Primitive Computer Functions

Transcription

1 Primitive Computer Functions Basic computer devices are currently binary switching devices. These devices are often called logic devices because their switching behavior can be described by logic or truth tables or by what is often called Boolean logic. A basic logic device is called a binary device, because inputs and outputs have one of 2 values. For convenience the values are named 0 and 1. The 0 and 1 represent a state of an input or an output or a state of a device. In real life there are no 0s and 1s in a circuit, but only signals, which have physical states HIGH and LOW. The actual devices switch between output states based on input states. A LOW signal may be Ground and a HIGH signal may be 5 Volt. Or more detailed LOW may be between 0 and 0.4Volt and HIGH between 2.8-5Volt. It is in physical realization not required to represent a logic state by a Voltage. In the early days of logic circuit design, as developed by Claude Shannon in his 1938 MIT M.Sc. thesis, logic states were represented by impedance of electrical relays. Furthermore, logic state 0 is often assumed to represent a low voltage and state 1 a high voltage. However, both positive logic and negative logic circuits are also physically used. Thus, it is the physical realization of logic state representation that ultimately determines how signals are further applied. The striking aspect of current switching technology is that even different underlying technology ultimately is applied to represent pretty much standard logic primitive functions in cryptography. (usually XOR and AND gates). The term logic is now so common that computer circuits are called logic circuits and are said to perform logic. However circuits merely process signals and do not perform logic. The term logic herein is merely descriptive and a model of what takes place in a circuit. How the description of a switching device relates to the physical switching is determined by physical interfaces between inputs and outputs of the switching devices. Commonly, the symbols 0 and 1 are assigned to input and output signals and are processed in accordance with that representation and the corresponding logic states of a device. However, that is not a requirement. This can be shown, for instance, in Matlab and other programming languages that process matrices in origin-1. Page 1 of 5

2 Binary Circuit Representation The following figure shows a representation of discrete switching circuits. Monadic Computer Functions of Inverters The left device shows an inverter, which performs in this case a monadic (one operand) operation: one input and one output. In general the binary inverter inverts an input: it changes a 0 into a 1 and a 1 into a 0. One notation is [0 1] [1 0]. This is illustrated in the figure below: The ordered states between brackets on the left of the arrow indicate input states and the states between brackets on the right show the corresponding output states. The position of a state between brackets indicates its relation to its corresponding input state. Positions are counted (in the above example) as starting at position 0. While commonly only a binary inverter [0 1] [1 0] is considered, there are in fact 3 more binary inverters: [0 1] [0 1] (Identity); [0 1] [0 0] (Always Off); and [0 1] [1 1] (Always On). Accordingly there are four 2-state inverters of which 2 (=2!) are reversible or invertible. A primitive monadic binary or 2-state computer function is the inverter [0 1] [1 0]. Dyadic Primitive Computer Functions There are 16 different binary (or 2-state) primitive dyadic (2 operand) computer functions. The following figure shows 4 truth tables of them: the AND, the NAND, the XOR and the EQUAL function. AND 0 1 NAND 0 1 XOR 0 1 EQ Page 2 of 5

3 The input in1 is provided by the top line of a table and input in2 by the left column. The inputs (in1,in2) form the address or determination of the output state of out. The AND function, in (0,1) notation, is identical to modulo-2 multiplication. The XOR function, in (0,1) notation, is identical to modulo-2 addition. It should be clear, from earlier description, that the actual circuits, like the XOR, do not perform a mathematical operation. They only switch between states and the similarity due to naming or representation may make it look like a modulo-2 operation. Properties of primitive computer functions Primitive computer functions have certain properties that do not depend on the representation of states (such as by 0 and 1 in the binary case.) Some of the properties are hidden and only show up in N-state cases with N>2. Monadic Primitives For instance in the case of binary inverters, only two of the four possible inverters are reversible of which one is the identity. In the 3-state case there are 27 different inverters of which 3! or 6 are reversible inverters: [0 1 2] [0 1 2] (Identity) (self reversing) [0 1 2] [0 2 1] (self reversing) [0 1 2] [1 0 2] (self reversing) [0 1 2] [1 2 0] (complete) [0 1 2] [2 0 1] (complete) [0 1 2] [2 1 0]. (self reversing) Of the 6 reversible inverters there are 2 complete reversible inverters. That is: a reversible inverter which does not transform any state to itself. Of the 6 reversible 3-state inverters 4 are self-reversing. That is applying these inverters twice will generate Identity. Other properties can be identified. For instance, some of the inverters may be considered to be 3-state multipliers if one considers the representation as 0,1 and 2 as actual values. For instance, the multiplier with a factor 2 modulo-3 is the inverter [0 1 2] [0 2 1]. (as 2*0 mod-3=0; 2*1 mod-3=2 and 2*2 mod- 3=1). One can see that beyond the binary case, there are many n-state inverters that are primitive computer functions that can be applied. One such application is the Finite Lab-transform (FLT) as explained elsewhere. Page 3 of 5

4 Dyadic Primitives N-state dyadic primitive operations have two operands and can be represented by an n-by-n matrix, often called a truth table. For convenience the operand values or input values are enumerated as 0, 1, 2,..., n-1, which is called representation in orgin-0. These truth tables, or switching function tables, when they pertain to computer functions have certain properties that are useful and are often independent of representation as 0, 1, etc. For instance, one can call a switching function table a matrix scn. The inputs in1 and in2, which in the binary case are 2-state inputs, to a device characterized by scn generate a 2-state output out. The output out of such a device is characterized by expression out=scn(in1,in2). If, for instance, scn is the truth table of an XOR device then XOR(0,0)=0; XOR(0,1)=1, etc. There are several properties of scn that are useful, such as: 1) commutatitivity or scn(a,b)=scn(b,a) in the binary case there are 8 commutative primitive functions 2) reversibility or: when c=scn(a,b) then b=scn -1 (a,c) and a=scn -1 (c,b) in the binary case there are two reversible functions XOR and EQ, which both are in effect self-reversing, or when c=scn(a,b) then a=scn(c,b). 3) associativity or: t1=scn(a,b) and out1=scn(t1,c); and t2=scn(a,c) and out2=scn(t2,b) and out1=out2. In the binary case there are 6 cases wherein the primitive functions are commutative and associative. 4) a zero-element z exists for which scn(a,z)=z for all a.. The AND function has element 0 as zero-element. The OR function has 1 as the zero-element. 5) a one-element e exists for which scn(a,e)=a for all a. The AND function has 1 as oneelement and the OR function has 0 as one-element. One can see that certain properties that are attributed to common functions such as the AND function, also exist in other binary functions. When N is greater than 2 There are 19,683 3-state dyadic primitive computer functions. There are over 4 billion 4-state dyadic primitive computer functions. There are over 6* state dyadic primitive computer functions. Page 4 of 5

5 The numbers grow exponentially with N. It becomes extremely difficult to find a dyadic n-state primitive computer function that has one or more desirable properties from these absurdly large numbers of possible functions. One may suspect that there are untold numbers of desirable functions for large values of N, but they are close to impossible to find. The fallback in real-life applications (such as in cryptography) is to use a known function, of which there are very few. There are known functions that one can use, just not many. The two best known primitive computer functions are the addition and multiplication modulo-n, which are widely used in cryptography. Other known primitive functions are the addition over Finite Field GF(n=2 k ), which is formed by bitwise XOR-ing of 2 words of k bits; and the multiplication over Finite Field GF(n=2 k ) which is a bitwise multiplication of two k-bit words modulo a (k+1) bit word, which is commonly represented by a primitive polynomial of degree k. That is it. In this universe of billions and trillions of primitive computer functions, only very few are used. The Finite Lab-transform (FLT) is a deterministic way to generate a novel and basically unknown N-state primitive computer function from a known primitive. Realization of Primitive Computer Functions Primitive computer functions are descriptions or models of basic n-state discrete computer devices. The best known of these basic devices are the binary switching devices. A primitive computer function can be realized by and in a combinational circuit. There are different ways to do that, as is well known. One may also realize a basic computer function by storing its truth table on an addressable memory, wherein the two inputs form the address that stores and provides the output. There is functionally no difference between these realizations. A third way is to use embedded computer circuitry, like ALU or co-processor or FPGA based circuitry, to execute primitive functions in a rule-based manner. The last realization is especially useful for large values of N and may apply BigInteger procedures. It is not necessary that an N-state signal is physically one of N voltages. A word of k bits that has one of N different states is as much an N-state signal as a signal that has one of N different discrete voltage values. Peter Lablans February 26, 2019 Page 5 of 5