Digital noise produced by a non discretized tent chaotic map

Microelectronic Engineering xxx (2013) xxx–xxx

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Digital noise produced by a non discretized tent chaotic map L. Palacios-Luengas a, G. Delgado-Gutiérrez a, M. Cruz-Irisson a, J.L. Del-Rio-Correa b, R. Vázquez-Medina a,⇑ a b

Instituto Politécnico Nacional, ESIME-Culhuacan, Santa Ana 1000, 04430 D.F., Mexico Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, 09340 D.F., Mexico

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Inverted tent chaotic map Pseudorandom noise generator Chaotic noise generator Binary sequences uniformly distributed

a b s t r a c t This paper shows a digital electronic system that produces uniformly distributed binary sequences using the inverted tent chaotic map (ITCM) without the scaling and discretization processes. The proposed system has been developed considering a numerical representation of floating point with a 64-bit precision format according to the standard IEEE-754. The proposed system has four important advantages: (i) the produced binary sequences are uniformly distributed and they satisfy 10 randomness tests defined in the NIST 800-22SP guide, (ii) the statistical behavior of the ITCM is not affected by the scaling and discretization processes; therefore, the chaotic map used is not modified, (iii) the statistical behavior of the ITCM does not have stability islands inside the chaotic region, although its control parameter is changed, as it occurs with the logistic chaotic map and (iv) the statistical behavior of the binary sequences is conducted by the control parameter and the skeleton of the bifurcation diagram of the ITCM, which can be considered as the security keys of the system. Ó 2013 Published by Elsevier B.V.

1. Introduction A pseudorandom noise generator (PRNG) can be implemented in hardware or software in order to generate analog or digital signals. These noise generators can be used in different applications and areas [1,2]. In the case of cryptographic systems, it is desirable to produce digital signals with apparently uniform statistical distributions, which have the largest period possible, and must satisfy the randomness tests proposed by the NIST 800-22SP guide [3]. The digital noise generation produces a great interest to investigate mathematical models that generate unpredictable and aperiodic signals whose statistical distribution can be similar to a uniform distribution. Some reported works related with the design of PRNG can be revised in [4–5]. The different alternatives to produce digital noise for cryptographic applications can be divided mainly into five types, and they are based on: (a) Linear Congruential [6], (b) Linear Feedback Shift Register [7], (c) cryptosystems [8] and cryptographic primitive functions [9,10], (d) combinations relatively complex of the above alternatives, and (e) chaotic maps [11–14]. In this paper, the used alternative to produce digital noise is based on chaotic maps. In particular, the inverted tent chaotic map (ITCM), described in Section 2 has been used in the structure of a generator of pseudorandom binary sequences (PBS). The ITCM is a function that produces real numbers sequences, which must be converted into binary sequences using some specific strategy. Particularly in this case, the used strategy does not consider the scal⇑ Corresponding author. Tel./fax: +52 55 5656 2058. E-mail address: [email protected] (R. Vázquez-Medina).

ing and discretization processes over the chaotic map, because these processes allow generating binary sequences through a non chaotic function, which is an approximation to the chaotic function. This approximation will be good or bad according with the discretization level. In this work, the scaling and discretization processes are avoided in order to conserve the statistical properties of the generated sequences, which, once they are generated with the desired statistical distribution function are converted into a binary format. Thus, the proposed system is considered an improvement regarding traditional techniques because it avoids the scaling and discretization processes. The strategy used to convert real numbers into binary numbers consists in defining a statistical partition which resolution depends on the number of bits used to convert each real number. This strategy is described in Section 3. In this paper the ITCM is used because it does not have stability islands as it occurs with the logistic chaotic map, therefore, when the chaotic region is reached the ITCM produces pseudorandom number sequences even when the control parameter changes. Section 4 shows the hardware implementation of the PBS generator. Finally, in Section 5, the statistical features of the binary sequences produced by the proposed system are evaluated using the randomness tests described in the NIST 800-22SP guide [3]. 2. Tent chaotic map There are several 1-D chaotic maps such as the logistic map [11], tent map [12], sine map [13], and Bernoulli map [14], which can be used to produce binary sequences. These maps are chaotic

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dynamic systems (CDS) that can be built with iterated functions. A typical 1-D chaotic map is the tent map, which is the kind of piecewise linear chaotic maps that has been extensively studied and used since it has great mathematical simplicity [15]. The classical iterated function governing this CDS is,

xnþ1 ¼ f ðl; xn Þ ¼

8 > < > : 2l

2l xn þ



1l 2



; a 6 xn 6 12  ðxn  1Þ þ 12l ; 12 < xn 6 b 

;

ð1Þ

where xn and xn+1 are the current and next values in the produced real numbers sequence; l is the control parameter and x0 is the initial condition, both are chosen arbitrarily but known and defined in the interval [a, b] in R. There are different variants of the tent chaotic map, some of these can be found in [16,17]. In particular, in this paper the variant used has been inspired in the research work of Nejati et al. [12]. In this work, the used alternative is named the inverted tent chaotic map (ITCM), which is a variant of the typical tent chaotic map and is described by Eq. 2 (see Fig. 1a),

      1 1l : xnþ1 ¼ sðl; xn Þ ¼ sn ðl; x0 Þ ¼ l 2xn   þ 2 2

ð2Þ

where s (l, xn): (0, 1) ? (0, 1) considering that n represent the iteration step, x0 is the initial condition, xn is the real number produced by the ITCM at the iteration n, l 2 (0, 1) in R (see Fig. 1a) and sn (l, x0) represents n iterations of s () applied on x0. According to the iterated function expressed by Eq. 2, the ITCM produces sequences whose behavior is dependent on the initial conditions and the control parameter. If the initial condition or the control parameter are changed, the produced sequence will be very different. This is a condition of the chaotic systems. In this context, the statistical distribution of a produced sequence by the ITCM can be determined by finding the frequency with which the different regions are visited. For this, the histogram of the sequence is calculated and this constitutes an approximation to the statistical distribution of the produced sequences. However, if a sequence of great length is considered, then a stationary statistical distribution can be obtained, which does not depend on the initial condition, and it only depends on the control parameter. Precisely, this is what is observed in a bifurcation diagram. The ITCM, being a chaotic dynamical system, has high dependence on initial conditions. Additionally, depending on the value of l, the ITCM can have two types of behavior, stable and unstable.

In the stable behavior, the ITCM always generates the same value, which also depends on l. In the unstable or chaotic behavior, the ITCM generates aperiodic numbers sequences with random appearance. The ITCM has a chaotic behavior if 0.5 < l < 1 and it has a stable behavior if l < 0.5. These behavior regions are shown by the bifurcation diagram in Fig. 1b. A bifurcation diagram, also named Feingenbaum’s diagram, is a tool that shows the behavior of the possible sequences that can be generated by a chaotic map as a function of its control parameter and it not considers the transient of the sequences. The bifurcation diagram is a tool derived from Chaos Theory, which shows a graph that illustrates the changes in the dynamic behavior of the chaotic map, demonstrating the phenomenon by which it comes to chaos. With this tool the regions of periodic or chaotic behavior are identified for the chaotic map [18]. The bifurcation diagram represents a summary of the statistical distribution functions of the number sequences generated by chaotic map as a function of l [19]. Now, to build a bifurcation diagram, the chaotic map is iterated considering different values of l in (0, 1) with a defined step Dl. A simple procedure that explains how to build a bifurcation diagram, avoiding the transient of the generated sequence, is the following: (a) define l = 0.0, (b) randomly selects an initial condition x0 in (0, 1) and the chaotic map must be iterated N times (e.g. N = 1000) to calculate the sequence {x1, x2, x3, ..., xN}, (c) The first 100 values of the sequence are discarded to ensure that the transient has been exceeded, (d) the remaining values of the sequences {x101, x102, x103, ..., xN} are plotted, (e) Increase the value of l a step Dl, that is, l l + Dl and the procedure must be repeated from (b) until l = 1.0. In this paper, the interest region happens when l 2 (0.7, 1.0) because for these values of l, the ITCM has only one interval in which the number sequence is produced. Whenever l 2 (0.5, 0.7) there is more than one interval where the produced sequence is concentrated (see Fig. 1b). Real number sequences with a good statistical distribution function can be produced by Eq. 2. If l ? 1, then the expected statistical distribution function of the produced number sequences will be very close to a uniform distribution function. Considering that the ITCM is a chaotic system and consequently it has high dependence on initial conditions, then the produced sequences will be different if the used initial condition is changed, but its statistical distribution function is the same at long term. A digital implementation of the ITCM is possible, several authors have proposed schemes that include the scaling and discretization processes over chaotic maps defined in the real numbers set, R, and the result of these processes is a new map defined in the natural numbers set, N, which is a non chaotic function [20,21]. That alternative produces binary sequences, which do not have a statistical behavior that is congruent with the one of the original chaotic map, since the rounding of numbers produced by the discretization process induces an error. The error is propagated and increased when the new map is iterated, and then the statistical behavior of the resulting sequence of real numbers is strongly affected [22]. The proposed system overcomes this problem.

3. Proposed system

Fig. 1. (a) Graphic expression of the ITCM considering x 2 (0,1) and (b) bifurcation diagram of the ITCM considering l 2 (0,1).

The proposed system is a hardware electronic device that generates digital noise sequences using the ITCM. These sequences are transmitted to a PC using the USB port, and for this process the next aspects need to be considered: (a) a secret key kl, which is formed by the concatenation of the control parameter l and the initial condition x0, (b) LB, the length of the binary sequence that will be produced, and (c) Sc, the starting command in its binary format (0 1 1 1 0 0 1). kl has a length of 128 bits conformed by 16 blocks of 8 bits each one, LB has a variable length that can take

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any number of 8-bits blocks, according to the size of the desired sequence. The system receives these three elements in the following order: 16 kl blocks, the required number of LB blocks and the Sc starting command, with which the system will begin to produce the binary sequences of the desired length. Then the sequence that must be introduced to the proposed system is determined by

D ¼ kl HnnHLB HnnHSc Hnn

ð3Þ

3

(b) A belonging function, P(w, Aj), which associates each number w 2 [a, b] to one integer j, which characterizes the subinterval Aj, where w specifically exists. The belonging function P is defined according to the following equation,

Pðw; Aj Þ ¼ jIAj ðwÞ:

ð7Þ

where H represents the concatenation function and nn represents a car return as separator.

where

3.1. Floating-point format

IAj ðwÞ ¼

The operations required by Eq. 2 are performed considering that each real number is represented in a floating-point format, according to the IEEE-754 standard [23] which considers a 64-bit format and it is structured in the next way: 1-bit used as the sign bit s, 11bits used to represent the exponent e and 52-bits used to represent the mantissa m. Fig. 2 shows the structure for a 64 bit number representation. The s bit is 1 when the real number is negative and 0 when it is positive. The real value assumed by a given 64-bit floating-point data with a given biased exponent e and a 52-bit fraction m is

(a) A scheme of binary codification: Bj = dec2Bin(j, bits), that expresses the integer j in a binary format of length bits and that is used in the conversion function conv(bj, yj) = dec2Bin(P(yj, Aj), bits)-1, which will be valid if and only if the binary number obtained bj P 0. The skeleton of the bifurcation diagram of the ITCM has been considered to establish the specific criterion in which the real numbers considered in the final sequence lies in the interval [a,b]. This skeleton is defined by the curves that limit the bifurcation diagram of the ITCM and they correspond to first iterations of the ITCM. Fig. 3 shows the skeleton of the bifurcation diagram of the ITCM corresponding to the first six iterations. In the proposed algorithm the following procedure has been established to build the statistical partition that can produce identically distributed binary numbers sequences.

v alue ¼ ð1Þs

1þ

52 X

! bi 2i

 2ðe1023Þ :

ð4Þ

i¼1

3.2. General description of the algorithm The scheme that represents the generation of binary sequences with good statistical features proposed in this paper consists of the following steps: (i) Generating a sequence of n real numbers, u(x0) = {x1, x2,. . ., xn}, through the ITCM, s (l, x), considering that the chaotic map will be iterated n times using specific values of the control parameter l 2 (0.7, 1) and the initial condition x0 2 (0, 1), (ii) Use the sequence u(x0) to obtain a subsequence of real numbers, uR(x0) = {y1, y2, . . ., yz}, which will be formed by all those real numbers yi 2 u(x0) and also yi 2 [a,b], where 1 < i < z and the limits a and b are defined according to the skeleton of bifurcation diagram of the ITCM in the following way:

a ¼ s1

b ¼ s3



l; 1 



  1þl 2

ð5Þ

  1þl 2

ð6Þ

l; 1 

where l 2 [lc,1] and lc is defined when a = b. (iii) convert the subsequence uR(x0) of real numbers yi to uB(x0) of real numbers yi to a sequence uB(x0) of binary numbers bi with 1 < i < z. This conversion is performed using the function conv(bi, yi), which converts the real number yi to the binary number bi. In this conversion process three additional elements are considered: (a) A statistical partition A = {A1, . . ., AM} with M = 2bits subintervals obtained from the interval [a, b] which is produced when the map s (l, x) is applied to the unitary interval and bits is the number of bits that are used to represent each real number in the sequence uR(x0),

Fig. 2. Floating point format for the numbers representation with a 64-bit double precision, according the IEEE 754 standard.



1;

if w 2 Aj

0;

if w R Aj

;

ð8Þ

(i) Let a and b be defined according to Eqs. (5) and (6), and bits is the number of bits used to convert each real number in the sequence uR(x0) to the binary number. (ii) Let Fk be the vector that contains the limits of the subintervals in the actual statistical partition A = {A1,. . .,Ak} for the definition interval of the ITCM considering a specific value of the control parameter l. (iii) Define the vector Fk for each value of k = 2, 3, . . ., bits +1 according to the following indications. (a) Fk(2i1) = Fk1(i), i = 1,. . ., (2k2 + 1), considering that F1(1) = a and F1(2) = b. In this step the vector that contains the limits of the subintervals is expanded to produce a thinner partition. (b) Fk(2i) = Fk(2i1) + (Qk,i1)Dk,i, i = 1,..,2k2 considering k k l(k1) Dk,i = (F (2i + 1)F (2i1))/S, S = 2 where l is the regular partition size, Qk,i is the value that satisfies Pk,i(Qk,i) P (1/2)k1 and Pk,i(Qk,i1) < (1/2)k1 and with an accumulated probability,

Fig. 3. Skeleton of the bifurcation diagram of the ITCM, corresponding to the first iterations, where sn ðl; x0 Þ considering n = 1, 2, 3, 4, 5 and 6 and x0 = 1((1 + l)/2).

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L. Palacios-Luengas et al. / Microelectronic Engineering xxx (2013) xxx–xxx

Pk;i ðqÞ ¼

q X nk;i j j¼1

nk;i ¼

S X

nk;i

;

nk;i r ;

ð9Þ

ð10Þ

r¼1

nk;i r ¼

L X

IBk;i ðymi Þ;

mi¼1

r

k k Bk;i r ¼ ðF ð2i  1Þ þ ðr  1ÞDk;i ; F ð2i  1Þ þ r Dk;i Þ;

ð11Þ

ð12Þ

where L = length(uR(x0)), IB(x) is defined according to Eq. 8 and it denotes the characteristic function of the subinterval Bi, Brk;i is the subinterval r with r = 1, 2, . . ., S in the regular partition defined inside the non-regular subinterval Akj , which has been calculated for the iteration k. In these steps the medium point in each subinterval is determined to produce two news subintervals. In this way, to define the statistical partition Ak ¼ fAk1 ; . . . ; AkN g where k k k Aj ¼ ½F ðjÞ; F ðj þ 1ÞÞ is a non-regular subinterval of Ak with j = 1,. . .,N, N = 2k1 and k = 2,. . .,bits. If k = 2, it is required to define the partition A2 ¼ fA21 ; A22 g two intervals must be considered: A21 ¼ ½F 2 ð1Þ; F 2 ð2ÞÞ and A22 ¼ ½F 2 ð2Þ; F 2 ð3Þ. If k = 3, it is required to define the partition A3 ¼ fA31 ; . . . ; A34 g four intervals must be considered: A31 ¼ ½F 3 ð1Þ; F 3 ð2ÞÞ, 3 3 3 3 3 3 3 A2 ¼ ½F ð2Þ; F ð3ÞÞ, A3 ¼ ½F ð3Þ; F ð4ÞÞ and A4 ¼ ½F 3 ð4Þ; F 3 ð5Þ, and so on.

Fig. 5. Statistical distribution obtained from sequences produced by s (l, x0) using l = 0.85, x0 = 0.75 and a statistical partition with: (a) 210 and (b) 213 intervals.

4. Hardware implementation of digital noise generator 4.1. General description The block diagram of the electronic system is shown in Fig. 4a, and the designed electronic circuit is shown in Fig. 4b. This system uses a floating point representation for non-integer numbers, which complies with the IEEE-754 standard for 64 bit precision [23]. The proposed system implemented in the 16-bit PICMicro™ microcontroller which was programmed using C language in the available development tool kit. The system is formed by two electronics modules (i) PICMicro™ module and (ii) the USB controller. For the transmission of data between

Fig. 6. Statistical distribution obtained from sequences produced by s (l, x0) using l = 0.85, x0 = 0.75 and a statistical partition with: (a) 215 and (b) 219 intervals.

Fig. 4. Global hardware architecture, (a) block diagram and (b) board digital noise generator.

these two modules a TDX signal and a RDX signal are used for transmitting and receiving data respectively. The synchronization of the system is made with a CLK signal from the USB controller to the PICMicro™ module. The USB controller is used for the communication with a PC through the USB port. In this way, the digital noise sequence produced by the electronic circuit can be transmitted to a PC for the evaluation of its statistical behavior using the randomness tests described in the NIST 800-22SP guide [3].

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L. Palacios-Luengas et al. / Microelectronic Engineering xxx (2013) xxx–xxx Table 1 Results of the randomness tests defined by the NIST 800-22SP guide when l = 0.85 and bits = 10 in the ITCM. Test

p-Value

p-Value

Result

2.1 2.2 2.3 2.5 2.6 2.7 2.9 2.14 2.15 2.16

Mono bit frequency test Block frequency test Runs test Binary matrix rank test Spectral test Non overla temp matching Maurers universal test Cumulative sums test Random excursions test Random excurs Vari test

0.4592999943 0.4986955028 0.3068217360 0.2746091367 0.9482627225 0.1087284422 0.8284422951 0.8579648814 0.4693938224 0.4691066781

True True True True True True True True True True

4.2. Evaluation of the pseudorandom binary sequences Using the procedure described in Section 3, pseudorandom binary sequences with good statistical features were obtained. Fig. 5 shows the statistical distribution function of a sequence produced by the proposed system when l = 0.85 using statistical partitions with 210 and 213 intervals that correspond to the conversion of real numbers into binary numbers using bits = 10 and bits = 13. Fig. 6 shows the statistical distribution function of a sequence produced by the proposed system when l = 0.85 using statistical partitions with 215 and 219 intervals that correspond to the conversion of real numbers into binary numbers using bits = 15 and bits = 19 respectively. Notice that the last produced statistical distribution function is very similar to the uniform statistical distribution function. The randomness of these sequences has been analyzed by NIST 800-22SP guide [3]. These randomness tests were applied to several sequences produced by the proposed system using different values of the control parameter l. However, as an example, Table 1 shows only the results of the randomness tests when the control parameter is l = 0.85. 5. Conclusions This paper presents an electronic circuit that generates digital noise in a low-cost 16-bit microcontroller using the inverted tent chaotic map (ITCM). The algorithm proposed produces digital noise sequences without the scaling and discretization processes. The algorithm is also supported by three conditions: (a) the identification of the chaotic region for the ITCM using its bifurcation diagram and its control parameter, (b) the bifurcation diagram skeleton defined by the first and third iteration of the ITCM and (c) the definition of a statistical partition over the domain of the ITCM considering non regular intervals, this partition depends of the number of bits used to convert each real number produced by the ITCM into a binary number. The identically distributed binary sequences produced by this algorithm were statistically evaluated using the randomness tests defined in the NIST 800-22SP guide [3]. This evaluation found that the proposed system should be improved to increase its complexity because even when the produced binary sequences have good statistical features, they could be gen-

5

erated by a cryptanalytic algorithm In this way, the proposed system is able to generate sequences of pseudorandom binary numbers, and its statistical distribution function depends of a wide set of possibilities if the control parameter l and the initial condition x0 are changed. The proposed system showed four important advantages: (i) the produced binary sequences satisfied 10 of 16 randomness tests defined in the NIST 800-22SP guide [3], (ii) the statistical behavior of the sequences generated by the ITCM was not affected by the scaling and discretization processes, which were avoided in the proposed system, (iii) the statistical behavior presented in the bifurcation diagram of the ITCM did not have stability islands inside the region of chaotic behavior even when its control parameter was changed, (iv) the statistical behavior of the binary sequences was conducted by the control parameter and the skeleton of the bifurcation diagram of the ITCM, both elements can be considered as the security keys of the system. Acknowledgments The authors thank the financial support of the SIP IPN 20130461 and ICYTDF 270/2010 projects. L. Palacios-Luengas (CVU-373990) and G. Delgado-Gutiérrez (CVU-372164) thank for the scholarship provided by CONACYT. References [1] C. Forastero, L.I. Zamora, D. Guirado, A.M. Lallena, Phys. Med. Biol. 55 (2010) 5213–5229. [2] M. Numminen, Random number generator United State Patent Application: 20090222501 - Class: 708250, Application date: 20090903. [3] A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, NIST SP 800–22 Rev 1a (2010). [4] J. Wu, M. O’Neill, Electron. Lett. 46 (2010) 988–990. [5] T. Addabbo, M. Alioto, A. Fort, S. Rocchi, V. Vignoli, J. Circuit. Syst. Compt. 19 (2010) 879–895. [6] M.N. Anyanwu, L.-Y. Deng, D. Dasgupta, Int. J. Comput. Sci. Sec. (IJCSS) 3 (2009) 186–200. [7] M. Elia, G. Morgari, M. Spicciola, On Binary sequences generated by self-clock controlled LFSR, On Proc. 19th Int. Symp. On Math. Theo. Of Networks and Systems (2010) 1275–1281. [8] A. Shamir, ACM Trans. Comput. Syst. 1 (1983) 38–44. [9] A. Boldyreva, V. Kumar, (2012) 187–202. [10] A. Gholipour, S. Mirzakuchaki, Int. J. Comput. Elec. Eng. 3 (2011) 896–899. [11] V. Patidar, K.K. Sud, N.K. Pareek, Informatica 33 (2009) 441–452. [12] H. Nejati, A. Beirami, Y. Massoud, A realizable modified tent map for true random number generation, Midwes. Symp. Circuit. (2008) 621–624. [13] J. Xu, P. Chargé, D. Fournier-Prunaret, A.K. Taha, K. Long, Sci. China Inf. Sci. 53 (2010) 129–136. [14] W.C. Saphir, H.H. Hasegawa, Phys. Lett. A 171 (1992) 317–322. [15] A. Medio, M. Linez, Cambridge University Press (2001) 216–217. [16] S. Callegari, G. Setti, P.J. Langlois, IEEE Int. Symp. Circuit. Syst. (1997) 781–784. [17] T. Habutsu, Y. Nishio, I. Sasase, S. Mori, LNCS (1991) 127–140. [18] R. Vázquez-Medina, Mexico (2008) 55–61. [19] H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals – New Frontiers of Science, Springer Science, USA, 2004. [20] M.S. Baptista, Phys. Lett. A 240 (1998) 50–54. [21] G. Jakimoski, L. Kocarev, Circuit. S-I: Fund. Theory Appl. 48 (2001) 163–169. [22] R. Vázquez-Medina, J.L. Del-Rio-Correa, C.E. Rojas-López, J.A. Díaz-Méndez, Lecture Notes in Computer Sciences 7209 (2012) 105–115. [23] IEEE Standards Board, IEEE Standard for floating-Point Arithmetic, ANSI/IEEE Std 754–1985, IEEE (1985).

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