JP2005527033A - Factorization and modular arithmetic methods - Google Patents

Abstract

A method for factoring a number in a non-binary computation scheme, more specifically, a method for factoring a number using a digital multi-state phase change material is provided.
The method includes providing an amount of energy characterizing a factor to be factored to a phase change material programmed according to the number of potential factors. The programming strategy sets the phase change material once whenever there are multiples of latent factors among the numbers to be factored. In carrying out this method, it is determined whether the latent factor is indeed a factor by counting the number of multiples and evaluating the state of the phase change material. A given volume of phase change material can be reprogrammed for different factors, or separate volumes of phase change material can be used for different factors. Parallel factorization is achieved on several latent factors by combining separate volumes of phase change material programmed according to different latent factors. Also included are methods of addition and joint calculation in modular arithmetic systems.

Description

  The present invention relates generally to computational methods directed to number factorization and modular arithmetic. More specifically, the present invention relates to performing factorization and modular arithmetic in a non-binary manner using digital multi-state phase change materials.

Related Application Information This application is a continuation-in-part of US application Ser. No. 10 / 144,319, filed on May 10, 2002.

Background Art The development of computers is usually considered as one of the most significant advances in the second half of the 20th century. Computers have simplified many aspects of daily life and have brought about significant productivity gains in the economy. Recent needs in image processing and complex computations are not met by significant improvements in microprocessor speed and memory storage density. Further advances in computers and future applications rely on human ability to process larger amounts of information more efficiently.

  Silicon is the heart of today's computer. The increase in computing power and speed was the result of a good understanding of the basic characteristics of silicon and the practical and effective use of those characteristics. Early progress was based on the construction of basic electronic components, such as transistors and diodes from silicon, and later progress began with the development of integrated circuits. Recent advances are a continuation of these trends, and now the emphasis is on miniaturization and the integration of more microelectronic devices on one chip. As devices become smaller, memory storage density increases, circuit density increases, and interaction time between devices on the same chip decreases.

  A unique feature of silicon-based computing devices is that they perform mathematical operations and other data processing purposes in binary. In binary computing, silicon, the computing medium, has two programming states that are used to represent and manipulate data. The two programming states are typically referred to as “0” and “1”, and the volume of silicon used to store “0” or “1” is typically referred to as a bit. Data including numbers and letters is converted to one or more series of “0” s and / or “1” s. Here, “0” or “1” is stored in the separated bits. Thus, a series of bits are programmed to store data by establishing the appropriate combination of “0” and “1”. Manipulating data includes bit operations that modify the state of the bits according to the desired computational purpose to produce an output. The generated output typically includes a series of bits that store a different “0” and / or “1” combination or sequence than was present at the beginning of the operation.

  Binary computers have proven to be extremely successful in many computing applications, such as automation, word processing, and basic mathematical calculations. As computing needs have expanded and more complex applications are envisioned, it has become apparent that traditional binary computation has many limitations. For example, the ability to continuously miniaturize silicon-based microelectronic devices is fundamental to increasing computation speed and using parallel computing heavily. However, concerns about whether or not miniaturization efforts can continue are becoming increasingly significant. This is because many people believe that practical and basic limitations cannot overcome barriers to miniaturization. Complex computing environments that require adaptability, interactivity, or a high degree of parallelism do not appear to be optimally achieved or possible with traditional binary methods.

  In order for the computer industry to expand and become involved in more applications and more complex computing environments, it is appropriate to change the way in which computers function.

DISCLOSURE OF THE INVENTION The present invention is generally directed to extending the range of computer capabilities by non-binary computation methods. The method of the present invention is designed for a computing medium capable of performing non-binary operations. Non-binary operations are implemented using a digital multi-state phase change material that can process more than two programming states. According to the calculation method of the present invention, the calculation can be performed using a calculation medium including three or more programming states. Calculations using more than two programming states can be achieved using the calculation method of the present invention.

  In one embodiment, a method of factorization is provided. The purpose of this method is to determine the multiplicative factor of the input number. The method includes programming the digital multi-state phase change material according to the latent multiplicative factor by establishing a number of programming states corresponding to the latent multiplicative factor. A numerical value is assigned to each programming state, and the transformation between different programming states corresponds to the number of mathematical operations associated with the programming state. The programming state is selected such that the programming state coupled to the latent multiplicative factor corresponds to the set state of the phase change material with energy separated from the reset state of the phase change material. Thus, when an amount of energy characterizing the latent multiplicative factor is applied to the reset state of the phase change material, the phase change material is set. Setting transformation is easily detected as a change in resistance of the phase change material. Factoring is achieved by analyzing the energy corresponding to the number to be factored in an amount that characterizes the latent multiplicative factor. The achievement counts the number of setting transformations (and the resets after each setting transformation) and ensures that there is no remainder when all the energy corresponding to the factored number is provided. Is done by doing.

  In one embodiment, a given volume of phase change material is used to test different latent multiplicative factors. The latent multiplicative factor test sequentially establishes the programming state according to one multiplicative factor, determines whether the latent multiplicative factor is indeed a factor, re-establishes the programming state accordingly, This is done by iterating over the multiplicative factor. A unique feature of multi-state phase change materials is the ability to define and redefine programming states, where a number of different programming states are placed between reset and set states with different energies As such, the programming state can be defined and redefined. Thus, it is possible to associate a set state with each of several different latent multiplicative factors.

  In an alternative embodiment, separate volumes of phase change material are used to test different multiplicative factors. In this embodiment, for each multiplicative factor in question, a separate volume of phase change material is used exclusively and programmed accordingly by establishing an appropriate number of programming states. In this embodiment, multiple latent multiplicative factors are tested by operating in series or in parallel on separate volumes of phase change material. Parallel operation provides an effective factorization of numbers in particular.

  The method of the present invention further provides computations in a modular arithmetic framework. These methods include calculating the remainder and congruence in an arithmetic system governed by the modulus.

DETAILED DESCRIPTION The present invention takes a step forward to overcome the limitations of conventional binary computing devices by providing a computing method suitable for using computing media that can operate in a non-binary manner. is there. The inventors of the present invention believe that non-binary computing methods provide an opportunity to improve functionality and extend the scope of computer applications. Non-binary computation is advantageous as described in co-pending patent application US patent application Ser. No. 10 / 144,319. This is because it increases the information storage density and increases the parallel processing of data.

  Achieving non-binary computation involves qualifying computational materials that can maintain more than two programming states and developing computational methods that can store and process data in a non-binary manner. Co-pending patent application US patent application Ser. No. 10 / 144,319 certifies phase change materials as examples of materials programmed to provide multi-state properties suitable for non-binary computation. The relevant attributes of phase change materials as a medium providing multi-state programming and computational capabilities are summarized below and described. Further details are described in co-pending patent application US patent application Ser. No. 10 / 144,319.

  Phase change materials suitable for non-binary operations include phases having at least a high resistance state and a differentially detectable low resistance state. As used herein, high and low resistance states refer to physical states characterized by high and low electrical resistance, respectively. Here, the electrical resistances in the high and low electrical resistance states are relative to each other and can be detected separately from each other. Examples of such phase change materials are commonly assigned US Pat. No. 5,166,758, US Pat. No. 5,296,716, US Pat. No. 5,524,711. , U.S. Pat. No. 5,536,947, U.S. Pat. No. 5,596,522, U.S. Pat. No. 5,825,046, U.S. Pat. No. 5,687,112. Has been. The disclosures of these patent documents are incorporated herein by reference. Exemplary phase change materials include one or more of the elements In, Ag, Te, Se, Ge, Sb, Bi, Pb, Sn, As, S, Si, P, O, and mixtures or alloys thereof. . In a preferred embodiment, the phase change material comprises a chalcogen element. Particularly preferred are phase change materials comprising chalcogen in combination with Ge and / or Sb. In other preferred embodiments, phase change materials include chalcogens and transition metals, such as Cr, Fe, Ni, Nb, Pd, Pt, or mixtures and alloys thereof.

  The phase change material is transformed from a high resistance state to a low resistance state when an effective amount of energy is applied. FIG. 1 disclosed herein is a plot of the electrical resistance of a phase change material as a function of energy or power. If the phase change material is initially in a high resistance state, application of a small amount of energy leaves the material in a high resistance state. This behavior is shown in the high resistance stable state region on the left side of FIG. However, if a sufficient amount of energy is applied, the phase change material is transformed from a high resistance state to a low resistance state. This transformation is indicated by a sharp reduction in electrical resistance just to the right of the high resistance steady state region of FIG. The transformation of the phase change material from the high resistance state to the low resistance state is hereafter referred to as a “setting” or “set” of the phase change material. The low resistance state generated by the setting is hereinafter referred to as the “set state” of the phase change material. The amount of energy sufficient to set the phase change material is hereafter referred to as “set energy” or “setting energy”. Note that the set energy is different in each state along the high resistance stable state.

  The right side of FIG. 1 corresponds to the behavior when the phase change material is set. Once set, the phase change material remains in a low resistance state and is affected by the application of power according to the post-setting region shown on the right side of FIG. This portion of the electrical resistance response curve is referred to as the analog or grayscale multi-bit portion of the curve in some of the aforementioned patent documents incorporated by reference. When energy is applied to the post-setting state of the phase change material, the electrical resistance changes. The change in electrical resistance is determined by the amount of energy applied and the rate at which the energy is applied. The rate at which energy is provided is now referred to as power and is an important factor in the behavior of phase change materials in the post-setting region.

  While not wishing to be bound by theory, the inventors of the present invention believe that the establishment of a low resistance set state in the setting transformation is a result of the formation of a continuous crystal path throughout the phase change material. . In the pre-setting region, the phase change material is considered to contain an amorphous phase component and possibly a crystalline phase component. The presence and relative amount of the crystalline phase depends on the preparation and processing conditions used to form the phase change material. For example, melting followed by rapid cooling appears to hinder crystallization, while melting followed by slow cooling appears to promote crystallization. The crystalline phase, if present in the pre-setting region, is interspersed in the amorphous phase and cannot provide a continuous path throughout the phase change material. Since the amorphous phase has a higher electrical resistance than the crystalline phase, if there is no continuous crystal network, the phase change material has a higher electrical resistance in the pre-setting region.

  The inventors of the present invention consider that when energy is applied in the high resistance stable state region before setting the electric resistance curve, the relative amount of the crystal phase in the phase change material increases. Assuming that a continuous crystal network is not formed, the amount of increase in crystal phase does not substantially affect the electrical resistance of the phase change material. The inventor of the present invention believes that the formation of a continuous crystal network occurs during setting transformation and that the reduction in electrical resistance associated with the setting transformation results from the presence of conductive paths throughout the continuous crystal phase. Since the continuous crystal phase has a lower resistance than the amorphous phase, if a continuous crystal network exists, the electrical resistance of the phase change material after setting is low.

  In the post-setting region, energy is applied to the low resistance set state, which may affect the crystal network. The addition of energy results in an exotherm and a temperature increase of the phase change material. When sufficient energy is applied to the phase change material, the continuous crystal network that existed at the time of setting can be melted or thermally broken. When melting occurs, subsequent cooling tends to produce phase change materials with different amounts or connectivity of crystalline phase components. Melting or thermal breakage of the crystal network breaks the conductive paths throughout the low resistance crystalline phase, thereby increasing the electrical resistance of the phase change material in the post-setting region. Melting or thermal fragmentation of the crystal network requires that sufficient energy remain at the melting or thermal fragmentation site to cause melting or thermal fragmentation. The heat dissipation process due to heat conduction, heat capacity, loss to the surroundings, etc. works to remove energy, thus preventing the melting or thermal fragmentation of the crystal network, so the rate of energy addition is high enough, Melting or thermal fragmentation must occur while compensating for the heat dissipation process. Thus, the energy or power rate becomes a significant consideration in the post-setting region of the electrical resistance curve.

  Depending on the power and state of the phase change material in the post-setting region of FIG. 1, an increase or decrease in electrical resistance may occur. This is because the behavior in the post-setting area is reversible. This reversibility is indicated by the two arrows in the post-setting region of FIG. 1 and is believed by the inventor of the present invention to reflect the reversibility of changing the nature of the crystal network of the phase change material. Power and electrical resistance may be coupled to each point in the post-setting region. If the applied power exceeds the power combined with the point indicating the phase change material in the post-setting region, the electrical resistance of the phase change material increases. Conversely, if the applied power is less than the power coupled to the point representing the phase change material in the post-setting region, the electrical resistance is reduced. The inventors of the present invention have found that the increase in electrical resistance in the post-setting region reflects the interruption or reconfiguration of the crystalline component of the phase change material, and the conductive path throughout the crystalline component is reduced in number, size, or capacity. I think to do. The decrease in electrical resistance in the post-setting region is considered to be the opposite.

  Reversibility is limited to the post-setting region of FIG. It is impossible to reverse the setting transformation by applying energy corresponding to the points in the high resistance steady state region of FIG. 1 prior to the setting transformation (ie, on the left). However, it is possible to restore the high resistance state of the phase change material by applying a sufficiently high power to the phase change material as indicated by the points in the post-setting region of FIG. The application of such power corresponds to moving in the right direction in FIG. 1 and does not correspond to the direction in which the setting transformation is reversed. As shown in the post-setting region of FIG. 1, the electrical resistance increases continuously when a continuously increasing amount of power is applied. To the far right of FIG. 1, when sufficient power is applied to drive the phase change material, the phase change material returns to the high resistance state and updates the high resistance stable state. The inventor of the present invention believes that the restoration of the high resistance stable state occurs when the power provided to the phase change material is sufficient to break the continuity of the crystalline components of the phase change material. Restoration of the high resistance stable state reduces the amount of crystalline component of the phase change material (eg, caused by melting and quenching processes that increase the amount of amorphous component), loss of continuity due to reconfiguration or redistribution of the crystalline component Or a combination of these.

  The power or rate of energy required to transform the phase change material from the low resistance set state to the high resistance state is now referred to as “Reset Power”, “Resetting Power”, “Reset Energy”, “Resetting”.・ It is called “energy”. The state of the phase change material when the application of reset energy is completed is now referred to as the “reset state”. The application of reset power “resets” the phase change material to create a high resistance reset state. The behavior observed upon further application of energy after resetting is similar to that described for the high resistance steady state region of FIG. The plot shown in FIG. 1 corresponds to one cycle of setting and resetting. The phase change material can be set and reset reproducibly over a number of cycles.

  Multiple programming states of non-binary computation are achieved using states along the high resistance steady state region shown on the left side of FIG. Applying an amount of energy less than that required to set (hereinafter referred to as subsetting energy amount) to a phase change material in a high resistance state will cause a physical modification that does not significantly change the electrical resistance of the material. Happens in. Applying an amount of subsetting energy to the phase change material indicated by the first point in the steady state region of FIG. 1 causes the phase change material to transform to the physical state indicated by the second point in the steady state region. . Here, the second point is to the right of the first point. By applying the subsetting energy amount, the phase change material cannot be transformed to the left of the high resistance stable state region. The reversible concept described for the post-setting analog multi-bit portion of FIG. 1 is not applicable to the pre-setting digital multi-state high resistance stable state. The point to the left of the first point in the high resistance steady state region is reached by setting the phase change material, resetting it, and then applying the appropriate amount of subsetting energy.

  As a result of the unidirectional response in the high resistance steady state region, the phase change material can accumulate and store energy quantities up to the set energy. Although the electrical resistance of the phase change material does not change in the high resistance stable state region, the phase change material advances predictively toward the low resistance set state by applying each subsetting energy. This behavior of the phase change material is utilized in the multi-state calculation method disclosed herein.

  In the preferred embodiment, the reset state is selected as the starting point in the high resistance steady state region of FIG. The interval of energy required to transform the phase change material from the reset state to the set state is hereinafter referred to as “reset state setting (or set) energy”. Since the reset state setting energy is reproducible and constant over many cycles of setting and resetting, it can be divided into two or more sub-intervals and different programming states of the phase change memory material Or the programming values can be defined by or coupled to different numbers or combinations of sub-interval energy applications. The reproducible unidirectional behavior of the phase change material in the high resistance steady state region of FIG. 1 provides the ability to establish a distinct programming state according to the cumulative amount of energy applied to the material in sub-interval quantities. Here, distinct programming states are defined for each cumulative energy amount that does not exceed the reset state set energy.

  A distinct set of programming states can be uniquely and reproducibly defined according to the cumulative amount of energy applied to the reset state by applying a series of subsetting energy in sub-intervals. For example, the initial programming state is coupled to applying a first partial interval energy amount to the reset state. Second application of the first sub-interval energy amount to the phase change material in the first programming state is used to define the second programming state. The second programming state may alternatively be coupled to applying a second partial interval energy amount that differs in magnitude from the first partial interval energy amount once to the reset state. Sequential application and combination of sub-interval energy quantities is used to define a series of programming states throughout the high resistance steady state region of FIG. Here, different programming states are distinguished according to the amount of energy relative to the reset state.

  The multiple sequences and combinations of applying energy in sub-intervals constitute an embodiment that establishes the programming state used by the calculation method of the present invention and general non-binary calculation methods. In some embodiments, only the state of the high resistance steady state region is included as a programming state. In other embodiments, the set state obtained from the transformation of the phase change material to the low resistance state is included as an additional programming state. The phase change material is set when the amount of accumulated energy applied to the reset state reaches the reset state setting energy. After setting, the behavior of the phase change material when further energy is applied is governed by the post-setting region of FIG.

  The phase change material used in the present invention has a continuum that extends from the reset state to the set state. States include a reset state, a series of intermediate states along a high resistance stable state, and a set state. The intermediate state and the set state are obtained by applying energy to the reset state. Thus, it can be seen that the state bound to the phase change material forms a continuum of states that can be distinguished by energy relative to the reset state. The programming state described so far is selected from this state continuum. The programming state is typically coupled to discrete data values such as integers and programming energy corresponding to the energy that must be applied to the phase change material reset state to transform the phase change material to the programming state. Have For example, a particular programming state has a programming energy corresponding to the amount of energy that must be added to the phase change material reset state to transform the phase change material to that programming state. Thus, the programming state programming energy is a measure of energy relative to the reset state, and each programming state is coupled to a unique programming energy.

  In the preferred embodiment, each programming state has a unique data value assigned or associated with it. In another preferred embodiment, consecutive integers are assigned or combined into programming states in order of increasing programming energy. In these embodiments, the number is stored by transforming the phase change material into a programming state coupled to the number. The unique correspondence between the numerical value and the programming state provides the ability to store any number in a range into one distinct state of the phase change material. Multiple numbers or one specific number of multiple digits are stored in multiple portions of the phase change material. Calculations are generally performed by providing energy that causes transformation in one or more programming states according to the calculation purpose.

  As described above, each application of the partial section energy amount causes the phase change material in the high resistance stable state to proceed toward the set state. The reset and intermediate states along the high resistance stable state have approximately the same electrical resistance and therefore cannot be distinguished on the basis of electrical resistance, but each state along the high resistance stable state is It is left so that it can be detected separately from the state. This is because different amounts of energy are required to set the phase change material indicated by different points along the high resistance stable state. For example, if an amount of energy is applied to transform the phase change material indicated by the first point along the high resistance stable state to the second point along the high resistance stable state, In order to transform the phase change material indicated by the second point to the set state, the amount of energy smaller than the amount of energy required to transform the phase change material indicated by the first point to the set state is required. The set state is a low resistance state that can be detected separately from the state along the high resistance stable state, and the amount of energy required to set the phase change material indicated by any point along the high resistance stable state is Being determinable, different points along the high resistance stable state can be distinguished, and each point along the high resistance stable state can be coupled to a different programming state and / or numerical value. As a result, by dividing the energy interval from the reset state to the set state by an appropriate number of partial intervals, any number of programming states can actually be defined in the high resistance stable state region of FIG. . As a result, the high resistance stable state region can be divided into a series of discrete programming states, which can be used for data storage and processing. Phase change materials are not limited to the same two programming states as current binary computers, but provide non-binary multi-state programming and computational capabilities.

  The number of programming states is the amount of energy interval coupled to the transformation from the reset state (or other starting state) to the set state of the phase change material (the amount of set energy in the reset state or starting state), It depends on the resolution with which this energy interval can be divided into sub-intervals and practical considerations, for example the computational or processing advantages associated with having a certain number of programming states. As used herein, a starting or initial state means an initial or lowest energy programming state along a high resistance stable state. The reset state is preferably set to the start state, and unless otherwise specified, the following description assumes that the reset state is the start state. However, it should be recognized that different states along the high resistance stable state may be selected as the starting state.

  The size of the energy interval between the reset state and the set state is affected by the chemical composition of the phase change material. The resolution at which the subinterval is defined depends on the energy resolution of the energy source used to program, transform, or read the phase change material. An energy source suitable for transforming a phase change material between states of the phase change material according to the calculation method of the present invention is described in detail in co-pending patent application US patent application Ser. No. 10 / 144,319. Explained. From a computational standpoint, the number of programming states affects the nature, speed, parallelism, or convenience of the computational method or algorithm. For example, if the number of programming states is selected as a multiple of 2, currently available binary algorithms can be adapted for use with the non-binary computation method of the present invention. For example, embodiments having programming states such as 4, 8, 16, 32, 64 are included in the present invention. Using ten programming states, a decimal based method used in daily life can be conveniently implemented and is also included in embodiments of the present invention. As will be described later, in the calculation method of the present invention, it is desirable to use a phase change material having a number of programming states corresponding to the latent multiplicative factor.

  Examples of phase change materials suitable for use in the calculation methods and algorithms of the present invention include US Pat. No. 5,166,758, US Pat. No. 5,296,716, US Pat. No. 524,711, US Pat. No. 5,536,947, US Pat. No. 5,596,522, US Pat. No. 5,825,046, US Pat. No. 5,687, 112, U.S. Pat. No. 5,912,839, U.S. Pat. No. 3,271,591, and U.S. Pat. No. 3,530,441. The disclosures of these patent documents are incorporated herein by reference. The volume of memory material can include a mixture of dielectric material and phase change material. An example of such a mixture is described in commonly assigned US Pat. No. 6,087,674. The disclosure of this patent document is incorporated herein by reference. Materials suitable for the calculation method of the present invention are typically one or more of the elements In, Ag, Te, Se, Ge, Sb, Bi, Pb, Sn, As, S, Si, P, O, Or mixtures and alloys thereof. In one preferred embodiment, the phase change material comprises a chalcogen. In another preferred embodiment, the phase change material comprises chalcogen and Ge. In a further preferred embodiment, the phase change material comprises Ge, chalcogen, and Sb. In the most preferred embodiment, the phase change material comprises Ge, Te, and Sb.

  The calculation method of the present invention includes providing or applying energy to the phase change material. The provision or application of energy may be accomplished by any source that can deliver a controlled amount of energy to the phase change material. The energy provided can take any form including electricity, light, and / or heat. The controlled amount of energy is hereafter referred to as the “pulse of energy” or “energy pulse”. A “set pulse” is a pulse of energy sufficient to set the phase change material and corresponds to providing an amount of setting energy. A “reset pulse” is a pulse of energy sufficient to reset the phase change material and corresponds to providing a resetting energy amount.

  In the calculation method of the present invention, a number of steps or operations are performed by providing the phase change material with an amount of energy characterizing the number. The characterizing energy depends on the reset state setting energy of the phase change material, the number of programming states, and the fraction of energy required to transform the phase change material from one programming state to the next. . In one embodiment, the energy that characterizes or corresponds to a number is an amount of energy proportional to that number. The energy sub-intervals required to progress from one programming state to the next programming state are now referred to as “programming interval”, “program interval”, “program interval energy”, “program pulse”, “incremental energy”. ”,“ Incremental interval ”, or equivalent names.

  As used herein, the progression from one programming state to the next programming state is the transformation of the phase change material from the current programming state to the next programming state defined along the electrical resistance curve of the phase change material. Means. The next programming state necessarily has a higher energy for the phase change material reset state than the current programming state and corresponds to the programming state to the right of the current programming state along the electrical resistance curve. To do. “Increment” or “apply incremental energy” and like terms refer to the phase change material from the current programming state to the programming state that is closest in energy and on the electrical resistance curve to the current programming state. It means to transform to the programming state on the right side of. Thus, incremental means that the phase change material is transformed one programming state at a time along the electrical resistance curve to the right.

  In one embodiment of the invention, the reset state represents the number 0, and uniform incremental energy is used to progress from one programming state to the next. As a result, all programming states are equally separated for energy, and each application of incremental energy corresponds to increasing the stored number of values by one. Thus, the number “1” is stored by applying incremental energy once to the reset state, the number “2” is stored by applying incremental energy twice to the reset state, and so on. Up to the maximum number that can be stored in the volume of memory material used for storage is stored. The application of uniform incremental energy is hereinafter referred to as “uniform incremental pulse” or equivalent terminology.

  Variant embodiments in which non-uniform programming intervals or non-uniform incremental intervals are used constitute additional embodiments of the invention. As used herein, a non-uniform programming interval means a sequence of programming intervals in which the programming state does not have an equal interval for energy. In other words, the incremental energy between adjacent programming states is different at different points along the high resistance stable state, for example, the incremental energy separating the first programming state and the second programming state is The incremental energy that separates the two programming states from the third programming state is different. The same applies hereinafter. In such embodiments, the incremental energy is not constant, but the incremental concept is sufficient to transform the phase change material from one programming state to the next higher energy programming state relative to the reset state. Means adding or providing the right amount of energy. Similarly, incrementing phase change material means providing energy to the phase change material to transform the phase change material from an initial programming state to a final programming state. Here, there is no programming state between the initial programming state and the final programming state. The initial and final programming states associated with the process of incrementing the phase change material are referred to herein as adjacent or continuous programming states.

  Establishing the programming state of the phase change material is referred to herein as a programming strategy. The programming strategy includes a predetermined sequence that applies energy in sub-intervals. In this sequence, the programming interval from one programming state to the next programming state and the cumulative applied energy from the reset state to each programming state are specifically defined. Thereby, a set of programming states is established in which each programming state has a unique programming energy relative to the reset state. The programming strategy can arbitrarily combine uniform or non-uniform program intervals extending from the reset state to the set state, and may or may not include the reset state or the set state as the programming state. Unless specified otherwise, the following description relates to programming strategies that use uniform program intervals. However, it should be recognized that this selection is made for convenience of explanation and is not intended to limit the practice of the invention.

  The possibility of assigning or combining data or numerical values to programming states is included in the programming strategy, with each programming state corresponding to a different data value or number. The calculation is performed by transforming the phase change material between the programming states of the phase change material. For example, converting the number of inputs to the number of outputs according to mathematical objectives is by providing energy appropriately to transform the phase change material from the programming state corresponding to the number of inputs to the programming state corresponding to the number of outputs. Achieved. Preferably, the value associated with the programming state is an integer. More preferably, the integer continues. Embodiments are contemplated where integer values are combined in an increasing or decreasing order of programming energy. For example, if the reset state is coupled to a data value of 0, application of one programming interval energy sets the material to the programming state coupled to a data value of 1 and application of the other programming interval energy is 2. Set the substance to the programming state bound to the two data values. The same applies hereinafter. Alternatively, data values can be combined into programming states in a decreasing order of programming energy. For example, if a set state is combined with a data value of 0, a programming state provided with one programming interval energy to obtain the set state is combined with a data value of 1. The same applies hereinafter. The combination of data values and programming states may be accomplished in a variety of ways that facilitate specific calculations or user preferences.

  The calculation method of the present invention is suitable when a non-binary calculation medium is used. As used herein, a non-binary computing medium refers to a material having more than two programming states. Binary computing media provides two programming states (typically states are called “0” and “1”), while non-binary computing media provide more than two programming states. For example, a three-state non-binary computing medium provides states called “0”, “1”, and “2”. Higher order non-binary computing media having four or more states may be defined similarly.

  In order to achieve the advantages of non-binary computation, it is necessary to formulate a computational method towards the multi-state nature of non-binary computing media. Addition, subtraction, multiplication, and division methods are described in co-pending US patent application Ser. No. 10 / 144,319. The method of the present invention is directed to number factorization and computation within a modular arithmetic framework. One method of factoring involves determining whether a particular number is another number of multiplicative factors. Another method of factoring involves determining a complete set of a particular number of multiplicative factors. In a preferred embodiment, a certain number of prime factors are determined. Modular arithmetic methods include determining the remainder, calculating the remainder, and classifying the number into a residue class.

Factoring Method One method of factoring involves determining whether a number is a multiplicative factor of another number. This method generally requires that you enter the number to be factored and the latent multiplicative factor, determine if the latent multiplicative factor is indeed a multiplicative factor, and use that determination as the output. return. The method includes defining a programming strategy for a phase change material whose programming state is established according to a latent multiplicative factor. Establishing the programming state includes defining a certain number of programming states for the phase change material so that the number of increments required to move from the reset state to the set state is equal to the latent multiplicative factor. The programming strategy includes specifying the programming energy for each programming state. Here, programming energy is the energy that must be provided to the reset state of the phase change material in order to transform the phase change material to the programmed state. Once the programming energy of each programming state is established, an incremental interval between adjacent programming states is also established and is necessary to transform phase change material from one programming state to the next. The energy increment is known.

  Once the programming strategy is defined for a particular latent multiplicative factor, a determination is made whether the latent multiplicative factor is indeed the number of factors for which factoring is desired. This determination includes resetting the phase change material and incrementing the phase change material a number of times equal to the number for which factoring is desired. This increment may be referred to herein as an increment according to the number for which factoring is desired. If the increment number is equal to the latent multiplicative factor, the phase change material is set. The set state of the phase change material has a lower electrical resistance than other states included in the programming strategy and is easily detected by measuring the electrical resistance. For example, each time the phase change material is incremented, the electrical resistance is measured to determine if the phase change material is in the set state. In the factorization method of the present invention, the phase change material is reset each time the phase change material is set for an increment according to the number for which factoring is desired. When the increment is complete, a determination is made whether the latent multiplicative factor is indeed a factor. At the completion of the increment according to the number that factoring is desired, if the phase change material is in the set state, the latent multiplicative factor is indeed a factor.

  As a result of the programming strategy used by this method, the end of the increment in the set state is a necessary condition for the factor. Each occurrence of a latent multiplicative factor in the number for which factoring is desired results in a phase change material setting. As a result, at the end of the increment, if the phase change material is in the set state, the latent multiplicative factor is a divisor, and thus the number of factors for which factoring is desired. If the latent multiplicative factor is indeed a factor, its cofactor (ie, the number multiplied to obtain the original number to be factored) counts the number of times the phase change material is set during the increment. Is determined by If the phase change material is in a state other than the set state, the latent multiplicative factor is not a factor.

  The factorization method of the present invention provides factorization according to a plurality of latent multiplicative factors. Different multiplicative factors are tested sequentially using one device programmed according to different multiplicative factors. The first multiplicative factor is tested using a first programming strategy based thereon. In that case, the number of increments required to transform the phase change material from the reset state to the set state is equal to the first multiplicative factor. The second multiplicative factor is tested in the same manner using a second programming strategy based on the second multiplicative factor. A computing medium that includes a phase change material provides the ability to program and reprogram a given volume of phase change material according to different multiplicative factors by defining and redefining the programming state. The number of programming states and the energy interval separating the programming states are altered for a given volume of phase change material to accommodate different multiplicative factors. By changing the programming strategy according to the range of latent multiplicative factors of interest, a given volume of phase change material can be used to determine the number factor. Thus, factorization is achieved using a single device that includes a phase change material.

  Multiple volumes of phase change material, each volume programmed according to different multiplicative factors, can also be used to achieve factorization. In this embodiment, multiple devices may be used to achieve factorization. The increment according to the factored number is performed on each of a series of devices programmed according to different multiplicative factors. The same number of increments are applied to each device. Incrementation is performed one increment at a time across all devices, and each device is incremented completely in consecutive full increments from device to device in the column, or increments across all devices and Fully incremented by a combination of multiple increments of individual devices. Similarly, each device has a dedicated energy source that provides an amount of incremental energy as dictated by the device's programming strategy, or variable to each of the plurality of devices as dictated by the programming strategy of each device. One shared energy source with the ability to provide an amount of energy may be used incrementally, or a combination of dedicated and shared energy sources may be used. Embodiments can range from parallel operating modes across all devices to operating modes that consider individual devices in series. By including a sufficient number of devices programmed according to the full range of latent multiplicative factors, all multiplicative factors of the number can be determined. In a preferred embodiment, a device programmed according to a prime factor is included and used to determine the prime factor of the number.

Example 1
In this embodiment, the factorization method described above is used to test whether Equation 4 is a factor of Equation 32. Thus, the factored number is 32 and the latent multiplicative factor is 4. The execution of this method begins with establishing a programming state according to a latent multiplicative factor. In this example, the programming strategy involves establishing the programming state of the phase change material such that four increments transform the phase change material from the reset state to the set state. One way to achieve the programming strategy is to set the set state to a programming state corresponding to 4 increments, the reset state to be a programming state corresponding to an increment of zero, and the three states between the reset state and the set state. Selecting the intermediate state as the programming state corresponding to 1, 2, 3 increments. The three intermediate states may be placed anywhere along the high resistance stable state of the electrical response curve of the phase change material. In the preferred embodiment, the three intermediate programming states are selected such that the incremental intervals between successive programming states are uniform. This embodiment is preferred because the energy source that provides the energy used to increment from one programming state to the next programming state need only provide a certain amount of energy. For example, if energy is provided in the form of energy pulses, a constant energy incremental pulse may be applied to transform the phase change material from one programming state to the next programming state. A non-uniform increment interval is within the scope of the present invention, but is not preferred because the energy provided by the energy source needs to change during the increment.

  FIG. 2 shows one programming strategy associated with the current example. FIG. 2 shows a portion of the electrical response curve of the phase change material, including a high resistance steady state and a steep decrease in resistance coupled to the setting transformation. The embodiment of FIG. 2 shows programming states separated by uniform energy intervals, so the incremental energy from one programming state to the next programming state is constant. The programming state is called with an integer starting from 0 and continuing. The state 0 is a reset state, the state 4 is a set state, and the states 1, 2, and 3 are three intermediate states. The programming state designation may be viewed as an integer data value that is assigned or bound to the programming state. In this example programming strategy, four energy increments are required to transform the phase change material from the reset state to the set state.

  To determine if 4 is a factor of 32, the phase change material having the programming state defined in FIG. 2 is transformed to the reset state and incremented 32 times. Meanwhile, each time the phase change material is set, the phase change material is reset. The provision of one incremental energy transforms the phase change material from programming state 0 to programming state 1. The second increment transforms the phase change material to programming state 2. The same applies hereinafter. Incrementing the phase change material in programming state 3 sets the phase change material to programming state 4. The setting of the phase change material is detected by measuring electrical resistance or other properties that characterize the set state. Electrical resistance is a preferred measurement means. Because it is convenient and makes it easy to identify the set state. The electrical resistance is measured after each increment of phase change material. When a set condition is detected, the phase change material is reset before further increments begin. In this example, when a fifth energy increment is added to the reset state, the phase change material transforms from programming state 0 to programming state 1 and further increments are initiated while resetting as necessary.

  When 32 increments are made, the state of the phase change material is evaluated. If the phase change material is in the set state, the latent multiplicative factor is indeed a factor. In this example, it is concluded that the phase change material is in the set state when incremented 32 times and 4 is a multiplicative factor of 32. The cofactor is also determined by counting the number of times the phase change material is set. In this example, the phase change material is set 8 times. Therefore, in this example, it can be seen that the cofactor of 4 is 8.

In one implementation of the method of Example 1 above, Ge ₂ Sb ₂ Te ₅ was used as the phase change material and incremental energy in the form of electrical energy pulses was used. An electrical energy pulse may be characterized by a pulse voltage or pulse duration. The pulse voltage is also called pulse height or pulse amplitude, and the pulse duration is also called pulse width. At a pulse voltage of 1.5V, applying a pulse having a duration of 320 ns transforms the phase change material from the reset state to the set state. Since 5 programming states (4 programming intervals) were used to test the latent multiplicative factor 4 and the programming states were chosen to be uniformly separated, 1.5V is applied to the energy applied for 80 ns. Programming states separated by the corresponding energy were selected. Application of a pulse of 1.5 ns and 80 ns provides sufficient energy to transform the phase change material from one programming state to the next. For example, applying an 80 ns pulse at 1.5 V to programming state 0 causes the phase change material to transform to programming state 1. The same applies hereinafter.

Example 2
In this example, it is determined whether 7 is a factor of 27. This embodiment may be completed in a manner similar to Example 1 described above. In this example, the number to be factored is 27 and the latent multiplicative factor is 7. Thus, to determine whether 7 is a factor of 27, the phase change material must be programmed according to the latent multiplicative factor 7. A suitable programming strategy is to define the programming state such that seven increments are required to transform the phase change material from the reset state to the set state. The reset state is selected as programming state 0, the set state is selected as programming state 7, and the six intermediate programming states are called 1, 2, 3, 4, 5, 6. The higher number of programming states in Example 2 than in Example 1 means that for a particular phase change material, the average energy interval between programming states is lower in Example 2 than in Example 1. . The appropriate incremental interval is achieved by adjusting the energy so that the energy provided by the energy source matches the desired energy interval that separates the selected programming state. For example, for an energy source that provides energy in the form of current or voltage pulses, the energy content per pulse can be varied by adjusting the pulse height and / or duration.

  After establishing the programming state according to the latent multiplicative factor 7, an increment according to the factored number begins. In this example, 27 increments of phase change material are required. The phase change material is reset. Incrementing the phase change material seven times transforms the phase change material to the set state, at which time the phase change material is reset and further increments and resets occur as needed until 27 increments are provided. In this example, after 21 increments, the phase change material is in the set state and has been reset 3 times since starting the increment. The additional six incremental pulses transform the phase change material to programming state 6. Thus, testing the resistance of the phase change material after 27 increments reveals that the phase change material is not in the set state and concludes that 7 is not a factor of 27. For example, if an integer value is combined into the programming state so that programming state 1 corresponds to the number 1, it can be seen that 6 is the remainder that occurs when 27 is divided by 7. Further explanation of the remainder is given later when describing the modular arithmetic method of the present invention.

Example 3
In this example, multiple volumes of phase change material are used to test multiple latent multiplicative factors. Each volume of phase change material can be thought of as a device computing medium programmed according to different latent multiplicative factors. By incrementing each such device according to the number to be factored, it can be determined which of the plurality of latent multiplicative factors are truly multiplicative factors. As an example, a multiplicative factor of 12 is determined.

  In one embodiment, the 12 multiplicative factor may be determined by including a device programmed according to a latent multiplicative factor having a value of 12 or less. In this embodiment, twelve devices programmed according to equations 1-12 are used, each of which is incremented 12 times. As described above, a device or volume of phase change material programmed according to a certain number, eg, X, requires X increments to be transformed from the reset state to the set state. As previously mentioned, each time a device reaches the set state during an increment, the device is reset prior to further increments. A device that is in the set state when the increment is complete corresponds to a number that is a true multiplicative factor of twelve. In this example, it can easily be seen that the devices programmed according to 1, 2, 3, 4, 6, 12 are in the set state at the completion of the increment and are therefore 12 multiplicative factors. The number of times each of the devices programmed according to the multiplicative factor is set during the increment can be counted to determine a cofactor that is coupled to each multiplicative factor. It is determined that the device programmed according to 5, 7, 8, 9, 10, 11 is in a state other than the set state at the completion of the increment and is therefore not a 12 multiplicative factor.

  Each device of the plurality of devices included in this embodiment may be fully incremented on an individual basis (in serial mode) before the next device is incremented, with further increments to one device. Prior to provision, an increment may be applied to all devices (parallel mode), or a combination of serial and parallel modes may be used in increments. A separate dedicated energy source may be used for each device, and these energy sources may be configured to provide incremental energy as dictated by the device's programming strategy. One energy source that can provide variable incremental energy can be used, and a combination of dedicated and variable energy sources can also be used.

  If the same phase change material is used for each of the plurality of devices, the energy interval separating uniform programming states will be different for devices programmed according to different latent multiplicative factors. For example, a device programmed according to Equation 9 will have a smaller uniform energy interval than a device programmed according to Equation 5. In this case, if energy is provided in the form of a current or voltage pulse with a fixed pulse amplitude, the same number of incremental pulses may be provided to each device. In this case, the pulse width is changed depending on the latent multiplicative factor used to establish the programming strategy for each device.

  Example 3 provides one embodiment directed to qualifying a set of multiplicative factors of numbers and considers multiplicative factors up to (including) the number to be factored. Alternative embodiments are possible.

  In other embodiments, it is recognized that the number is its own factor, 1 is the factor of all numbers, and there are no other multiplicative factors that exceed half the value of the number. For example, in the case of Equation 12, the maximum factor excluding 12 is 6. Thus, by considering a device programmed according to a number starting from 2 and ending with half of the factored number, the number factor is determined more efficiently. In the odd case, it is recognized that 2 is not a factor and the largest factor does not exceed one third of the factored number. Thus, the number of devices that need to be considered is reduced to improve the efficiency of the factorization method.

Factoring Method to Prime Factors The foregoing embodiments and disclosures determine whether one number is a multiplicative factor of another number and determine a complete set or subset of multiplicative factors of a number. A factorization method including this will be described. The present invention includes an associated method in which devices programmed according to prime numbers only are considered. According to number theory, it is known that an arbitrary number can be expressed as a product of only prime numbers. Thus, considering only prime numbers as latent multiplicative factors provides a way to factor numbers into prime numbers. It is possible using the method of the present invention to factorize numbers into prime numbers, identifying which prime numbers are number factors and completely factoring numbers into products containing only prime numbers.

  As described above, each of the plurality of latent multiplicative factors can use one device to change the programming strategy according to different latent multiplicative factors, or use multiple devices to use different latent multiplicative factors. Each device can be tested according to a genetic factor, or by using a combination of these. The method of factoring into prime numbers includes only devices programmed according to prime numbers. Except for being limited to prime numbers, the implementation of the method of factoring into prime numbers is similar to the case of the factorization method described above. Since the number 1 is a factor of all numbers, it is not necessary to consider 1 in particular, considering that 1 is a prime number.

Example 4
In this embodiment, the prime factor of Equation 21 is determined using a plurality of devices. Each of these devices includes a phase change material programmed according to a different prime number. In this embodiment, prime numbers that do not exceed one half of the number to be factored are considered. Thus, in this embodiment, devices programmed according to 2, 3, 5, 7 are considered. The next prime number 11 is greater than half of 21, so a prime number greater than 11 is not a factor of 21.

  To perform 21 prime factorization, the device programmed according to the numbers 2, 3, 5, 7 is incremented 21 times and reset as needed during the increment. A device that is in the set state at the completion of the increment corresponds to a prime factor of 21. Incremental execution in this embodiment indicates that the devices according to 3 and 7 are in the set state at the completion of the increment, and the devices programmed according to 2 and 5 are not in the set state. Thus, it is concluded that 3 and 7 are 21 prime factors.

  If the prime number itself is analyzed according to the embodiment of Example 4, there will be no device in the set state at the completion of the increment. Such a result leads to the conclusion that the original number is a prime number or an equivalent conclusion that the number has only itself and one as a factor. If 13 is the number for which factoring is desired and the procedure described in Example 4 is used, the device programmed according to 2, 3, 5 will be incremented 13 times, and at the completion of the increment, set There will be no devices in the state. Similarly, if a device programmed according to prime numbers up to 13 is used in the method, the conclusion is reached that 13 has only itself and 1 as a factor.

  As described above, the cofactor of each prime factor is determined by counting the number of times a device or volume of phase change material programmed according to the prime factor is set during the increment. The product of each prime factor and its cofactor is equal to the original number for which prime factorization is desired. Even when a device programmed according to a prime number is used, the cofactor may not be a prime number. Thus, further consideration is necessary to fully factorize a number into a product containing only prime numbers.

  These further considerations were not necessary in Example 4 described above. This is because each of the 21 recognized prime factors (3 and 7) had cofactors (7 and 3 respectively) that were also prime numbers. The method described in Example 4 may be used to determine whether a particular prime number of a number is a multiplicative factor, but does not completely factor a number into prime factors. Here, “complete factorization” means identifying a unique set of prime numbers having a product equal to the number to be factored. In Example 4, complete factorization occurred because the product of the recognized prime factors (3 × 7) is equal to the number to be factored (21).

  Fully factoring a number such as 18 into a prime number illustrates the need for consideration beyond that shown in Example 4. In order to factor 18 using devices programmed according to prime numbers in the framework of Example 4, include devices or volumes of phase change material programmed according to prime numbers 2, 3, 5, 7 is required. When the increment of each device is completed according to Equation 18, it is found that only 2 and 3 are 18 prime factors. Since 2 * 3 is not equal to 18, the result in this state is not a complete factorization of 18 to a prime number. (For convenience, the notation * is used herein to mean multiplicative mathematical operations. For example, the product of 2 and 3 is expressed as 2 * 3, 2 * 3 means 2 times 3 .) Instead, the only prime numbers of 18 are 2 and 3, but in the complete prime factorization of 18, the conclusion is that one or both of factors 2 and 3 must appear more than once. Thus, a complete prime factorization of a number includes the qualification of all prime factors and the qualification of the number of times each prime factor occurs in a product of prime numbers equal to the number for which the prime factorization is sought.

  Full prime factorization requires further consideration of the cofactors obtained in the first factorization according to prime numbers. For 18, prime factors 2 and 3 are identified and have cofactors 9 and 6, respectively. As described above, the cofactor is determined by counting the number of setting transformations that occur in the device programmed according to the certified prime factor. Since the cofactor is not a prime number, it is clear that a complete prime factorization of 18 was not achieved when the prime number was first considered. It is necessary to second determine the cofactor of the cofactor obtained for the first recognized prime factor. Since both 2 * 9 and 3 * 6 are equal to 18, the second determination of prime factors is made with respect to either cofactors 9 and 6.

  The second decision is simplified because it is known that the first consideration has qualified all prime factors present in the number to be factored into prime numbers. Thus, the cofactor is factored using a device programmed according to the recognized prime factor. In case of 18, the cofactor 9 or cofactor 6 is selected for further consideration, and this further consideration only needs to include devices programmed according to the certified prime factors 2 and 3. As described above for devices programmed according to 2 and 3, if cofactor 9 is selected and tested, it is found that 2 is not a prime factor but 3 is a prime factor with cofactor 3. Since cofactor 3 is a prime number, factoring 18 into prime numbers is complete, and the result is 18 = 2 * 3 * 3. In this result, 2 is the prime factor recognized in the first determination of the prime factor, 3 is the prime factor recognized in the second determination of the prime factor, and 3 is the cofactor determined in the second determination of the prime factor. is there. Therefore, it can be seen that the prime number 3 exists twice in the prime factorization of 18.

  Determining the cofactor prime factor is performed several times in succession to perform a complete prime factorization of any number of inputs. Prime factorization can be thought of as determining the number of prime factors that are successively smaller. The prime factor of the input number is determined at the first factor or level of factorization, the cofactor of one prime factor identified by the first factor is used in the next factor, and so on. To be discovered. Thus, each level of consideration includes a smaller number of analyzes than the prior level of consideration of the stage scheme. Each decision level is completed with the cofactor found in the previous decision. At the level where the cofactor is found as a prime number, complete prime factorization is achieved. In the case of Equation 18, two consideration levels were included. At the first level of consideration, numbers 2 and 3 were identified as prime factors, and at the second level of consideration, cofactor 9 was further considered and was found to include prime factor 3 with cofactor 3. Since the cofactor 3 is a prime number, only two consideration levels are necessary for the equation 18. More complex cases are analyzed as well.

  An alternative way to complete prime factorization is to compare the product of prime factors obtained after one level of consideration with the input number and see if they are equal. For example, copending patent application US patent application Ser. No. 10 / 144,319 describes a method of multiplication using phase change materials. If the product of prime factors obtained after one level of consideration is equal to the number of inputs, complete prime factorization has been achieved. Otherwise, further consideration is necessary. Further considerations may include other consideration levels of prime factors as described above, or dividing the input number by the product of the prime factors obtained after the initial consideration level. Methods for removing using phase change materials are disclosed, for example, in co-pending patent application US patent application Ser. No. 10 / 144,319. If the result of the division is a prime number, prime factorization is achieved. If not, the result of the division will be further considered by factoring or division and will continue to be considered until prime factorization is achieved. This alternative method may also be used to generally factor numbers into factors that include non-prime numbers.

  To illustrate the above, consider the example of prime factorization of number 18. As described above, after one level of factorization, 2 and 3 are identified as 18 prime factors. Multiplication of 2 and 3 yields 6. Since 6 is not equal to 18, further consideration is necessary to complete the prime factorization. According to the division method described above, the original number 18 is divided by 6 to obtain 3. Thus, the result of the division is that 3 is another factor of 18. Since 3 is a prime number, one execution of the division method completes the prime factorization, resulting in 18 being factored into 2 * 3 * 3.

  In the factorization and prime factorization methods disclosed herein, it is often desirable to determine a factor or a cofactor of a prime factor. As previously mentioned, the cofactor of the factor may be determined by counting the number of times the volume of phase change material programmed according to the factor is set while being incremented according to the input number. Setting transformation counting may be accomplished using an external counter or by incrementing the counting register by one each time a phase change material is set. A counting register containing a phase change material is described in co-pending patent application US patent application Ser. No. 10 / 144,319. The counting register may have the same or different number of programming states as the volume of phase change material used as a computing medium by the method of the present invention. For a certain number of factorizations, the cofactor may be a large number and may exceed the number of programming states available in the counting register. One way to deal with such cofactors is to use multiple counting registers. In that case, each of several registers may be used to store one digit of the multi-digit cofactor. Multi-digit number storage is described in co-pending US patent application Ser. No. 10 / 144,319.

Modular Arithmetic Method In an example such as Example 2 described so far, if it is determined that the latent multiplicative factor is not a factor, the result of the factorization method is still useful interpretively. A mathematical subsection known as modular arithmetic takes into account the remainder that results when one number is divided by another. As described above, factorization is achieved when the remainder is zero. Non-zero remainder does not indicate factorization but is important in modular arithmetic calculations. Modular arithmetic is described in many mathematical textbooks and is described in part below, to provide background information and context for the modular arithmetic methods disclosed herein.

  The remainder is an important quantity in modular arithmetic. In this specification, the remainder means the number that remains when one number is divided by another number. For example, when 19 is divided by 7, a remainder of 5 is obtained. For example, when 34 is divided by 9, a remainder of 7 is obtained. The general result from number theory, known as divisor, provides a remainder whenever an integer is divided by another integer. According to the division method, an arbitrary integer a (dividend) is represented by a = q * b + r in relation to the integer b (divisor). Here, q is an integer (quotient), and r is a non-negative integer having a value smaller than the magnitude of the divisor b. According to number theory, q and r are unique for a given dividend and divisor.

  In modular arithmetic, integers are expressed in remainder with respect to the modulus of the arithmetic system. The modulus is similar to the divisor in the division context. For example, the remainder on division by 7 may be used to define an arithmetic system with a modulus equal to 7. Such an arithmetic system is also referred to as a modulo 7 system. In a modulo 7 system, the integer is represented by the remainder on the division by 7. In the above example where 19 is divided by 7, a remainder of 5 has been determined. Thus, in a modulo 7 system, number 19 is represented by a remainder of 5. This result may be further expressed in the form of a joint relationship 19≡5 mod 7. Here, the mathematical symbol ≡ represents congruence. Similar considerations apply to arithmetic systems with arbitrary moduli, and general congruence relations may be written in the form a≡r mod m. Here, m is the modulus, a is an integer, and r is the remainder of the result when a is divided by m. The remainder r is also called congruence of the integer a in the modulo m system. The remainder r can also be considered as a remainder generated when the dividend a is divided by the divisor m.

  In a modulo m system, the remainder r must have an integer value from 0 to m-1, and all integers a must have r's allowed value of 1. For example, in a modulo 5 system, r is 0, 1, 2, 3, 4 and all integer values a have a remainder r selected from 0, 1, 2, 3, 4. Examples of congruence relationships in modulo 5 include 19≡4 mod 5, 22≡2 mod 5, 46≡1 mod 5, and the like. A given value of the remainder r is necessarily combined with several values of the integer a. For example, 19≡4 mod 5, 54≡4 mod 5, 799≡4 mod 5, etc. Integers having the same remainder b in a modulo m arithmetic system are called the same equivalence class or remainder class. For example, 19, 54, 799 are the same equivalence classes in a modulo 5 system. A complete equivalence class includes all integers that have the same remainder in a particular modulo system. The number of different equivalence classes in a modulo m system is m. For example, a modulo-3 system has three equivalence classes defined by three possible residue values 0, 1, and 2.

  The determination of the remainder r is one of basic calculations in modular arithmetic. As illustrated in Example 2 above, the multi-state computing medium may be used to provide a remainder. As previously mentioned, phase change materials are preferred multi-state computing media in the context of the method of the present invention. When incrementing is completed during the execution of one factorization method described above, it is known that if the phase change material is not in the set state, the programmed number of phase change materials is not a factor of the original input number. . At the completion of the increment, the remainder may be determined by reading the phase change material and determining what programming state it is in. In Example 5 below, an example of a method for determining the congruence of input numbers in a modular arithmetic system is provided.

Example 5
In this example, 16 congruences in the modulo 6 system are determined. This calculation is also called a joint calculation. In this joint calculation, the amount r of the joint relation 16≡r mod 6 is obtained. To complete the joint decision, the phase change material is programmed according to modulus 6. As described above, programming is accomplished through a strategy, where the programming state is defined such that six increments are required to transform the phase change material from the reset state to the set state. The reset state is coupled to zero increments, the set state is coupled to six increments, and additional states may be defined along the high resistance stable state corresponding to 1, 2, 3, 4, 5 increments. . The energy separation between states may be uniform or non-uniform. A phase change material so programmed may be referred to as a modulo 6 phase change material.

For the purpose of modular arithmetic, integer values are combined with programming states to facilitate interpretation of the state of the phase change material. The reset state is coupled to the value 0, and the programming state corresponding to one increment is coupled to the value 1. The same applies hereinafter. Since the modulo 6 system provides a remainder of 5 or less, the set state (in this example, the programming state corresponding to 6 increments) is appropriately coupled to the value 0. (If the set state is bound to 6, the meaning of the phase change material in the set state is interpreted as 6 mod 6. Since 0 is congruent to 6 mod 6, collisions by binding the value 0 to the set state are (This coupling is consistent with using the set state as a factorization criterion as described above.)
Once programmed according to the desired modulus, the phase change material is used to determine any congruence within the programmed modulus. In this example, 16 congruences in a modulo 6 system must be determined. The decision proceeds by incrementing the phase change material according to the number for which the joint relationship is sought. In this example, 16 increments are provided to the phase change material, which is reset each time it is transformed to the set state. The phase change material is transformed to the reset state prior to incrementing, if necessary. As described above, the electrical resistance of the phase change material is measured during increments to evaluate when the phase change material is transformed to the set state. In this example, application of six energy increments transforms the phase change material from the reset state to the set state, at which point the phase change material is reset prior to further increments. In this example, application from increments 7 to 12 transforms the phase change material to the second set state. The phase change material is reset again and the last four of the required 16 increments are provided to the phase change material. Increment 13 transforms the phase change material to the programming state coupled to value 1, increment 14 transforms the phase change material to the programming state coupled to value 2, and increment 15 is the programming coupled to value 3. Transform the phase change material to a state, and increment 16 transforms the phase change material to the programming state coupled to the value 4. Thus, upon completion of the required 16 increments, the phase change material is in the programming state coupled to the value 4. This last programming state provides the desired remainder or congruence, and it is concluded that the congruence relationship determined in this example is 16≡4 mod 6.

  The calculation of congruence in Example 5 requires the determination of the programming state in which the phase change material is present when the increment is complete. This last programming state is determined by reading the phase change material. If the phase change material is in the set state at the end of the increment, the reading is achieved by measuring the electrical resistance. In this case, the remainder is 0 and the congruence is completed by making r equal to 0. This case corresponds to the case where the modulus m is a multiplicative factor of a.

  When the modulus m is not a multiplicative factor of a, the phase change material is in the last programming state other than the set state at the end of the increment. This last programming state may be determined by reading the phase change material. This reading is described in co-pending patent application US patent application Ser. No. 10 / 144,319, which requires incrementing the phase change material from the last programming state achieved during the execution of the joint calculation. is there. The last programming state may be determined by (1) incrementing until the phase change material is set, (2) counting the required number of increments, and (3) subtracting the required increments from the modulus of the calculation. For example, in Example 5 above, two increments are required to transform the phase change material from the last programming state to the set state. The subtraction of 2 from the modulus 6 gives 4. Therefore, it is concluded that the last programming state is the programming state coupled to Equation 4, and the remainder obtained by the joint calculation of Example 5 is 4. In one embodiment described in copending patent application US patent application Ser. No. 10 / 144,319, the counting included in step 2 includes the step of each of the energy provided to the last programming state of the phase change material. This is accomplished by incrementing the counting register once for the increment. In this embodiment, assuming that the counting register has the same number of programming states as the phase change material used in the joint calculation, the difference represented by the subtraction of step 3 sets the counting register. Corresponds to the number of increments required. In a preferred embodiment, the counting register includes a phase change material programmed according to the modulus used in the joint calculation.

  As described above, the congruence calculation for the modulus m separates all sets of integers into congruence classes or remainders that have a common remainder for division by m. For example, in a modulo 5 system, there are five remainder classes corresponding to the remainders 0, 1, 2, 3, 4 and all integers are included in only one of these remainder classes. For example, the residue class 2 has 2, 7, 12, 17,. . . , And the residue class 3 is 3, 8, 13, 18,. . . including. The same applies hereinafter. The method of the present invention for determining the congruence of input numbers in a modular arithmetic system is applied to each of several input numbers and is used to classify a set of numbers into a residue class.

  Addition in a modular arithmetic system is also completed using the phase change material according to the present invention. The modular arithmetic system is closed in terms of addition, and adding one element of the modular arithmetic system to the other elements of the modular arithmetic system necessarily provides a result that is also a member of the modular arithmetic system. As an example, consider the modulo 5 system described above. The modulo 5 system has the numbers 0, 1, 2, 3, 4 as members. As mentioned above, these numbers may be considered as possible remainders or residue classes existing in a modulo-5 arithmetic system. Addition in modular arithmetic is completed in a manner similar to normal decimal addition, with the exception that the result is limited to members of the modular arithmetic system. Adding 1 in the modulo 5 system to 2 in the modulo 5 system equals 3 in the modulo 5 system. This addition is expressed as 1 mod 5 + 2 mod 5 = 3 mod 5.

  Special consideration is needed when the sum of the three members of the modular arithmetic system is equal to or exceeds the modulus of the modular arithmetic system. For example, in a modulo 5 system, the addition of 3 and 4 requires proper processing. This is because the expected result 7 in normal decimal arithmetic exceeds the modulus 5 of the modular arithmetic system. These cases are handled appropriately by converting the expected decimal result to a modulo-5 equivalent. Therefore, 3 mod 5 + 4 mod 5 = 7 mod 5 = 2 mod 5. Result 2 is a member of the modulo-5 arithmetic system and is therefore a reasonable result. General decimal addition in the framework of modular arithmetic systems is similarly accomplished by converting decimal numbers to modulo equivalence and adding. As an example, the addition of 19 and 37 in a modulo 5 system is added as 19 mod 5 + 37 mod 5 = 4 mod 5 + 2 mod 5 = 6 mod 5 = 1 mod 5 The calculation is completed by first adding the decimal numbers and then converting the result to the modulo-5 equivalent. That is, 19 mod 5 + 37 mod 5 = 56 mod 5 = 1 mod 5

  Addition in a modular arithmetic system is easily accomplished by the present invention by programming the phase change material according to the modulus of the modular arithmetic system, resetting the phase change material, and incrementing the phase change material according to the number to be summed. . For example, in the above example where 19 and 37 are added in a modulo-5 arithmetic system, one may start by programming the phase change material according to modulus5. As described above, this programming includes defining the programming state such that five increments are required to transform the phase change material from the reset state to the set state. Thus, a total of six programming states are defined including a reset state, a set state, and four intermediate states. Addition is achieved by the phase change material being reset, incrementing according to each of the two numbers to be summed, and resetting as necessary as described above each time the phase change material is transformed to the set state. Is done. In this example, the phase change material is first incremented 19 times to provide programming state 5 (programming state corresponding to equation 4), and then the phase change material is programmed state 2 (programming state corresponding to equation 1). Incremented 37 times to provide a final state.

  Addition in a modular arithmetic system can also be considered in relation to residue classes. This is because it is the remainder that ultimately determines the sum of two numbers in modular arithmetic. In one example described above, it was found that 1 mod 5 + 2 mod 5 = 3 mod 5. This result can be interpreted in relation to the residue class as follows. That is, the sum of the number of residue classes 1 and the number of residue classes 2 inevitably generates the number of residue classes 3 in the modulo 5 system. As an example, 16 is the remainder class 1, 32 is the remainder class 2, and the sum 16 + 32 = 48 is the remainder class 3. Addition may be interpreted in the same way with other cosets and other modular arithmetic systems.

  The disclosure set forth herein is illustrative and is not intended to limit the practice of the invention. Many equivalents and minor variations are assumed to be within the scope of the invention. Limiting the scope of the invention are the following claims, including all equivalents, combined with the previous disclosure.

FIG. 6 is a plot of the electrical resistance of a representative phase change material as a function of provided energy or power. This plot includes the high resistance stable state before setting on the left side and the post-setting region on the right side. For purposes of illustration, representative states of both regions are shown. FIG. 4 is a portion of an electrical resistance plot of a phase change material used to test whether number 4 is a multiplicative factor of an input number, as described in Example 1 herein. The electric resistance in the high resistance stable state before setting and in the set state is shown. The names of the five programming states are indicated by symbols.

Claims (26)

A method of factoring the number of inputs,
Providing a volume of a digital multi-state phase change material, wherein the phase change material has a plurality of states, the state being applied by applying an amount of energy corresponding to a reset state, a set energy of the reset state; A set state obtained from a reset state, the set state having a lower resistance than the reset state, and one or more intermediate states having substantially the same resistance as the reset state, wherein the set of the reset state One or more intermediate states obtained from the reset state by applying an amount of energy less than energy,
Providing a latent multiplicative factor of the input number;
Programming the phase change material according to the latent multiplicative factor, the programming comprising defining a programming state, wherein the programming state is selected from the plurality of states of the phase change material, and the programming state is the reset state And the set state, wherein the number of programming states is one greater than the latent multiplicative factor,
Transforming the phase change material to the reset state;
A. Incrementing the phase change material, the increment comprising providing sufficient energy to transform the phase change material to a different one of the programming states;
B. Repeating incremental step A until the phase change material is transformed to the set state,
C. Resetting the phase change material;
Repeating the steps A, B, C until the number of increments of the phase change material is equal to the number of inputs.   The method of claim 1, wherein the incremental step A further comprises measuring a resistance of the phase change material.   The method of claim 2, wherein the measurement is completed after the phase change material is transformed to the different one of the programming states.   Further, the method includes determining whether the latent multiplicative factor is a multiplicative factor, the determining step determining the resistance of the phase change material after the phase change material is incremented a number of times equal to the input number. The method of claim 1, comprising measuring.   The method of claim 1, further comprising counting the number of times the phase change material has been transformed to the set state.   The phase change material is one or more elements selected from the group consisting of In, Ag, Te, Se, Ge, Sb, Bi, Pb, Sn, As, S, and P, or a mixture or alloy thereof The method of claim 1 comprising:   The method of claim 1, wherein the phase change material comprises a chalcogen.   The method of claim 7, wherein the chalcogen is a mixture of Te and Se.   The method of claim 7, wherein the phase change material further comprises Ge.   The method of claim 7, wherein the phase change material further comprises Sb.   The method of claim 7, wherein the phase change material further comprises an element of a transition metal.   The method of claim 11, wherein the transition metal is selected from the group consisting of Cr, Fe, Ni, Nb, Pd, and Pt.   The method of claim 1, wherein the programming states are separated by uniform energy intervals.   The method of claim 1, wherein the incremental step A is accomplished by providing energy in the form of electrical energy.   The method of claim 14, wherein the electrical energy comprises a current or voltage pulse.   The method of claim 1, wherein the incremental step A is accomplished by providing energy in the form of light energy.   The method of claim 1, wherein the incremental step A is accomplished by providing energy in the form of thermal energy. A method of factoring the number of inputs,
Providing a plurality of volumes of digital multi-state phase change material, each of the volumes of phase change material having a plurality of states, wherein the state is an amount of energy corresponding to a reset state, the set energy of the reset state A set state obtained from the reset state by applying a set state having a lower resistance than the reset state, and one or more intermediate states having substantially the same resistance as the reset state, Including one or more intermediate states obtained from the reset state by applying an amount of energy less than the set energy of the reset state;
Providing a plurality of latent multiplicative factors of the input number;
Applying the method of claim 1 to the volume of phase change material, each of the volumes of phase change material being programmed according to a different one of the latent multiplicative factors.   The method of claim 18, wherein each of the plurality of latent multiplicative factors is a prime number.   Further, determining which of the prime numbers is a multiplicative factor of the input number, the determining step comprising measuring an electrical resistance of the volume of the phase change material programmed according to the prime number. The method of claim 19.   The method further comprises calculating a cofactor of a prime number determined to be a multiplicative factor of the input number, wherein the calculating step includes the multiplicative factor of the prime number during the step of applying the method of claim 1. 21. The method of claim 20, comprising counting the number of times the volume of phase change material programmed according to is set.   The method of claim 21, further comprising: factoring the cofactor according to the method of claim 18.   The method of claim 21, further comprising factorizing the cofactor according to the method of claim 19. A method for determining congruence of input numbers in a modular arithmetic system,
Providing a volume of a digital multi-state phase change material, wherein the phase change material includes a plurality of states, the state being reset by applying an amount of energy corresponding to a reset state, the set energy of the reset state A set state obtained from a state, the set state having a lower resistance than the reset state, and one or more intermediate states having substantially the same resistance as the reset state, wherein the set state of the reset state Including one or more intermediate states obtained from the reset state by applying an amount of energy less than energy;
Providing the modulus of the modular arithmetic system;
Programming the phase change material according to the modulus, the programming includes defining a programming state, wherein the programming state is selected from the plurality of states of the phase change material, the programming state being the reset state and the Including set states, the number of programming states being one more than the modulus,
Transforming the phase change material to the reset state;
A. Incrementing the phase change material, the increment comprising providing sufficient energy to transform the phase change material to a different one of the programming states;
B. Repeating incremental step A until the phase change material is transformed to the set state,
C. Resetting the phase change material;
Repeating steps A, B and C until the number of increments of the phase change material is equal to the number of inputs,
Reading the phase change material, the reading comprising:
Counting the number of increments required to transform the phase change material into the set state;
Subtracting the number of increments from the modulus. A method of adding a first number and a second number in a modular arithmetic system,
Providing a volume of a digital multi-state phase change material, wherein the phase change material has a plurality of states, the state being applied by applying an amount of energy corresponding to a reset state, a set energy of the reset state; A set state obtained from a reset state, the set state having a lower resistance than the reset state, and one or more intermediate states having substantially the same resistance as the reset state, wherein the set of the reset state One or more intermediate states obtained from the reset state by applying an amount of energy less than energy,
Providing the first number and the second number;
Providing the modulus of the modular arithmetic system;
Programming the phase change material according to the modulus, the programming includes defining a programming state, the programming state being selected from the plurality of states of the phase change material, wherein the programming state is the reset state and the set The number of programming states is one more than the modulus,
Transforming the phase change material to the reset state;
A. Incrementing the phase change material, the increment comprising providing sufficient energy to transform the phase change material to a different one of the programming states;
B. Repeating incremental step A until the phase change material is transformed to the set state,
C. Resetting the phase change material;
Repeating steps A, B, and C until the number of increments of the phase change material is equal to the first number,
D. Incrementing the phase change material, the increment comprising providing sufficient energy to transform the phase change material to a different one of the programming states;
E. Repeating the incremental step D until the phase change material is transformed into the set state,
F. Resetting the phase change material;
Repeating the steps D, E, and F until the number of increments of the phase change material is equal to the second number. And reading the phase change material, the reading comprising:
Counting the number of increments required to transform the phase change material into the set state;
26. The method of claim 25, comprising subtracting the number of increments from the modulus.

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