# 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).

### 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.