ADC采样真随机数发生器的熵估计方法研究
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"基于ADC采样的真随机数发生器的熵估计"
在信息安全和加密系统中，真随机数发生器（TRNGs）扮演着至关重要的角色。这些TRNGs生成的随机数是许多密码算法和协议安全性的基石。当前，评估特定TRNG结构的安全性时，基于随机模型的熵估计被认为是一种推荐的方法。此外，生成速率也是TRNG性能的重要指标，为了提高比特率，通常会使用模拟数字转换器（ADC）来采样噪声信号。
然而，针对ADC采样进行的熵估计尚未成为研究焦点。这篇研究论文，"基于ADC采样的真随机数发生器的熵估计"，由Yuan Ma、Tianyu Chen、Jingqiang Lin、Jing Yang和Jiwu Jing共同撰写，并发表在2019年11月的IEEETRANSACTIONSONINFORMATIONFORENSICSANDSECURITY第14卷第11期上。
论文中，作者提出了一种新的熵估计方法，专门针对使用ADC采样的TRNG。他们通过扩展现有的模型，设计了一个等效模型来估计由ADC采样获得的转换样本中任意单个比特的熵。这种方法旨在更准确地评估ADC采样过程中产生的随机性，从而确保生成的随机数具有足够的不可预测性和均匀性，这是衡量TRNG安全性的重要标准。
在传统的随机数生成中，熵是衡量信息不确定性的一个度量，高熵意味着更高的随机性。对于一个理想的TRNG，其输出应具有最大可能的熵，这样才能保证生成的随机数序列难以预测，符合密码学的安全需求。ADC的引入可以显著提高采样速率，但同时也带来了新的挑战，即如何在高速采样下保持随机数的质量。
论文进一步探讨了在不同参数设置下，ADC采样对熵的影响，以及如何调整这些参数以优化TRNG的性能。此外，作者还可能进行了实验验证，以证明他们的熵估计方法的有效性和准确性，这通常包括与现有方法的比较，以及对实际系统中生成的随机数序列的统计分析。
这篇研究对于理解ADC采样在TRNG中的应用，以及如何对其进行有效的熵估计，提供了深入的理论和实践指导。这对于设计更安全、更高效的随机数发生器，特别是在高速通信和加密应用中，具有重要的实际意义。
MA et al.: ENTROPY ESTIMATION FOR ADC SAMPLING-BASED TRNGs 2889
or bit-rate entropy with heavy complexity. Baudet et al. [4]
and our work also adopted the latter approach to avoid the
influence of the correlation on entropy estimation.
B. Stochastic Model Based on Phase Evolution
We summarize the entropy estimation approach presented
in [4] for the EO-TRNG. The stochastic model utilizes a
Wiener process (ϕ(t))
t∈R
(i.e., one-dimensional Brownian
motion) to describe the evolution of the phase with drift μ>0
and volatility σ
2
> 0. Specifically, for an initial time t
0
and
phase ϕ(t
0
), the phase ϕ(t) follows a Gaussian distribution
N with mean ϕ(t
0
) + μt and variance σ
2
t at any time t,
where t = t − t
0
is the sampling interval.
In the sampling process, when the sampling point lies in
the high voltage level of the sampled signal, a random bit ‘1’
is outputted; otherwise, ‘0’ is outputted. From the aspect of
phase, the random variable B(t) of sampling bit at time t is
expressed as follows with 1-periodic phase ϕ(t) mod 1. It is
called conventional sampling process in this paper.
B(t) =
⎧
⎪
⎨
⎪
⎩
1,ϕ(t) mod 1 ∈]
1
2
, 1[;
0,ϕ(t) mod 1 ∈]0,
1
2
[;
0or1,ϕ(t) mod 1 ∈{0,
1
2
}.
The quality of sampling bit depends on two independent
parameters: the sampling interval t and the volatility σ
2
> 0.
Obviously, the increase of either of the two parameters makes
the improvement of the entropy of generated bits. Baudet
et al. [4] defined quality factor Q = σ
2
t representing the
accumulation of the phase jitter during the time interval t.
Then, the expression of the joint entropy for the n-bit output
b is deduced in [4]:
H
n
=
b∈{0,1}
n
− p(b) log
p(b)
= n −
32(n − 1)
π
4
ln(2)
cos
2
(2πν)e
−4π
2
Q
+ O(e
−6π
2
Q
), (1)
where ν = μt represents the frequency ratio of the sampled
signal to the sampling one.
Nevertheless, the authors emphasized that the bit-rate
entropy cannot be calculated directly from Formula (1),
because calculated H
n
is not uniform in n. Following the
similar idea in [3], Baudet et al. also provided a lower bound
of the entropy:
H
B(t)|ϕ(0)
=
1
0
H
B(t)|ϕ(0) = x
dx
=−
1
0
[ p
1,x
(t) log
2
p
1,x
(t)
+
1 − p
1,x
(t)
log
2
1 − p
1,x
(t)
]dx
= 1 −
4
π
2
ln(2)
e
−4π
2
Q
+ O(e
−6π
2
Q
), (2)
where p
1,x
(t) = p
B(t) = 1|ϕ(0) = x
.
The derivation for p
1,x
(t) is introduced as follows. For the
sampling bit B(t) = 1(orB(t) = 0), a probability function
g
1
(x) (or g
0
(x)) is defined with a given value x of the phase
at time t:
g
1
ϕ(t) = x
=
⎧
⎪
⎨
⎪
⎩
1, x mod 1 ∈]
1
2
, 1[;
0, x mod 1 ∈]0,
1
2
[;
1
2
, x mod 1 ∈{0,
1
2
}.
The Fourier coefficient of g
1
(x) (or g
0
(x)) is expressed as γ
1
(or γ
0
), i.e., γ
1
(h) =
1
0
g
1
(x)e
−2πihx
dx,wherei is imaginary
unit and h ∈ Z.
From g
0
(x) + g
1
(x) = 1, it follows γ
1
(0) = γ
0
(0) =
1
2
and
γ
0
(h) =−γ
1
(h) for h = 0. Therefore, the exact expression of
p
1,x
(t) is
p
1,x
(t) =
h∈Z
γ
1
(−h)e
−2πih(μt+x)
e
−2π
2
σ
2
th
2
. (3)
It is noted that, the introduction of g
1
(·) and g
0
(·) provides
a universality of the above entropy computation method. One
can apply this method to the entropy estimation of other digi-
tization processes by rewriting the functions, and substituting
them into the computation process to get a lower bound of the
entropy.
C. Foundational Concepts of ADC
ADC is applied to convert an analog signal into a digital
signal/code at the sampling point. Several considerable factors
of ADC are listed as follows.
• Sampling rate: the number of converting samples per sec-
ond.
• Resolution. An ADC with a resolution M in bits means
the number of discrete values that the ADC can produce
over the range of analog values, which is usually a power
of two, namely 2
M
. The discrete values can represent the
range from 0 to 2
M
− 1 (i.e., unsigned integer) or from
−2
M−1
to 2
M−1
− 1 (i.e., signed integer). In this paper,
we use the former representation. The voltage resolution
of an ADC is given by V
FSR
/2
M
,whereV
FSR
is the full
scale voltage range. V
FSR
is given by V
FSR
= V
Ref_Hi
−
V
Ref_Low
,whereV
Ref_Hi
and V
Ref_Low
are the upper and
lower voltages, respectively. The quantization principle of
an ADC is generally described as follows. An M-bit ADC
converts a sampled voltage V
real
(∈[V
Ref_Low
, V
Ref_Hi
])
to an appropriate numerical representation that is (2
M
−
1) · (V
real
− V
Ref_Low
)/V
FSR
in M-bit binary form.
• Effective number of bits (ENOB): a certain number of bits,
in the digitized sample, actually converted by an ADC
when the ADC samples an analog signal. The ENOB
is limited by the signal-to-noise ratio (SNR). The rest
bits (except for the bits corresponding to ENOB) in the
samples are produced by the inherent noise in ADCs.
An ideal ADC has a property that the ENOB is equal to
its resolution. In this paper, we first assume that the ADC
is ideal in the model establishment. Note that, the real
resolution of a practical ADC can be treated as the integer
part of ENOB. Therefore, this assumption does not affect
the availability of the proposed stochastic model.
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