2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Research on True Random Numbers with a Key FeatureShuichi Aono† , Yoshifumi Nishio† and Hideharu Aruga∗†

Department of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {aoichi, nishio}@ee.tokushima-u.ac.jp∗Leisure Electronics Technology Co.
Hitotsubashi Building 4F, 2-6-3 Hitotsubashi Chiyoda-ku, Tokyo, 101-0003 Japan
Phone:+81-35-210-4759, Fax:+81-35-210-4753
Email: h aruga@letech.co.jp

1. Introduction

two successive numbers in the chaotic sequence. The relation
clearly shows the shape of the map.

The encryption technology has the function of defending information, and becomes the element of other information security technologies like the digital signature etc.
Random numbers play an essential role in various encryption schemes [1][2]. However, pesudo-random numbers canbe forecasted, because the characteristics of pseudo-random
numbers depend on the type and the initialization of the function. On the other hand, true random numbers based on aphysical phenomenon can be said as ideal random number
sequences because it is unpredictable.
In this research, we embed a kind of key feature into true
random numbers. We use true random numbers generated
from the true random number generator invented by LeisureElectronics Technology Co. We replace one number per every
Figure 1: Chaotic sequence generated by the Logistic Map
N random numbers by a number of the Logistic Map, which(α = 4.0).
is known as the simplest map generating chaotic sequence,and investigate various properties of the created sequence.
3. True Random Number with Key Feature
2. Logistic Map
In this research, we use true random numbers generated
In this research, we use the Logistic Map, which is known from the true random number generator invented by Leisureas the simplest map generating chaotic sequence. The Logis- Electronics Technology Co. This true random numbers gentic Map is expressed by the following equation:
erator uses pure thermal noise for the random source. Thegenerated random numbers are experimentally confirmed toz(t + 1) = αz(t)(1 − z(t)).
(1) be truly random. The generation rate is 20MB per second.
Moreover, this random numbers generator can automaticallyHere, α is a control parameter changing the behavior of the detect unexpected noises, accidental failures in the circuit andgenerated sequence. In this research, we use the parameter intentional attacks, because the statistical characteristics of
α = 4.0.
the generated random numbers are always examined in real
Figure 1 shows an example of chaotic sequences generated time.
by the Logistic Map. Figure 2 shows the relation between
In this research, we embed a kind of key feature into true- 61 -One true random number per every N is replaced by thenumber generated by the Logistic Map. The value of the previous random number is used for an initial value of the Logistic Map.
4. Simulated Results
Simulated results are shown in Figs. 4 and 5. Figure 4
shows the created sequence for the case of N = 5. Figure 5
shows the relations between two successive numbers in thecreated sequence for the cases of N = 5 and N = 15.

Figure 2: Relation between two successive numbers in thechaotic sequence (α = 4.0).

random numbers generated from this true random numbergenerator. The meaning of providing a key feature is not giving a seed to pseudo-random numbers generators, but to embed information into the random number sequence. Though
true random numbers cannot be reproduced again, the random number sequence with the same information can be reproduced by giving a key feature. For example, if we can embed two different keys into the random number sequence andalso can extract the key from the sequence, the true random
numbers might be utilized as a carrier of some information.

Figure 4: Created sequence.

This created sequence looks similar to the true random
numbers in Fig. 1. However, when we analyze the relation
between two successive numbers, we can clearly observe thatthe shape of the Logistic Map is embedded into the sequence
as a key feature.
Next, we investigate the distribution of the created sequence. Figure 6 shows the distribution of the sequence. The
We use the Logistic Map to give a key feature. Figure 3 horizontal axis shows the value of the sequence, the verticalaxis shows the number of the sequence. When the value ofshows the method of embedding a key feature.
N is small, that is, the amount of the added chaotic value islarge, the distribution of the sequence becomes non-uniform.
True Random Numbers :Further, we investigate the runs within the created sequence.
A run is a monotonic subsequence; Run up is increas0.9 0.1 0.3 0.6 0.2 0.2 0.3 0.7 0.8 0.5 0.1 0.3 0.2
ing subsequences and Run down is decreasing subsequences.
per every N numbers
The expected distribution of the number of runs for a truly
random sequence is known as follows [1].
Insert one number generated by the Logstic Mapz (t+1) = α z(t)( 1 - z(t) )rd∗ =z(t) : previous random number

Created Sequence :0.9 0.1 0.4 0.6 0.2 0.6 0.3 0.7 0.8 0.5 0.1 0.4 0.2

2n(d2 + 3d + 1) 2(d3 + 3d2 − d − 4)−(d + 3)!
(d + 3)!

(2)where rd∗ is the expected number of runs with the length dand n is the length of the total sequence. The expected distribution is compared against the sample distributions using
the standard χ2 formula. By executing the run test, we confirmed that the created sequence does not pass the run test for
N < 29.

Figure 3: Making true random number with key feature.

- 62 -

Figure 6: Distribution of the sequence.
5. Conclusion(a)

In this research, we embedded a kind of key feature intotrue random numbers and investigated various properties of
the created sequence.
As our future subject, we are considering to insert chaotic
sequence into true random numbers in an irregular manner.
References[1] O. Miyatake and K. Wakimoto, “Random Number andMonte Carlo Method,” Morikita Shuppan Co., 1978.
[2] D.E. Knuth, “Art of Computer Programming, Volume
2: Seminumerical Algorithms, 3rd Edition,” AddisonWesley, 1997.

(b)
Figure 5: Relations between two successive numbers in thecreated sequence. (a) N = 5. (b) N = 15.

- 63 -

- 64 -