Navajo code was declassified, and the following year the code talkers held their first reunion.

Ron Rivest, one of the inventors of RSA, thinks that restricting cryptography would be foolhardy: It is poor policy to clamp down indiscriminately on a technology just because some criminals might be able to use it to their advantage. For example, any U.S. citizen can freely buy a pair of gloves, even though a burglar might use them to ransack a house without leaving fingerprints. Cryptography is a data-protection technology, just as gloves are a hand-protection technology. Cryptography protects data from hackers, corporate spies, and con artists, whereas gloves protect hands from cuts, scrapes, heat, cold, and infection. The former can frustrate FBI wiretapping, and the latter can thwart FBI fingerprint analysis. Cryptography and gloves are both dirt-cheap and widely available. In fact, you can download good cryptographic software from the Internet for less than the price of a good pair of gloves.

The first known European book to describe the use of cryptography was written in the thirteenth century by the English Franciscan monk and polymath Roger Bacon. Epistle on the Secret Works of Art and the Nullity of Magic included seven methods for keeping messages secret, and cautioned: “A man is crazy who writes a secret in any other way than one which will conceal it from the vulgar.

Propose to an Englishman any principle, or any instrument, however admirable, and you will observe that the whole effort of the English mind is directed to find a difficulty, a defect, or an impossibility in it. If you speak to him of a machine for peeling a potato, he will pronounce it impossible: if you peel a potato with it before his eyes, he will declare it useless, because it will not slice a pineapple.

Those “weird evolutions” enabled by a CTC, Ringbauer notes, would have remarkable practical applications, such as breaking quantum-based cryptography through the cloning of the quantum states of fundamental particles. “If you can clone quantum states,” he says, “you can violate the Heisenberg uncertainty principle,” which comes in handy in quantum cryptography because the principle forbids simultaneously accurate measurements of certain kinds of paired variables, such as position and momentum. “But if you clone that system, you can measure one quantity in the first and the other quantity in the second, allowing you to decrypt an encoded message.

The correct use of a strong cipher is a clear boon to sender and receiver, but the misuse of a weak cipher can generate a very false sense of security.

Alan Turing was another cryptanalyst who did not live long enough to receive any public recognition. Instead of being acclaimed a hero, he was persecuted for his homosexuality. In 1952, while reporting a burglary to the police, he naively revealed that he was having a homosexual relationship. The police felt they had no option but to arrest and charge him with “Gross Indecency contrary to Section 11 of the Criminal Law Amendment Act 1885.” The newspapers reported the subsequent trial and conviction, and Turing was publicly humiliated. Turing’s secret had been exposed, and his sexuality was now public knowledge. The British Government withdrew his security clearance. He was forbidden to work on research projects relating to the development of the computer. He was forced to consult a psychiatrist and had to undergo hormone treatment, which made him impotent and obese. Over the next two years he became severely depressed, and on June 7, 1954, he went to his bedroom, carrying with him a jar of cyanide solution and an apple. Twenty years earlier he had chanted the rhyme of the Wicked Witch: “Dip the apple in the brew, Let the sleeping death seep through.” Now he was ready to obey her incantation. He dipped the apple in the cyanide and took several bites. At the age of just forty-two, one of the true geniuses of cryptanalysis committed suicide.

The three Britons had to sit back and watch as their discoveries were rediscovered by Diffie, Hellman, Merkle, Rivest, Shamir and Adleman over the next three years.

Lyndon Johnson and Richard Nixon were guilty of unjustified wiretaps, and President John F. Kennedy conducted dubious wiretaps in the first month of his presidency.

Even the Vatican, probably the second most active center of cryptanalysis, would send Soro seemingly impenetrable messages that had fallen into its hands. In 1526, Pope Clement VII sent him two encrypted messages, and both were returned having been successfully cryptanalyzed.

The Vigenère cipher was called “le chiffre indéchiffrable,” but Babbage broke it;

He was a rather quirky worker, and he didn’t really fit into the day-to-day business of GCHQ. But in terms of coming up with new ideas he was quite exceptional. You had to sort through some rubbish sometimes, but he was very innovative and always willing to challenge the orthodoxy.

We would be in real trouble if everybody in GCHQ was like him, but we can tolerate a higher proportion of such people than most organizations. We put up with a number of people like him. Figure 66 James Ellis. (photo credit 6.4) One of Ellis’s greatest qualities was his breadth of knowledge.

in the 1980s it was only government, the military and large businesses that owned computers powerful enough to run RSA. Not surprisingly, RSA Data Security, Inc., the company set up to commercialize RSA, developed their encryption products with only these markets in mind.

Zimmermann believed that everybody deserved the right to the privacy that was offered by RSA encryption, and he directed his political zeal toward developing an RSA encryption product for the masses.

Zimmermann employed a neat trick that used asymmetric RSA encryption in tandem with old-fashioned symmetric encryption.

In June 1991 he took the drastic step of asking a friend to post PGP on a Usenet bulletin board. PGP is just a piece of software, and so from the bulletin board it could be downloaded by anyone for free. PGP was now loose on the Internet.

For decades, ENIAC, not Colossus, was considered the mother of all computers.

NSA employs more mathematicians, buys more computer hardware, and intercepts more messages than any other organization in the world. It is the world leader when it comes to snooping.

In the 1970s, banks attempted to distribute keys by employing special dispatch riders who had been vetted and who were among the company’s most trusted employees.

dispatch riders would race across the world with padlocked briefcases, personally distributing keys to everyone who would receive messages from the bank over the next week. As business networks grew in size, as more messages were sent, and as more keys had to be delivered, the banks found that this distribution process became a horrendous logistical nightmare, and the overhead costs became prohibitive.

A large amount of identically encrypted material provides a cryptanalyst with a correspondingly larger chance of identifying the key.

the Germans therefore took the clever step of using the day key settings to transmit a new message key for each message.

many shipwrecks and engineering disasters were blamed on faulty tables. These mathematical tables were calculated by hand, and the mistakes were simply the result of human error. This caused Babbage to exclaim, “I wish to God these calculations had been executed by steam!” This marked the beginning of an extraordinary endeavor to build a machine capable of faultlessly calculating the tables to a high degree of accuracy. In 1823 Babbage designed “Difference Engine No. 1,” a magnificent calculator consisting of 25,000 precision parts, to be built with government funding. Although Babbage was a brilliant innovator, he was not a great implementer. After ten years of toil, he abandoned “Difference Engine No.

Vigenère’s work culminated in his Traicté des Chiffres (“A Treatise on Secret Writing”), published in 1586. Ironically, this was the same year that Thomas Phelippes was breaking the cipher of Mary Queen of Scots. If only Mary’s secretary had read this treatise, he would have known about the Vigenère cipher, Mary’s messages to Babington would have baffled Phelippes, and her life might have been spared.

the Vigenère cipher belongs to a class known as polyalphabetic, because it employs several cipher alphabets per message. The polyalphabetic nature of the Vigenère cipher is what gives it its strength, but it also makes it much more complicated to use. The additional effort required in order to implement the Vigenère cipher discouraged many people from employing it.

French listening posts learned to recognize a radio operator’s fist. Once encrypted, a message is sent in Morse code, as a series of dots and dashes, and each operator can be identified by his pauses, the speed of transmission, and the relative lengths of dots and dashes. A fist is the equivalent of a recognizable style of handwriting.

copies were passed to the cryptanalysts, who sat in little kiosks, ready to tease out the meanings of the messages. As well as supplying the emperors of Austria with invaluable intelligence, the Viennese Black Chamber sold the information it harvested to other powers in Europe. In 1774 an

it can be mathematically proved that it is impossible for a cryptanalyst to crack a message encrypted with a onetime pad cipher. In other words, the onetime pad cipher is not merely believed to be unbreakable, just as the Vigenère cipher was in the nineteenth century, it really is absolutely secure

it requires a great deal of time, effort and money to create a random key. The best random keys are created by harnessing natural physical processes, such as radioactivity, which is known to exhibit truly random behavior. The cryptographer could place a lump of radioactive material on a bench, and detect its emissions with a Geiger counter. Sometimes the emissions follow each other in rapid succession, sometimes there are long delays—the time between emissions is unpredictable and random. The cryptographer could then connect a display to the Geiger counter, which rapidly cycles through the alphabet at a fixed rate, but which freezes momentarily as soon as an emission is detected.

the hotline between the presidents of Russia and America is secured via a onetime pad cipher.

impenetrability of the Navajo code was all down to the fact that Navajo belongs to the Na-Dene family of languages, which has no link with any Asian or European language.

a Navajo verb is conjugated not solely according to its subject, but also according to its object. The verb ending depends on which category the object belongs to: long (e.g., pipe, pencil), slender and flexible (e.g., snake, thong), granular (e.g., sugar, salt), bundled (e.g., hay), viscous (e.g., mud, feces) and many others. The verb will also incorporate adverbs, and will reflect whether or not the speaker has experienced what he or she is talking about, or whether it is hearsay. Consequently, a single verb can be equivalent to a whole sentence, making it virtually impossible for foreigners to disentangle its meaning.

Major General Howard Conner, “without the Navajos, the marines would never have taken Iwo Jima.

It is quite possible that British Intelligence demanded that Babbage keep his work secret, thus providing them with a nine-year head start over the rest of the world.

imagined a machine whose internal workings could be altered so that it could perform all the functions of all conceivable Turing machines. The alterations would be made by inserting carefully selected tapes, which transformed the single flexible machine into a dividing machine, a multiplying machine, or any other type of machine. Turing called this hypothetical device a universal Turing machine because it would be capable of answering any question that could logically be answered. Unfortunately, it turned out that it is not always logically possible to answer a question about the undecidability of another question, and so even the universal Turing machine was unable to identify every undecidable question.

Turing knew of Babbage’s work, and the universal Turing machine can be seen as a reincarnation of Difference Engine No. 2. In fact, Turing had gone much further, and provided computing with a solid theoretical basis,

The fundamental weakness of the Vigenère cipher is its cyclical nature. If the keyword is five letters long, then every fifth letter of the plaintext is encrypted according to the same cipher alphabet. If the cryptanalyst can identify the length of the keyword, the ciphertext can be treated as a series of five monoalphabetic ciphers, and each one can be broken by frequency analysis. However, consider what happens as the keyword gets longer. Imagine a plaintext of 1,000 letters encrypted according to the Vigenère cipher,

The fate of the Polish nation had depended on Rejewski, and he did not disappoint his country. Rejewski’s attack on Enigma is one of the truly great accomplishments of cryptanalysis.

By depriving Rejewski of the keys, Langer believed he was preparing him for the inevitable time when the keys would no longer be available. He knew that if war broke out it would be impossible for Schmidt to continue to attend covert meetings, and Rejewski would then be forced to be self-sufficient. Langer thought that Rejewski should practice self-sufficiency in peacetime, as preparation for what lay ahead.

Rejewski had vastly simplified the task of finding the day key by divorcing the problem of finding the scrambler settings from the problem of finding the plugboard settings. On their own, both of these problems were solvable. Originally, we estimated that it would take more than the lifetime of the universe to check every possible Enigma key. However, Rejewski had spent only a year compiling his catalogue of chain lengths, and thereafter he could find the day key before the day was out. Once he had the day key, he possessed the same information as the intended receiver and so could decipher messages just as easily.

quantum cryptography is a system that ensures the security of a message by making it hard for Eve to read accurately a communication between Alice and Bob. Furthermore, if Eve tries to eavesdrop then Alice and Bob will be able to detect her presence. Quantum cryptography therefore allows Alice and Bob to exchange and agree upon a onetime pad in complete privacy, and thereafter they can use this as a key to encrypt a message.

Quantum cryptography is an unbreakable system of encryption.

When Hermann Göring visited Warsaw in 1934, he was totally unaware of the fact that his communications were being intercepted and deciphered. As he and other German dignitaries laid a wreath at the Tomb of the Unknown Soldier next to the offices of the Biuro Szyfrów, Rejewski could stare down at them from his window, content in the knowledge that he could read their most secret communications.

Babbage’s successful cryptanalysis of the Vigenère cipher was probably achieved in 1854, soon after his spat with Thwaites, but his discovery went completely unrecognized because he never published it. The discovery came to light only in the twentieth century, when scholars examined Babbage’s extensive notes. In the meantime, his technique was independently discovered by Friedrich Wilhelm Kasiski, a retired officer in the Prussian army. Ever since 1863, when he published his cryptanalytic breakthrough in Die Geheimschriften und die Dechiffrir-kunst (“Secret Writing and the Art of Deciphering”), the technique has been known as the Kasiski Test, and Babbage’s contribution has been largely ignored.

He took the messages to a local brewer, who wrapped them in a leather packet, which was then hidden inside a hollow bung used to seal a barrel of beer. The brewer would deliver the barrel to Chartley Hall, whereupon one of Mary’s servants would open the bung and take the contents to the Queen of Scots. The process worked equally well for getting messages out of Chartley Hall.

When ships carrying COMSEC material came into dock, crypto-custodians would march onboard, collect stacks of cards, paper tapes, floppy disks, or whatever other medium the keys might be stored on, and then deliver them to the intended recipients.

one-way functions are sometimes called Humpty Dumpty functions. Modular arithmetic, sometimes called clock arithmetic in schools, is an area of mathematics that is rich in one-way functions. In modular arithmetic, mathematicians consider a finite group of numbers arranged in a loop,

alphabetic scripts tend to have between 20 and 40 characters (Russian, for example, has 36 signs, and Arabic has 28).

Syllabic scripts occupy the middle ground, with between 50 and 100 syllabic characters. Beyond these two facts, Linear B was an unfathomable mystery.

scripts that rely on semagrams tend to have hundreds or even thousands of signs (Chinese has over 5,000).

Histaiaeus shaved the head of his messenger, wrote the message on his scalp, and then waited for the hair to regrow. This was clearly a period of history that tolerated a certain lack of urgency.

Secret communication achieved by hiding the existence of a message is known as steganography

Chinese wrote messages on fine silk, which was then scrunched into a tiny ball and covered in wax. The messenger would then swallow the ball of wax.

The first microdot to be spotted by the FBI was in 1941, following a tip-off that the Americans should look for a tiny gleam from the surface of a letter, indicative of smooth film.

first ever military cryptographic device, the Spartan scytale, dating back to the fifth century B.C. The scytale is a wooden staff around which a strip of leather or parchment is wound,

significance of the key, as opposed to the algorithm, is an enduring principle of cryptography. It was definitively stated in 1883 by the Dutch linguist Auguste Kerckhoffs von Nieuwenhof in his book La Cryptographie militaire: “Kerckhoffs’ Principle: The security of a cryptosystem must not depend on keeping secret the crypto-algorithm. The security depends only on keeping secret the key.

The letters a and l are the most common in Arabic, partly because of the definite article al-, whereas the letter j appears only a tenth as frequently.

had the cipher machines been used properly—without repeated message keys, without cillies, without restrictions on plugboard settings and scrambler arrangements, and without stereotypical messages which resulted in cribs—it is quite possible that they might never have been broken at all.

the First World War was the chemists’ war, because mustard gas and chlorine were employed for the first time,

Second World War was the physicists’ war, because the atom bomb was detonated.

the Third World War would be the mathematicians’ war, because mathematicians will have control over the next great weapon of war—information. Mathematicians have been responsible for developing the codes that are currently used to protect military information.

destitute.

At the same time as acquiring knowledge, the Islamic civilization was able to disperse it, because it had procured the art of paper-making from the Chinese. The manufacture of paper gave rise to the profession of warraqin, or 'those who handle paper,' human photocopying machines who copied manuscripts and supplied the burgeoning publishing industry. At its peak, tens of thousands of books were published every year, and in just one suburb of Baghdad there were over a hundred bookshops. As well as such classics as Tales from the Thousand and One Nights, these bookshops also sold textbooks on every imaginable subject, and helped to support the most literate and learned society in the world.

Had the Arabs merely been familiar with the use of the mono-alphabetic substitution cipher, they would not warrant a significant mention in any history of cryptography. However, in addition to employing ciphers, the Arab scholars were also capable of destroying ciphers. They in fact invented cryptanalysis, the science of unscrambling a message without knowledge of the key. While the cryptographer develops new methods of secret writing, it is the cryptanalyst who struggles to find weaknesses in these methods in order to break into secret messages. Arabian cryptanalysts succeeded in finding a method for breaking the monoalphabetic substitution cipher, a cipher that had remained invulnerable for several centuries.

Cryptanalysis could not be invented until a civilization had reached a sufficiently sophisticated level of scholarship in several disciplines, including mathematics, statistics, and linguistics. The Muslim civilization provided an ideal cradle for cryptanalysis, because Islam demands justice in all spheres of human activity, and achieving this requires knowledge, or ilm. Every Muslim is obliged to pursue knowledge in all its forms, and the economic success of the Abbasid caliphate meant that scholars had the time, money, and materials required to fulfil their duty. They endeavoured to acquire knowledge of previous civilizations by obtaining Egyptian, Babylonian, Indian, Chinese, Farsi, Syriac, Armenian, Hebrew and Roman texts and translating them into Arabic. In 815, the Caliph of Ma'mun established in Baghdad the Bait al-Hikmah ('House of Wisdom'), a library and centre for translation.

Al-Kindi's technique, known as frequency analysis, shows that it is unnecessary to check each of the billions of potential keys. Instead, it is possible to reveal the contents of a scrambled message simply by analysing the frequency of the characters in the ciphertext.

In addition to a greater understanding of secular subjects, the invention of cryptanalysis aslo depended on the growth of religious sholarship. Major theological schools were established in Basra, Kufa and Baghdad, where thelogians scrutinized the revelations of Muhammad as contained in the Koran. The theologians were interested in establishing the chronology of the revelations, which they did by counting the frequencies of words contained in each revelation. The theory was that certain words had evolved relatively recently, and hence if a revelation contained a high number of these newer words, this would indicate that it came later in the chronology. Theologians also studied the Hadith, which consists of the Prophet's daily utterances. They tried to demonstrate that each statement was indeed attributable to Muhammad. This was done by studying the etymology of words and the structure of sentences, to test whether particular texts were consistent with the linguistic patterns of the Prophet.

The French had handed the information from Schmidt to the Poles because they believed it to be of no value, but the Poles had proved them wrong.

Rejewski had no idea of the day key, and he had no idea which message keys were being chosen, but he did know that they resulted in this table of relationships. Had

The best random keys are created by harnessing natural physical processes, such as radioactivity, which is known to exhibit truly random behavior. The

necessity is the mother of invention, then

The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography by Simon Singh.

In an open immeasurable non-linear System of Systems - the Quantum Realm; the Quantum Fields, Quantum Entanglement, Quantum Engineering, Quantum Cryptography, Quantum Intelligence, Quantum Thinking, Quantum Algorithms, Quantum Chaos phenomena are a few of many (to be defined) interconnected elements of this Universal System.

It has been said that the First World War was the chemists’ war, because mustard gas and chlorine were employed for the first time, and that the Second World War was the physicists’ war, because the atom bomb was detonated. Similarly, it has been argued that the Third World War would be the mathematicians’ war, because mathematicians will have control over the next great weapon of war—information.

In fact, Britain had captured thousands of Enigma machines, and distributed them among its former colonies, who believed that the cipher was as secure as it had seemed to the Germans. The British did nothing to disabuse them of this belief, and routinely deciphered their secret communications in the years that followed. Meanwhile,

In April 1977, Rivest, Shamir and Adleman spent Passover at the house of a student, and had consumed significant amounts of Manischewitz wine before returning to their respective homes some time around midnight. Rivest, unable to sleep, lay on his couch reading a mathematics textbook. He began mulling over the question that had been puzzling him for weeks—is it possible to build an asymmetric cipher? Is it possible to find a one-way function that can be reversed only if the receiver has some special information? Suddenly, the mists began to clear and he had a revelation. He spent the rest of that night formalizing his idea, effectively writing a complete scientific paper before daybreak. Rivest had made a breakthrough, but it had grown out of a yearlong collaboration with Shamir and Adleman, and it would not have been possible without them. Rivest finished off the paper by listing the authors alphabetically; Adleman, Rivest, Shamir. The next morning, Rivest handed the paper to Adleman, who went through his usual process of trying to tear it apart, but this time he could find no faults. His only criticism was with the list of authors. “I told Ron to take my name off the paper,” recalls Adleman. “I told him that it was his invention, not mine. But Ron refused and we got into a discussion about it. We agreed that I would go home and contemplate it for one night, and consider what I wanted to do. I went back the next day and suggested to Ron that I be the third author. I recall thinking that this paper would be the least interesting paper that I will ever be on.” Adleman could not have been more wrong. The system, dubbed RSA (Rivest, Shamir, Adleman) as opposed to ARS, went on to become the most influential cipher in modern cryptography.

Codebreakers are linguistic alchemists, a mystical tribe attempting to conjure sensible words out of meaningless symbols.

Although quantum computers can in principle be built, we don’t know how hard it will be or when that might happen, if at all. And so far, it looks really hard. As of early 2017, the record holder is a machine that is able to keep 14 (fourteen!) qubits stable for only a few milliseconds, whereas we’d need to keep millions of qubits stable for weeks in order to break any crypto.

Although it is not known who first realized that the variation in the frequencies of letters could be exploited in order to break ciphers, the earliest known description of the technique is by the ninth-century scientist Abū Yūsūf Ya’qūb ibn Is-hāq ibn as-Sabbāh ibn ‘omrān ibn Ismaīl al-Kindī.

Jack Good, a veteran of Bletchley, commented: “Fortunately the authorities did not know that Turing was a homosexual. Otherwise we might have lost the war.

One of the features of the Enigma machine was its inability to encipher a letter as itself, which was a consequence of the reflector. The letter a could never be enciphered as A, the letter b could never be enciphered as B, and so on.

Turing needed more staff, but his requests had been blocked by Commander Edward Travis, who had taken over as Director of Bletchley, and who felt that he could not justify recruiting more people. On October 21, 1941, the cryptanalysts took the insubordinate step of ignoring Travis and writing directly to Churchill.

war, instead of finishing in 1945, would have ended in 1948 had the Government Code and Cypher School not been able to read the Enigma cyphers and produce the Ultra intelligence.” During this period of delay, additional lives would have been lost in Europe, and Hitler would have been able to make greater use of his V-weapons, inflicting damage throughout southern England. The historian David Kahn summarizes the impact of breaking Enigma: “It saved lives. Not only Allied and Russian lives but, by shortening the war, German, Italian, and Japanese lives as well. Some people alive after World War II might not have been but for these solutions. That is the debt that the world owes to the codebreakers; that is the crowning human value of their triumphs.

The German military were equally unenthusiastic, because they were oblivious to the damage caused by their insecure ciphers during the Great War. For example, they had been led to believe that the Zimmermann telegram had been stolen by American spies in Mexico, and so they blamed that failure on Mexican security. They still did not realize that the telegram had in fact been intercepted and deciphered by the British, and that the Zimmermann debacle was actually a failure of German cryptography.

I told Ron to take my name off the paper,” recalls Adleman. “I told him that it was his invention, not mine. But Ron refused and we got into a discussion about it.

I recall thinking that this paper would be the least interesting paper that I will ever be on.” Adleman could not have been more wrong. The system, dubbed RSA (Rivest, Shamir, Adleman) as opposed to ARS, went on to become the most influential cipher in modern cryptography.

diagonally polarized photons are in a quantum quandary when confronted by a vertical Polaroid filter.

half of them at random will be blocked, and half will pass through, and those that do pass through will be reoriented with a vertical polarization.

Because a quantum computer deals with 1’s and 0’s that are in a quantum superposition, they are called quantum bits, or qubits (pronounced “cubits”). The advantage of qubits becomes even clearer when we consider more particles.

250 qubits, it is possible to represent roughly 1075 combinations, which is greater than the number of atoms in the universe. If it were possible to achieve the appropriate superposition with 250 particles, then a quantum computer could perform 1075 simultaneous computations,

the development of a fully operational quantum computer would imperil our personal privacy, destroy electronic commerce and demolish the concept of national security. A quantum computer would jeopardize the stability of the world.