Scalable true random number generator using adiabatic superconductor logic

Table 1 describes the truth table of an XOR gate, where a And b are the inputs and x (=ab) is the output. From now on a And b they are assumed to be uncorrelated random bits (a random bit becomes 0 or 1 stochastically with equal probability).

Table 1 Truth table of an XOR gate.

Logical point of view

More importantly, XORing two random bits results in another random bit34 as follows: a And b are random bits, so the four possible input combinations [(A, B)  {(0, 0), (0, 1), (1, 0), (1, 1)}] appear randomly. consequentially, x becomes 0 or 1 randomly (i.e., x is also a random bit), since x includes the same number of 0s as the number of 1s in the truth table. Here we discuss the correlation regarding a, bAnd x calculating mutual information35, which quantifies the correlation between probability variables. Mutual information between a And x it is given by

$$Ileft(A;Xright)=Hleft(Aright)+Hleft(Xright)-Hleft(A,Xright)$$

(1)

where is it h(a) And h(x) are the logical entropy (i.e. the Shannon entropy of logical states)35 or a And xrespectively, and h(a, x) is the joint logical entropy of a And x. h(a) it is given by

$$Hleft(Aright)=-sum_{a}Pleft(aright){text{ln}}Pleft(aright)$$

(2)

where is it a takes on a value a with the probability P(a). According to table 1, a{0, 1} and P(0) = P(1) = 0.5, which givesh( a) = ln2. Likewise,h(x) = − ΣxP(x) lnP( x) = ln2, eh( a,x) = − Σa,xP( a,x) lnP( a, x) = 2ln2. Consequentially,I( a; x) = ln2 + ln2 − 2ln2 = 0, which indicates this a And x are not related to each other, i.e. the value of cannot be said a from a given value of x, and viceversa. likewise,I( b; x) = 0 andI( a; b) = 0. Therefore, there is no correlation between any pair of a, bAnd x. However, a, bAndx are related since, if you know the values ​​of any two of the three ( a, bAnd x), one can tell the value of the other. This is quantified by the mutual information between a, bAndx as follows:

$$Ileft(A;B;Xright)=Hleft(Aright)+Hleft(Bright)+Hleft(Xright)-Hleft(A,B right)-Hleft(A,Xright)-Hleft(B,Xright)+Hleft(A, B,Xright)$$

(3)

h(a) =h( b) =h( x) = ln2, eh( a,b) = h(a, x) = h( b,x) = h(a, b,x) = 2ln2. consequentially, I( a; B; x) = − ln2. The above discussion indicates that an XOR gate can augment two unrelated random bits ( a And b) to uncorrelated random three bits (a,bAndx), where the correlation appears only when all ofa,bAnd x are considered together.

thermodynamic point of view

In physical systems, random number generation is related to thermodynamics because logical entropy is related to (thermodynamic) entropy: in the quasi-static limit, Δh = ΔSt = βQ35.36where Δh is the logical change in entropy of the system, Δ St is the entropy change of the system, β is the inverse temperature, eQis the heat absorbed by the system. For example, a TRNG AQFP27 generates a random bit (that is, Δh= ln2) increasing entropy through heat absorption (i.e., ΔSt= βQ= ln2)25. Therefore, we explore random number generation using XOR gates from a thermodynamic perspective.

We first derive the thermodynamic relations for a logic gate with two uncorrelated random inputs (aAndb) and an output (x). From EQ. (3), the total change in logic entropy during a logic operation is given by

$$Delta Hleft(A, B,Xright)=Delta Hleft(A,Bright)+Delta Hleft(A,Xright)+Delta Hleft(B ,Xright)-Delta Hleft(Aright)-Delta Hleft(Bright)-Delta Hleft(Xright)+Delta Ileft(A;B;X right)$$

(4)

The inputs do not change during a logical operation, so Δ h(a) = Δh(b) = Δh( a,b) = 0. Also, the total logical change in entropy is related to heat absorption. Thus, in the quasi-static limit (that is, assuming that the logical operation is performed without energy dissipation), Eq. (4) becomes

$$Delta Hleft(A, B,Xright)=Delta {H}_{text{eff}}left(Xright)+Delta Ileft(A;B;X right)=beta Q$$

(5)

where Δheff( x) = Δh(a, x) + Δh(b, x ) – Δ h ( x) is the logical effective entropy change of x ; d heff( x ) becomes ln2 when xis a random bit that is uncorrelated withaorb. Conventional logic gates operate deterministically and do not include entropy-increasing processes such as heat absorption; Like this, Q= 0 and Eq. (5) boils down to

$$Delta {H}_{text{eff}}left(Xright)=-Delta Ileft(A;B;Xright)$$

(6)

This equation shows that even if a logic gate does not include entropy-increasing processes, the logic gate can generate a random bit by producing information about each other.

Figure 1 shows the change in logical entropy and mutual information related to an XOR gate. a Andbare random bits, so that h(a) = h(b) = ln2. In the initial state (Fig. 1a), xis not generated yet and therefore h(x) = 0, which results in h( a, x) = h( b , x ) = ln2 eI( a; b; x) = 0. In the final state (Fig. 1b), xis calculated by a And b: h(x ) = ln2. As mentioned above, any pair of a,bAnd xare not related to each other, but a,bAnd xthey are related; Like this, h(a,x) =h(b, x) = 2ln2 e I( a; b; x ) = − ln2. Therefore, Δheff( x ) = -Δ I(a;b;x) = ln2, which indicates that the XOR gate generates a random bit yielding information about each other, and that XOR-based random number generation agrees with thermodynamics.

Figure 1

Logical entropy and mutual information on an XOR gate. (a) initial state, where the outputxit was not generated. (b) final state, wherex(a random bit) is generated by producing information about each otherI(a;b;x).

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