Myanmar is proving to be a major test of strategic economic cooperation between the United States and Japan – one which reveals wider foreign policy differences between the allied nations. Those differences make substantial collaboration towards realizing a “free and open Indo-Pacific” (FOIP) in specific countries difficult, despite joint strategic interests and aligned high-level political visions.
In Japan’s pursuit of a policy response to China’s infrastructure Belt and Road Initiative (BRI), economic cooperation with partners in third countries to beef up the financing on offer has emerged as a key component. Strategic commercial cooperation is arguably most opportune and necessary in Myanmar because the country is at a fraught stage in its democratization, set into motion in 2010 when it pivoted to form closer ties with countries like the United States and Japan in order to reduce economic over-dependence on China.
However, despite similar headline commitments under FOIP by the United States and Japan to promote rule of law and needs-based economic development, the extent of cooperation in Myanmar has been limited to Japanese trading houses subcontracting to General Electric and U.S. aluminum can manufacturer Ball investing in the Japan-led Thilawa Special Economic Zone (SEZ). Why has the type of economic cooperation much touted by their FOIP strategies come to so little fruition in Myanmar? Is there a way forward?
Cooperation in Myanmar is revealing substantial differences in foreign policy between the United States and Japan
A major issue preventing cooperation is the dire state of U.S. economic relations with Myanmar. Japanese officials and business executives view, perhaps not wrongly, the costs of cooperating with U.S. government bodies and corporations to outweigh potential benefits. A prerequisite for substantial joint economic cooperation is the United States lifting sanctions, to allow for its own economic ties with Myanmar to flourish.
The United States has not put garments and textiles – Myanmar’s biggest potential export product to the United States – on its Generalized System of Preferences, a scheme that eliminates tariffs for goods imported from developing countries. The specter of targeted sanctions against perpetrators of the Rohingya crisis, despite not being fully enforced, poses a reputational risk to U.S. investors.
Furthermore, the United States has not lifted Section 312 of the USA Patriot Act, placing burdensome due diligence requirements on the movement of trade or investment finance in or out of Myanmar. Other than Myanmar, only North Korea, Iran, and Cuba also have this restriction placed upon them, resulting in the absence of American financial institutions there. With the United States unable to freely trade with and invest in Myanmar, economic cooperation with Japan is a non-starter.
The United States places primary importance to human rights – Japan gives primacy to rule of law and development
This is not just a practical matter but points to a wider foreign policy divergence. The United States places primary importance on the human rights agenda as a prerequisite to economic ties. Japan, however, gives primacy to the rule of law and economic development, viewing them as the basis from which to solve human rights problems. Disengagement means no leverage. Japanese ambassador to Myanmar Ichiro Maruyama recently reiterated strong disagreement with international pressure and sanctions on Myanmar – previously condemning it in strong language (for a diplomat) as “utter nonsense”. A Japanese business leader in Myanmar expressed to me his lack of faith in U.S. companies’ commitment to the country.
Japan, by contrast, has a robust presence in Myanmar, with a 3.5 percent share of investment into the country and strong trading ties. It led the establishment of the enormously successful Thilawa SEZ near Yangon, which acts as a “laboratory” to test regulatory reforms to then be rolled out across the rest of the country. Toyota recently announced that it will assemble pick-up trucks there, likely bringing auxiliary industries with it. Japan’s development bank has strategically taken a stake in the Dawei SEZ, the ocean outlet which lies west of the Thai capital Bangkok. Japanese investments, particularly in transportation infrastructure, are beginning to spread north from Yangon to Mandalay, where Chinese influence is stronger.
Japan’s influence goes beyond commerce. Along with 57 staff deployed in Myanmar’s ministries, Japan has established the Myanmar-Japan Joint Initiative (MJJI), an exclusively bilateral dialogue that aims to provide technical assistance to promote a favorable business environment. Japan has established these dialogues across Southeast Asia, focusing on detailed regulations and rule-making for areas such as visas, imports and exports, tax, industrial policy and insurance – a purportedly tedious task that distinguishes Japan from China in its economic offering.
It makes sense for the United States to reinforce and complement Japan’s economic and diplomatic efforts
The United States and Japan share deep strategic interests in promoting responsible economic development in Myanmar, a gateway to the Indian Ocean sandwiched between China and India, to ensure that it does not become economically over-dependent on China. Therefore, it makes much sense for the United States to reinforce and complement Japan’s economic and diplomatic efforts, following its lead.
Steps towards improving economic cooperation – and ultimately building Myanmar’s economic resilience – are not untenable. First, each country has actions that they can individually take to make collaboration more feasible. The United States should place Myanmar as centrally important to its Asia strategy, and promote deeper business ties, as encouraged by the top Asia official in the National Security Council. This should be achieved via the removal of restrictions where politically possible and by the American Chamber of Commerce doubling down to systematically explain troubles faced by its businesses to relevant Myanmar government agencies. Despite strong resistance, Japan should be more open to exploratory dialogue and should improve information sharing with other embassies and chambers of commerce.
A second lesson is that to tackle BRI regionally, the United States and Japan have to find creative ways to collaborate on the country level in a manner that benefits development, not merely blocking or out-doing China. In Myanmar, there are four good places to begin. First, search for opportunities to collaborate on Myanmar’s agriculture industry, including construction of transportation infrastructure to reach markets. Second, look for synergies to build capacity in banking, infrastructure finance, procurement and so on. Third, encourage U.S, investment into Thilawa and Dawei. Fourth, jointly assist Naypyidaw, its capital, with building policies and regulation that encourage investment.
A major investment by China, perhaps in Kyaupkyu, the terminus for an oil and gas pipeline to Kunming and potential site of a deep-sea port and SEZ, may jolt the United States and Japan into joint action. But after all, the U.S.-Japan alliance treaty mandates them to “encourage economic collaboration” in Article II – before it ever mentions military cooperation or U.S. bases – and China has shown through BRI that a strategic economic vision is best built from the ground up, project by project. Sustained, long-term efforts, rather than reactionary pushbacks, to strengthen economic cooperation from the bottom up would better benefit Myanmar’s development, as well as U.S. and Japanese businesses.
Over the last few decades, the word ‘quantum’ has become increasingly popular. It is common to find articles, reports, and many people interested in quantum mechanics and the new capabilities and improvements it brings to the scientific community. This topic not only concerns physics, since the development of quantum mechanics impacts on several other fields such as chemistry, economics, artificial intelligence, operations research, and undoubtedly, cryptography.
This post begins a trio of blogs describing the impact of quantum computing on cryptography, and how to use stronger algorithms resistant to the power of quantum computing.
This post introduces quantum computing and describes the main aspects of this new computing model and its devastating impact on security standards; it summarizes some approaches to securing information using quantum-resistant algorithms.
All of this is part of Cloudflare’s Crypto Week 2019, now fasten your seatbelt and get ready to make a quantum leap.
What is Quantum Computing?
Back in 1981, Richard Feynman raised the question about what kind of computers can be used to simulate physics. However, some physical phenomena, such as quantum mechanics, cannot be simulated using a classical computer. Then, he conjectured the existence of a computer model that behaves under quantum mechanics rules, which opened a field of research now called quantum computing. To understand the basics of quantum computing, it is necessary to recall how classical computers work, and from that shine a spotlight on the differences between these computational models.
Fellows of the Royal Society: John Maynard Smith, Richard Feynman & Alan Turing
In 1936, Alan Turing and Emil Post independently described models that gave rise to the foundation of the computing model known as the Post-Turing machine, which describes how computers work and allowed further determination of limits for solving problems.
In this model, the units of information are bits, which store one of two possible values, usually denoted by 0 and 1. A computing machine contains a set of bits and performs operations that modify the values of the bits, also known as the machine’s state. Thus, a machine with N bits can be in one of 2ᴺ possible states. With this in mind, the Post-Turing computing model can be abstractly described as a machine of states, in which running a program is translated as machine transitions along the set of states.
A paper David Deutsch published in 1985 describes a computing model that extends the capabilities of a Turing machine based on the theory of quantum mechanics. This computing model introduces several advantages over the Turing model for processing large volumes of information. It also presents unique properties that deviate from the way we understand classical computing. Most of these properties come from the nature of quantum mechanics. We’re going to dive into these details before approaching the concept of quantum computing.
Superposition
One of the most exciting properties of quantum computing that provides an advantage over the classical computing model is superposition. In physics, superposition is the ability to produce valid states from the addition or superposition of several other states that are part of a system.
Applying these concepts to computing information, it means that there is a system in which it is possible to generate a machine state that represents a (weighted) sum of the states 0 and 1; in this case, the term weighted means that the state can keep track of “the quantity of” 0 and 1 present in the state. In the classical computation model, one bit can only store either the state of 0 or 1, not both; even using two bits, they cannot represent the weighted sum of these states. Hence, to make a distinction from the basic states, quantum computing uses the concept of a quantum bit (qubit) — a unit of information to denote the superposition of two states. This is a cornerstone concept of quantum computing as it provides a way of tracking more than a single state per unit of information, making it a powerful tool for processing information.
Classical computing – A bit stores only one of two possible states: ON or OFF.Quantum computing – A qubit stores a combination of two or more states.
So, a qubit represents the sum of two parts: the 0 or 1 state plus the amount each 0/1 state contributes to produce the state of the qubit.
In mathematical notation, qubit \( | \Psi \rangle \) is an explicit sum indicating that a qubit represents the superposition of the states 0 and 1. This is the Dirac notation used to describe the value of a qubit \( | \Psi \rangle = A | 0 \rangle +B | 1 \rangle \), where, A and B are complex numbers known as the amplitude of the states 0 and 1, respectively. The value of the basic states is represented by qubits as \( | 0 \rangle = 1 | 0 \rangle + 0 | 1 \rangle \) and \( | 1 \rangle = 0 | 0 \rangle + 1 | 1 \rangle \), respectively. The right side of the term contains the abbreviated notation for these special states.
Measurement
In a classical computer, the values 0 and 1 are implemented as digital signals. Measuring the current of the signal automatically reveals the status of a bit. This means that at any moment the value of the bit can be observed or measured.
The state of a qubit is maintained in a physically closed system, meaning that the properties of the system, such as superposition, require no interaction with the environment; otherwise any interaction, like performing a measurement, can cause interference on the state of a qubit.
Measuring a qubit is a probabilistic experiment. The result is a bit of information that depends on the state of the qubit. The bit, obtained by measuring \( | \Psi \rangle = A | 0 \rangle +B | 1 \rangle \), will be equal to 0 with probability \( |A|^2 \), and equal to 1 with probability \( |B|^2 \), where \( |x| \) represents the absolute value of \(x\).
From Statistics, we know that the sum of probabilities of all possible events is always equal to 1, so it must hold that \( |A|^2 +|B|^2 =1 \). This last equation motivates to represent qubits as the points of a circle of radius one, and more generally, as the points on the surface of a sphere of radius one, which is known as the Bloch Sphere.
The qubit state is analogous to a point on a unitary circle.The Bloch Sphere by Smite-Meister – Own work, CC BY-SA 3.0.
Let’s break it down: If you measure a qubit you also destroy the superposition of the qubit, resulting in a superposition state collapse, where it assumes one of the basics states, providing your final result.
Another way to think about superposition and measurement is through the coin tossing experiment.
Toss a coin in the air and you give people a random choice between two options: heads or tails. Now, don’t focus on the randomness of the experiment, instead note that while the coin is rotating in the air, participants are uncertain which side will face up when the coin lands. Conversely, once the coin stops with a random side facing up, participants are 100% certain of the status.
How does it relate? Qubits are similar to the participants. When a qubit is in a superposition of states, it is tracking the probability of heads or tails, which is the participants’ uncertainty quotient while the coin is in the air. However, once you start to measure the qubit to retrieve its value, the superposition vanishes, and a classical bit value sticks: heads or tails. Measurement is that moment when the coin is static with only one side facing up.
A fair coin is a coin that is not biased. Each side (assume 0=heads and 1=tails) of a fair coin has the same probability of sticking after a measurement is performed. The qubit \( \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle \) describes the probabilities of tossing a fair coin. Note that squaring either of the amplitudes results in ½, indicating that there is a 50% chance either heads or tails sticks.
It would be interesting to be able to charge a fair coin at will while it is in the air. Although this is the magic of a professional illusionist, this task, in fact, can be achieved by performing operations over qubits. So, get ready to become the next quantum magician!
Quantum Gates
A logic gate represents a Boolean function operating over a set of inputs (on the left) and producing an output (on the right). A logic circuit is a set of connected logic gates, a convenient way to represent bit operations.
The NOT gate is a single-bit operation that flips the value of the input bit.
Other gates are AND, OR, XOR, and NAND, and more. A set of gates is universal if it can generate other gates. For example, NOR and NAND gates are universal since any circuit can be constructed using only these gates.
Quantum computing also admits a description using circuits. Quantum gates operate over qubits, modifying the superposition of the states. For example, there is a quantum gate analogous to the NOT gate, the X gate.
The X quantum gate interchanges the amplitudes of the states of the input qubit.
The Z quantum gate flips the sign’s amplitude of state 1:
Another quantum gate is the Hadamard gate, which generates an equiprobable superposition of the basic states.
Using our coin tossing analogy, the Hadamard gate has the action of tossing a fair coin to the air. In quantum circuits, a triangle represents measuring a qubit, and the resulting bit is indicated by a double-wire.
Other gates, such as the CNOT gate, Pauli’s gates, Toffoli gate, Deutsch gate, are slightly more advanced. Quirk, the open-source playground, is a fun sandbox where you can construct quantum circuits using all of these gates.
Reversibility
An operation is reversible if there exists another operation that rolls back the output state to the initial state. For instance, a NOT gate is reversible since applying a second NOT gate recovers the initial input.
In contrast, AND, OR, NAND gates are not reversible. This means that some classical computations cannot be reversed by a classic circuit that uses only the output bits. However, if you insert additional bits of information, the operation can be reversed.
Quantum computing mainly focuses on reversible computations, because there’s always a way to construct a reversible circuit to perform an irreversible computation. The reversible version of a circuit could require the use of ancillary qubits as auxiliary (but not temporary) variables.
Due to the nature of composed systems, it could be possible that these ancillas (extra qubits) correlate to qubits of the main computation. This correlation makes it infeasible to reuse ancillas since any modification could have the side-effect on the operation of a reversible circuit. This is like memory assigned to a process by the operating system: the process cannot use memory from other processes or it could cause memory corruption, and processes cannot release their assigned memory to other processes. You could use garbage collection mechanisms for ancillas, but performing reversible computations increases your qubit budget.
Composed Systems
In quantum mechanics, a single qubit can be described as a single closed system: a system that has no interaction with the environment nor other qubits. Letting qubits interact with others leads to a composed system where more states are represented. The state of a 2-qubit composite system is denoted as \(A_0|00\rangle+A_1|01\rangle+A_2|10\rangle+A_3|11\rangle \), where, \( A_i \) values correspond to the amplitudes of the four basic states 00, 01, 10, and 11. This qubit \( \tfrac{1}{2}|00\rangle+\tfrac{1}{2}|01\rangle+\tfrac{1}{2}|10\rangle+\tfrac{1}{2}|11\rangle \) represents the superposition of these basic states, both having the same probability obtained after measuring the two qubits.
In the classical case, the state of N bits represents only one of 2ᴺ possible states, whereas a composed state of N qubits represents all the 2ᴺ states but in superposition. This is one big difference between these computing models as it carries two important properties: entanglement and quantum parallelism.
Entanglement
According to the theory behind quantum mechanics, some composed states can be described through the description of its constituents. However, there are composed states where no description is possible, known as entangled states.
The entanglement phenomenon was pointed out by Einstein, Podolsky, and Rosen in the so-called EPR paradox. Suppose there is a composed system of two entangled qubits, in which by performing a measurement in one qubit causes interference in the measurement of the second. This interference occurs even when qubits are separated by a long distance, which means that some information transfer happens faster than the speed of light. This is how quantum entanglement conflicts with the theory of relativity, where information cannot travel faster than the speed of light. The EPR paradox motivated further investigation for deriving new interpretations about quantum mechanics and aiming to resolve the paradox.
Quantum entanglement can help to transfer information at a distance by following a communication protocol. The following protocol examples rely on the fact that Alice and Bob separately possess one of two entangled qubits:
The superdense coding protocol allows Alice to communicate a 2-bit message \(m_0,m_1\) to Bob using a quantum communication channel, for example, using fiber optics to transmit photons. All Alice has to do is operate on her qubit according to the value of the message and send the resulting qubit to Bob. Once Bob receives the qubit, he measures both qubits, noting that the collapsed 2-bit state corresponds to Alice’s message.
Superdense coding protocol.
The quantum teleportation protocol allows Alice to transmit a qubit to Bob without using a quantum communication channel. Alice measures the qubit to send Bob and her entangled qubit resulting in two bits. Alice sends these bits to Bob, who operates on his entangled qubit according to the bits received and notes that the result state matches the original state of Alice’s qubit.
Quantum teleportation protocol.
Quantum Parallelism
Composed systems of qubits allow representation of more information per composed state. Note that operating on a composed state of N qubits is equivalent to operating over a set of 2ᴺ states in superposition. This procedure is quantum parallelism. In this setting, operating over a large volume of information gives the intuition of performing operations in parallel, like in the parallel computing paradigm; one big caveat is that superposition is not equivalent to parallelism.
Remember that a composed state is a superposition of several states so, a computation that takes a composed state of inputs will result in a composed state of outputs. The main divergence between classical and quantum parallelism is that quantum parallelism can obtain only one of the processed outputs. Observe that a measurement in the output of a composed state causes that the qubits collapse to only one of the outputs, making it unattainable to calculate all computed values.
Although quantum parallelism does not match precisely with the traditional notion of parallel computing, you can still leverage this computational power to get related information.
Deutsch-Jozsa Problem: Assume \(F\) is a function that takes as input N bits, outputs one bit, and is either constant (always outputs the same value for all inputs) or balanced (outputs 0 for half of the inputs and 1 for the other half). The problem is to determine if \(F\) is constant or balanced.
The quantum algorithm that solves the Deutsch-Jozsa problem uses quantum parallelism. First, N qubits are initialized in a superposition of 2ᴺ states. Then, in a single shot, it evaluates \(F\) for all of these states.
(note that some factors were omitted for simplicity)
The result of applying \(F\) appears (in the exponent) of the amplitude of the all-zero state. Note that only when \(F\) is constant is this amplitude, either +1 or -1. If the result of measuring the N qubit is an all-zeros bitstring, then there is a 100% certainty that \(F\) is constant. Any other result indicates that \(F\) is balanced.
A deterministic classical algorithm solves this problem using \( 2^{N-1}+1\) evaluations of \(F\) in the worst case. Meanwhile, the quantum algorithm requires only one evaluation. The Deutsch-Jozsa problem exemplifies the exponential advantage of a quantum algorithm over classical algorithms.
Quantum Computers
The theory of quantum computing is supported by investigations in the field of quantum mechanics. However, constructing a quantum machine requires a physical system that allows representing qubits and manipulation of states in a reliable and precise way.
The DiVincenzo Criteria require that a physical implementation of a quantum computer must:
Be scalable and have well-defined qubits.
Be able to initialize qubits to a state.
Have long decoherence times to apply quantum error-correcting codes. Decoherence of a qubit happens when the qubit interacts with the environment, for example, when a measurement is performed.
Use a universal set of quantum gates.
Be able to measure single qubits without modifying others.
Quantum computer physical implementations face huge engineering obstacles to satisfy these requirements. The most important challenge is to guarantee low error rates during computation and measurement. Lowering these rates require techniques for error correction, which add a significant number of qubits specialized on this task. For this reason, the number of qubits of a quantum computer should not be regarded as for classical systems. In a classical computer, the bits of a computer are all effective for performing a calculation, whereas the number of qubits is the sum of the effective qubits (those used to make calculations) plus the ancillas (used for reversible computations) plus the error correction qubits.
Current implementations of quantum computers partially satisfy the DiVincenzo criteria. Quantum adiabatic computers fit in this category since they do not operate using quantum gates. For this reason, they are not considered to be universal quantum computers.
Quantum Adiabatic Computers
A recurrent problem in optimization is to find the global minimum of an objective function. For example, a route-traffic control system can be modeled as a function that reduces the cost of routing to a minimum. Simulated annealing is a heuristic procedure that provides a good solution to these types of problems. Simulated annealing finds the solution state by slowly introducing changes (the adiabatic process) on the variables that govern the system.
Quantum annealing is the analogous quantum version of simulated annealing. A qubit is initialized into a superposition of states representing all possible solutions to the problem. Here is used the Hamiltonian operator, which is the sum of vectors of potential and kinetic energies of the system. Hence, the objective function is encoded using this operator describing the evolution of the system in correspondence with time. Then, if the system is allowed to evolve very slowly, it will eventually land on a final state representing the optimal value of the objective function.
Currently, there exist adiabatic computers in the market, such as the D-Wave and IBM Q systems, featuring hundreds of qubits; however, their capabilities are somewhat limited to some problems that can be modeled as optimization problems. The limits of adiabatic computers were studied by van Dam et al, showing that despite solving local searching problems and even some instances of the max-SAT problem, there exists harder searching problems this computing model cannot efficiently solve.
Nuclear Magnetic Resonance
Nuclear Magnetic Resonance (NMR) is a physical phenomena that can be used to represent qubits. The spin of atomic nucleus of molecules is perturbed by an oscillating magnetic field. A 2001 report describes successful implementation of Shor’s algorithm in a 7-qubit NMR quantum computer. An iconic result since this computer was able to factor the number 15.
Nucleus spinning induced by a magnetic field, Darekk2 – CC BY-SA 3.0NRM Spectrometer by UCSB
Superconducting Quantum Computers
One way to physically construct qubits is based on superconductors, materials that conduct electric current with zero resistance when exposed to temperatures close to absolute zero.
The Josephson effect, in which current flows across the junction of two superconductors separated by a non-superconducting material, is used to physically implement a superposition of states.
When a magnetic flux is applied to this junction, the current flows continuously in one direction. But, depending on the quantity of magnetic flux applied, the current can also flow in the opposite direction. There exists a quantum superposition of currents going both clockwise and counterclockwise leading to a physical implementation of a qubit called flux qubit. The complete device is known as the Superconducting Quantum Interference Device (SQUID) and can be easily coupled scaling the number of qubits. Thus, SQUIDs are like the transistors of a quantum computer.
SQUID: Superconducting Quantum Interference Device. Image by Kurzweil Network and original source.
The quantum zoo website tracks problems that can be solved using quantum algorithms. As of mid-2018, more than 60 problems appear on this list, targeting diverse applications in the area of number theory, approximation, simulation, and searching. As terrific as it sounds, some easily-solvable problems by quantum computing are surrounding the security of information.
Grover’s Algorithm
Tales of a quantum detective (fragment). A couple of detectives have the mission of finding one culprit in a group of suspects that always respond to this question honestly: “are you guilty?”.
The detective C follows a classic interrogative method and interviews every person one at a time, until finding the first one that confesses.
The detective Q proceeds in a different way, First gather all suspects in a completely dark room, and after that, the detective Q asks them — are you guilty? — A steady sound comes from the room saying “No!” while at the same time, a single voice mixed in the air responds “Yes!.” Since everybody is submerged in darkness, the detective cannot see the culprit. However, detective Q knows that, as long as the interrogation advances, the culprit will feel desperate and start to speak louder and louder, and so, he continues asking the same question. Suddenly, detective Q turns on the lights, enters into the room, and captures the culprit. How did he do it?
The task of the detective can be modeled as a searching problem. Given a Boolean function \( f\) that takes N bits and produces one bit, to find the unique input \(x\) such that \( f(x)=1\).
A classical algorithm (detective C) finds \(x\) using \(2^N-1\) function evaluations in the worst case. However, the quantum algorithm devised by Grover, corresponding to detective Q, searches quadratically faster using around \(2^{N/2}\) function evaluations.
The key intuition of Grover’s algorithm is increasing the amplitude of the state that represents the solution while maintaining the other states in a lower amplitude. In this way, a system of N qubits, which is a superposition of 2ᴺ possible inputs, can be continuously updated using this intuition until the solution state has an amplitude closer to 1. Hence, after updating the qubits many times, there will be a high probability to measure the solution state.
Initially, a superposition of 2ᴺ states (horizontal axis) is set, each state has an amplitude (vertical axis) close to 0. The qubits are updated so that the amplitude of the solution state increases more than the amplitude of other states. By repeating the update step, the amplitude of the solution state gets closer to 1, which boosts the probability of collapsing to the solution state after measuring.
Image taken from D. Bernstein’s .
Grover’s Algorithm (pseudo-code):
Prepare an N qubit \(|x\rangle \) as a uniform superposition of 2ᴺ states.
Update the qubits by performing this core operation. $$ |x\rangle \mapsto (-1)^{f(x)} |x\rangle $$ The result of \( f(x) \) only flips the amplitude of the searched state.
Negate the N qubit over the average of the amplitudes.
Repeat Step 2 and 3 for \( (\tfrac{\pi}{4}) 2^{ N/2} \) times.
Measure the qubit and return the bits obtained.
Alternatively, the second step can be better understood as a conditional statement:
IF f(x) = 1 THEN
Negate the amplitude of the solution state.
ELSE
/* nothing */
ENDIF
Grover’s algorithm considers function \(f\) a black box, so with slight modifications, the algorithm can also be used to find collisions on the function. This implies that Grover’s algorithm can find a collision using an asymptotically less number of operations than using a brute-force algorithm.
The power of Grover’s algorithm can be turned against cryptographic hash functions. For instance, a quantum computer running Grover’s algorithm could find a collision on SHA256 performing only 2¹²⁸ evaluations of a reversible circuit of SHA256. The natural protection for hash functions is to increase the output size to double. More generally, most of symmetric key encryption algorithms will survive to the power of Grover’s algorithm by doubling the size of keys.
The scenario for public-key algorithms is devastating in face of Peter Shor’s algorithm.
Shor’s Algorithm
Multiplying integers is an easy task to accomplish, however, finding the factors that compose an integer is difficult. The integer factorization problem is to decompose a given integer number into its prime factors. For example, 42 has three factors 2, 3, and 7 since \( 2\times 3\times 7 = 42\). As the numbers get bigger, integer factorization becomes more difficult to solve, and the hardest instances of integer factorization are when the factors are only two different large primes. Thus, given an integer number \(N\), to find primes \(p\) and \(q\) such that \( N = p \times q\), is known as integer splitting.
Factoring integers is like cutting wood, and the specific task of splitting integers is analogous to using an axe for splitting the log in two parts. There exist many different tools (algorithms) for accomplishing each task.
For integer factorization, trial division, the Rho method, the elliptic curve method are common algorithms. Fermat’s method, the quadratic- and rational-sieve, leads to the (general) number field sieve (NFS) algorithm for integer splitting. The latter relies on finding a congruence of squares, that is, splitting \(N\) as a product of squares such that $$ N = x^2 – y^2 = (x+y)\times(x-y) $$ The complexity of NFS is mainly attached to the number of pairs \((x, y)\) that must be examined before getting a pair that factors \(N\). The NFS algorithm has subexponential complexity on the size of \(N\), meaning that the time required for splitting an integer increases significantly as the size of \(N\) grows. For large integers, the problem becomes intractable for classical computers.
The many different guesses of the NFS algorithm are analogous to hitting the log using a dulled axe; after subexponential many tries, the log is cut by half. However, using a sharper axe allows you to split the log faster. This sharpened axe is the quantum algorithm proposed by Shor in 1994.
Let \(x\) be an integer less than \(N\) and of the order \(k\). Then, if \(k\) is even, there exists an integer \(q\) so \(qN\) can be factored as follows.
This approach has some issues. For example, the factorization could correspond to \(q\) not \(N\) and the order of \(x\) is unknown, and here is where Shor’s algorithm enters the picture, finding the order of \(x\).
The internals of Shor’s algorithm rely on encoding the order \(k\) into a periodic function, so that its period can be obtained using the quantum version of the Fourier transform (QFT). The order of \(x\) can be found using a polynomial number quantum evaluations of Shor’s algorithm. Therefore, splitting integers using this quantum approach has polynomial complexity on the size of \(N\).
Shor’s algorithm carries strong implications on the security of the RSA encryption scheme because its security relies on integer factorization. A large-enough quantum computer can efficiently break RSA for current instances.
Alternatively, one may recur to elliptic curves, used in cryptographic protocols like ECDSA or ECDH. Moreover, all TLS ciphersuites use a combination of elliptic curve groups, large prime groups, and RSA and DSA signatures. Unfortunately, these algorithms all succumb to Shor’s algorithm. It only takes a few modifications for Shor’s algorithm to solve the discrete logarithm problem on finite groups. This sounds like a catastrophic story where all of our encrypted data and privacy are no longer secure with the advent of a quantum computer, and in some sense this is true.
On one hand, it is a fact that the quantum computers constructed as of 2019 are not large enough to run, for instance, Shor’s algorithm for the RSA key sizes used in standard protocols. For example, a 2018 report shows experiments on the factorization of a 19-bit number using 94 qubits, they also estimate that 147456 qubits are needed for factoring a 768-bit number. Hence, there numbers indicates that we are still far from breaking RSA.
What if we increment RSA key sizes to be resistant to quantum algorithms, just like for symmetric algorithms?
Bernstein et al. estimated that RSA public keys should be as large as 1 terabyte to maintain secure RSA even in the presence of quantum factoring algorithms. So, for public-key algorithms, increasing the size of keys does not help.
A recent investigation by Gidney and Ekerá shows improvements that accelerate the evaluation of quantum factorization. In their report, the cost of factoring 2048-bit integers is estimated to take a few hours using a quantum machine of 20 million qubits, which is far from any current development. Something worth noting is that the number of qubits needed is two orders of magnitude smaller than the estimated numbers given in previous works developed in this decade. Under these estimates, current encryption algorithms will remain secure several more years; however, consider the following not-so-unrealistic situation.
Information currently encrypted with for example, RSA, can be easily decrypted with a quantum computer in the future. Now, suppose that someone records encrypted information and stores them until a quantum computer is able to decrypt ciphertexts. Although this could be as far as 20 years from now, the forward-secrecy principle is violated. A 20-year gap to the future is sometimes difficult to imagine. So, let’s think backwards, what would happen if all you did on the Internet at the end of the 1990s can be revealed 20 years later — today. How does this impact the security of your personal information? What if the ciphertexts were company secrets or business deals? In 1999, most of us were concerned about the effects of the Y2K problem, now we’re facing Y2Q (years to quantum): the advent of quantum computers.
Post-Quantum Cryptography
Although the current capacity of the physical implementation of quantum computers is far from a real threat to secure communications, a transition to use stronger problems to protect information has already started. This wave emerged as post-quantum cryptography (PQC). The core idea of PQC is finding algorithms difficult enough that no quantum (and classical) algorithm can solve them.
A recurrent question is: How does it look like a problem that even a quantum computer can not solve?
These so-called quantum-resistant algorithms rely on different hard mathematical assumptions; some of them as old as RSA, others more recently proposed. For example, McEliece cryptosystem, formulated in the late 70s, relies on the hardness of decoding a linear code (in the sense of coding theory). The practical use of this cryptosystem didn’t become widespread, since with the passing of time, other cryptosystems superseded in efficiency. Fortunately, McEliece cryptosystem remains immune to Shor’s algorithm, gaining it relevance in the post-quantum era.
Post-quantum cryptography presents alternatives:
As of 2017, the NIST started an evaluation process that tracks possible alternatives for next-generation secure algorithms. From a practical perspective, all candidates present different trade-offs in implementation and usage. The time and space requirements are diverse; at this moment, it’s too early to define which will succeed RSA and elliptic curves. An initial round collected 70 algorithms for deploying key encapsulation mechanisms and digital signatures. As of early 2019, 28 of these survive and are currently in the analysis, investigation, and experimentation phase.
Cloudflare’s mission is to help build a better Internet. As a proactive action, our cryptography team is preparing experiments on the deployment of post-quantum algorithms at Cloudflare scale. Watch our blog post for more details.
After a busy year, Facebook’s VR arm is returning to San Jose, Calif. on September 25 and 26 for the sixth annual Oculus Connect.
Oculus has had a transformative year with the release of its Quest and Rift S headsets, turning the high-end gaming company into one more focused on meeting the needs of mainstream consumers. Oculus Connect 6 will give the company an opportunity to hit a stride on content and software optimizations, without the specter of missing hardware features hanging heavy.
“With Quest and Rift S bringing more people into VR than ever before, OC6 is the perfect moment to think bigger, build smarter, and realize the true potential of what we’re creating together,” the company wrote in a short blog post.
For developers, this could be a more contentious meeting as Facebook’s top virtual reality hardware product remains a walled garden with only certain content permitted in the store. Apple has shifting its efforts over the past two years to nabbing top game developers and offering less monetary support to indies that are experimenting in VR for the first time.
In the teaser post, the company is already highlighting that one of the main announcements will be a first-person combat title created by Respawn Entertainment, the maker of Apex Legends.
The summer solstice is one such day of the year when the night is the shortest throughout. This day receives most sunlight throughout the year. This is officially the first day of summer, and this is when the Sun would be at its pinnacle throughout the daytime. Even though the 21st century has seen us shift away from the vast beauties of nature, it wouldn’t go amiss to say that summer and winter solstices and autumn and spring equinoxes are four such phenomena that receive a lot of attention and adulation throughout the ages. It also wouldn’t be wrong to mention that the body would receive much energy, positive in nature, that would help it. Since history repeats itself, it is inevitable that the display and worship of these natural events that took place centuries ago would be present again, in this present century. This is a way to connect to nature and understand the dichotomy or the binary between man and nature. But, there are four such rituals that do not take time, to fully enjoy the summer solstice: 1. Witness the Sunrise Wake up before dawn. Watch the sky as it gets illuminated by the dynamic vestiges of the morning sun. The amount of positive energy that would reach you is tumultuous as these rays are pure and powerful. The body is a solar conduit, and these rays invigorate it. 2. Activation of the Solar Body Due to the similarity between the force which keeps us alive and the force which keeps this Earth moving, and in orbit, watching the Sunrise and feeling the Sun rays hit every node of your body, can be refreshing. As these Sun rays would always balance the equilibrium between the Earth and the body. The body is solar, and so is the Earth. 3. Introspection This is the time to introspect unto yourself where you stand in life as a human. The journey you undertake can be understood at this stage because it allows you to reflect upon your actions and the plans that you would go through later in your life. 4. Go Outside Needlessly informative, go outside. Have a picnic. Camp in the wood. Feel the complete Sun rays hit you and bring you from the deep, dark corner to a place which is bright, and optimistic. FOLLOW US ON INSTAGRAM HERE
Boeing is moving its Space and Launch headquarters to Titusville, Florida ahead of the completion and first launch of its Starliner spacecraft, planned for later this year.
Boeing to Move Space and Launch Headquarters to Florida
Boeing announced today that it was moving the headquarters for is Space and Launch division from Arlington, Virginia to Titusville, Florida, near Kennedy Space Center on the so-called “Space Coast”.
“Looking to the future, this storied Florida space community will be the center of gravity for Boeing’s space programs as we continue to build our company’s leadership beyond gravity,” said Leanne Caret, president and CEO of Boeing Defense, Space & Security. “The time is right for us to locate our space headquarters where so much of our space history was made over the past six decades and where so much history awaits.”
Boeing says that the move will not affect facilities in other states like Texas and California, where Boeing has major operations. The move is necessary though for several reasons, the company said. Boeing has several space projects coming to fruition in the near future, such as the crewed CST-100 Starliner, and its other launch vehicles like the uncrewed X-37B are seeing an increased workload, necessitating the move to the country’s decades-old hub for space related activity.
“Boeing has been a dominant presence on the Space Coast for six decades, and this move represents a continuation of that legacy and future commitment,” said Jim Chilton, Boeing senior vice president of Space and Launch. “Expanding our Boeing presence on the Space Coast brings tremendous value for our commercial and government space programs through focused leadership, strategic investment, customer proximity and additional contributions to the vitality of the region.”
So, you’ve heard about this thing called artificial intelligence. It’s changing the world, you’ve been told. It’s going to drive your car, grow your food, maybe even take your job. You’ll be forgiven for having some questions about this chaotic, AI-driven world that’s predicted to unfold.
Gregory Barber covers cryptocurrency, blockchain, and artificial intelligence for WIRED.
First off, it’s true that AI is overhyped. But it’s improving rapidly, and in some ways catching up to the hype. Part of that is a natural evolution: AI improves at a given task when it learns from new data, and the world is producing more data every second. New techniques developed in academic labs and at tech companies lead to jumps in performance, too. That’s led to cars that can drive themselves in some situations, to medical diagnoses that have beaten the accuracy of human doctors, and to facial recognition that’s reliable enough to unlock your iPhone.
AI, in other words, is getting really good at some specific tasks. “The nice thing about AI is that it gets better with every iteration,” AI researcher and Udacity founder Sebastian Thrun says. He believes it might just “free humanity from the burden of repetitive work.” But on the lofty goal of so-called “general” AI intelligence that deftly switches between tasks just like a human? Please don’t hold your breath. Preserve those brain cells; you’ll need them to out-think the machines.
In the meantime, AI’s biggest impact may come from democratizing the capabilities that we have now. Tech companies have made powerful software tools and data sets open source, meaning they’re just a download away for tinkerers, and the computing power used to train AI algorithms is getting cheaper and easier to access. That puts AI in the hands of a (yes, precocious) teenager who can develop a system to detect pancreatic cancer, and allows a group of hobbyists in Berkeley to race (and crash) their DIY autonomous cars. “We now have the ability to do things that were PhD theses five or 10 years ago,” says Chris Anderson, founder of DIY Drones (and a former WIRED editor-in-chief).
But there are plenty of side effects to making cutting-edge technology available to all. Deepfakes, for example—AI-generated videos meant to look like real footage—are now accessible to anyone with a laptop. It’s easier than ever for any company, not just Facebook, to wield AI to target ads or sell your data at scale. And with AI burrowing into the fiber of every business and inching deeper into government, it’s easy to see how automated bias and privacy compromises could become normalized swiftly. As Neha Narula, head of MIT’s Digital Currency Initiative asks, “What are the controls that can be put in place so that we still have agency, that we can still shape it and it doesn’t shape us too much?”
Find out more in the video above, a new documentary by WIRED, directed by filmmaker Chris Cannucciari and supported by McCann Worldgroup.
What if you held a big party for 200 people and one of the guests you most wanted to see RSVPed but never showed up? This was the scenario with North Korea’s absence at the sixth Ulaanbaatar Dialogue (UBD) on Northeast Asian Security, a 1.5 level forum for officials and academics, which I attended from June 5-6 in the Mongolian capital. Nonetheless, Mongolia succeeded in making its case as a meaningful interlocutor on North Korean issues and a participant in Northeast Asian economic integration efforts, such as ongoing discussions about expanding the use of wind and solar power in a regional power grid.
Although Mongolia was considered as a venue for one of the summits between U.S. President Donald Trump and North Korean leader Kim Jong Un, it was eventually not selected. Of course, it was not unexpected that North Korea would prefer an authoritarian host to a fledgling democracy that had made a transition from socialism. Nevertheless, Mongolia has played an important, if often overlooked, role over the years as a facilitator of Northeast Asian diplomacy with North Korean officials. As Foreign Minister Damdin Tsogtbaatar put it, Mongolia has the potential to be a “bridge for peace” in Northeast Asia, due to its own unique history as a socialist state and more recent development as a democracy.
Mongolia is also one of a few countries to enjoy good relations with both South and North Korea. Mongolia’s relations with South Korea have deep historical and cultural roots, and democratization in both countries has deepened their mutual affinity. Alicia Campi, a former diplomat and scholar of Mongolia’s foreign policy, notes that its longstanding bilateral relations with North Korea are “underappreciated.” Despite the differences in their trajectory after Mongolia’s democratic transition, the two countries have retained mutually beneficial economic ties, including the provision of North Korean guest workers (until sanctions prohibited this in 2018). Then-President Tsakhiagiin Elbegdorj used the occasion of a state visit to Pyongyang in 2013 to offer his country as a mediator in the nuclear crisis (as well as to praise democracy during his speech at Kim Il Sung university). The annual Ulaanbaatar Dialogue began in 2014 as a means of encouraging regionwide security discussion and reducing distrust among the parties in the aftermath the collapse of the Six-Party Talks.
Elbegdorj was the first foreign leader to meet Kim Jong Un, and his successor, President Khaltmaagiin Battulga, extended an invitation to the North Korean leader to visit Mongolia. In December 2018, North Korean Foreign Minister Ri Yong Ho visited Ulaanbaatar to celebrate 70 years of diplomatic relations between the two countries. Mongolian officials have also participated in a number of less public mediation efforts, helping to facilitate the return of Japanese abductees from North Korea and assisting South Korea in resettling North Korean refugees. As a nuclear weapons-free state and a small developing country surrounded by stronger powers, Mongolian officials believe their experience is highly relevant to ongoing discussions of security on the Korean Peninsula. To this end, at the recent UBD some proposed Mongolia’s participation in future multilateral talks on the nuclear crisis, a position that Russian officials supported in the past.
Apart from North Korea’s no show at the June 2019 UBD, the other hot topic in Ulaanbaatar was whether or not Mongolia should seek full membership in the Shanghai Cooperation Organization (SCO), which both Russia and China have encouraged. Membership was restricted to China, Russia, and the Central Asian states until 2017, when India and Pakistan both joined. Mongolia has been an observer in the SCO since 2004. Some Mongolian officials contend that full membership would enhance trust between Mongolia, Russia, and China, and potentially add new dynamism to their trilateral economic cooperation plans. Others argue that such a move might compromise Mongolia’s “third neighbor” policy and that Mongolia does not share the same concerns as other SCO members over terrorism, extremism, and separatism. Moreover, at a fraught time in U.S. relations with both Russia and China, Mongolia’s membership in the SCO might be construed in Washington and other Western capitals as anti-NATO, despite Mongolia’s history of military cooperation with it. Certainly, Mongolia’s participation last summer in the major Russian military exercise, Vostok, along with a contingent of Chinese forces, raised some eyebrows.
While India has been able to navigate between membership in the SCO and its partnership with the United States and other democracies, it does not face the same economic or geopolitical pressures as landlocked Mongolia, which seeks to balance sustainable development with independence from its two powerful neighbors. Despite expectations of a shift in Mongolia’s position, Battulga’s attendance at the SCO summit in Bishkek did not lead to any change from his country’s observer status. Admitting that SCO membership remained controversial at home, the Mongolian president noted that “Mongolia is exploring levels of increase of its participation” in the organization and supported the additional opportunities at the Bishkek summit for observer states and international organizations to join in the discussions with member states.
Presidents Xi Jinping, Vladimir Putin, and Battulga met on the sidelines of the summit to discuss their trilateral cooperation in the framework of the China-Mongolia-Russia-Economic Corridor established as a part of the Belt and Road in 2014. In their individual statements, each president provided a different vision of what this corridor should involve. Given the lack of consensus among the three, it is not surprising that the corridor has made little progress so far, not even in achieving mutually acceptable feasibility studies, not to mention completing planned projects.
Mongolia has enthusiastically supported the trilateral economic agenda with its two neighbors, but bilateral issues have stymied its progress. On the one hand, the slow pace of Sino-Russian regional cooperation has held up trilateral plans for road and rail connections via Mongolia. For example, the bridge from Blagoveshchensk, Russia, to Heihe, China, on the books since 1995 and at long last constructed earlier this month, will be a key link in these new transit routes once road and rail connections are completed. The possibility of a second Sino-Russian gas pipeline transiting Mongolian territory depends on the protracted Sino-Russian negotiations over routing and pricing, as well as China’s view of pipelines transiting third countries as an energy security risk, a concern likely to color its view of a Northeast Asian energy grid as well. Other key areas of China-Mongolia-Russia trilateral cooperation (simplifying customs clearance and rail logistics) also need to be negotiated bilaterally.
On the other hand, the deepening Sino-Russian political partnership makes Mongolia’s effort to balance a good relationship with each of its two neighbors — with the goal of avoiding economic dependence on either one — all the more difficult. Although Battulga, responding to anti-Chinese sentiment in the Mongolian public, appeared to be tilting more toward Moscow in his first year in office, a series of meetings with Xi since 2018 have sought to rebalance Mongolian foreign policy, given the inescapable fact of the country’s considerable reliance on trade and investment from China. Nonetheless, excessive dependence on China for trade and investment creates new vulnerabilities, as economic stagnation in China diminishes its demand for Mongolian minerals, a fact that already worsened Mongolia’s economic woes earlier in the decade. The challenge will be for Mongolia to implement some of its creative foreign policy thinking, which looks beyond the immediate pressures of the two large neighbors and seeks to put Mongolia on the map through a variety of multilateral initiatives as well as its “third neighbor” policy.
Elizabeth Wishnick is a Professor of Political Science at Montclair State University and a Senior Research Scholar at the Weatherhead East Asian Institute, Columbia University.
Mila Kunis and Ashton Kutcher hit back at split rumours with funniest video
The couple tied the knot in 2015
Mila Kunis and Ashton Kutcher have found the perfect way to shut down reports they are splitting up. The couple, who tied the knot in 2015, filmed a hilarious video that showed them poking fun at the rumours printed by one tabloid magazine. Mila held up a photo of the magazine cover, while Ashton asked his wife: “Babe, what’s happening? What’s going on?”
“It’s over between us!” the Black Swan actress dramatically replied. “It’s over between us? Oh my god, what are we going to do?” Ashton asked. “Yeah… I felt suffocated!” Mila continued. “You felt suffocated by me? I was just so overbearing, wasn’t I!” Ashton quipped.
Mila and Ashton poked fun at the rumours
Pausing, Mila added: “Also… I took the kids.” “Oh, you took the kids? I don’t get the kids anymore?” Ashton exclaimed. “But… you had a very dark secret exposed!” his wife continued. “Oh my gosh, what was the dark secret?” Ashton asked, to which Mila shook her head and said: “I don’t know! I only have this photo…” The magazine cover claimed Ashton, 41, and Mila, 35, were going through a “$400 million divorce shocker”. One of the dramatic coverlines read: “The lie that destroyed everything.”
The couple are the doting parents to four-year-old daughter Wyatt Isabelle and son Dimitri Portwood, two. Speaking previously to Marie Claire, Mila has said of being a mum: “What motherhood shows you is how selfless you can get. I’m ragged tired. Who cares? My kids are healthy, I’m happy.”
The couple have two children together
Ashton, meanwhile, has spoken about how different raising a girl and a boy can be. “It’s weird because girls advance apparently faster than boys,” the actor said in 2017 on the Tonight Show with Jimmy Fallon. “My daughter is like two and a half, she speaks three languages, she’s got like Russian and Spanish and English. She actually understands and speaks them.” He quipped: “The boy is just like ‘thuck.’ Just like a chunky, ‘BAHHHH.’ He sounds like a pterodactyl, and he’s not coming around.”
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It was the outfit which marked Sandy Olson’s transition from prim and proper school girl to sex symbol in Grease.
Now, you could own the famous leather jacket and skin-tight pants worn by Olivia Newton-John in an one of the world’s most iconic cinema moments.
California auction house Julien’s Auctions is advertising the outfit along with a huge list of 200 pieces of memorabilia and costumes worn during Newton-John’s acting and singing career.
‘Grease’ is getting a prequel retelling the origin story of how Danny Zuko and Sandy Olsson first met.
The leather outfit, worn as Newton-John sang “You’re the One that I Want” alongside John Travolta, is being described as “one of the highly anticipated moments of the auction” and is expected to fetch up to $US200,000 ($290,000)
A portion of the money raised will be donated to the 70-year-old actress’ Melbourne cancer centre, which provides services for cancer patients and research.
Other Grease memorabilia up for grabs includes Newton-John’s original script from the film, which is expected to fetch up to $US4000 ($5800), and a Grease 10-times platinum award, expected to be sold for up to $US2000 ($2902).
If you preferred Sandy’s “pink ladies” persona, a custom jacket presented to the actress by the cast and crew of Grease is expected to be sold for up to $US4000 ($5800).
Also going under the hammer are a pair of custom-made cream thigh-high boots and a pair of harem pants worn by Newton-John in Xanadu and a silk satin bodice worn on the cover of her 1982 Physical tour book.
Super fans may even want to bid on a white and gold velvet top and hot pants worn on The Merv Griffin Show in 1981 or a petal pink point d’espirit lace gown worn to the premiere of Grease at The Mann’s Chinese Theatre in Los Angeles, expected to sell for up to $US5000 ($7200).
The auction is taking place in Beverly Hills and online on November 2 and will follow exhibitions of the items in Ireland and the US.
Olivia Newton-John and John Travolta celebrated the event at a special screening.
Newton-John was diagnosed with her third bout of cancer in June 2017 and has a tumour in the base of her spine.
After several news outlets reported the star had “weeks, not months” left to live in January, the Grease actress said she was “doing great”.
“The rumours of my death have been greatly exaggerated, to quote a very famous quote,” she said in a video on her Facebook page.
Oprah Winfrey may prove that life can go full circle at times, because she hinted at rebooting “The Oprah Winfrey Show,” the daily talk show that made her a household name.
The syndicated show ran from 1986 to 2011 and taped in Chicago. Some of the episodes had big-name celebrities as guests, others had everyday people who told compelling stories and sometimes Winfrey made herself the subject matter — like in a 1988 episode when she revealed a massive weight loss.
Oprah Winfrey said she’d love to bring back her talk show. (Photo: Rodin Eckenroth / Getty Images Entertainment via Getty Images)
Winfrey also had special giveaway episodes where she’d give the studio audience extravagant gifts, which is where her famous “You get a car, you get a car” line came from.
Winfrey was in Toronto earlier this week promoting her new book “The Path Made Clear: Discovering Your Life’s Direction and Purpose” when the subject of her talk show came up.
And according to the 65-year-old, bringing it back would be a great idea but just on a different schedule than before. Winfrey also said she gets that talk show itch whenever a big news story unfolds.
“I would love to make that happen, let me tell you,” she told “ET Canada.” “But maybe not every day. For 25 years, it was perfect. The only time I missed it was during the election or when something really big happens in the news. I think, ‘Oh, gee, I wish I had a show.’”
