Liberty BASIC Programmer's Encyc - CryptographyWithLB103

<h1>Cryptography with Liberty BASIC : 103 RSA Algorithm</h1>
 Onur Alver (CryptoMan) 
<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/><hr />
<h1>RSA ALGORITHM</h1>
 
 
There is an excellent description about RSA here . https://www.di-mgt.com.au/rsa_alg.html 
 
<h2>Step by Step RSA Description</h2>
 
RSA Algorithm depends of the difficulty of factoring very large composite numbers. 
Remember, composite number n is a product of two or more prime numbers. For example 
for example 4,6,8,9,10,12,15.... are composite numbers. While 2,3,5,7,11,13... are 
prime numbers. Prime numbers are the root numbers of all other numbers in the universe. 
2 is only even prime number and all other prime numbers have to be odd. Prime numbers 
are only divisible by themselves without leaving any remainder. 
 
The difficulty is finding all of the prime numbers and there are infinite number of 
prime numbers. Especially, if you are trying to find huge prime numbers of hundred 
or more digits. 
 
When you select 15, it is easy to guess 3 and 5 as the prime factors of 15. In a strong 
RSA implementation we will choose much bigger prime numbers p and q such that n = p.q 
such as the following huge composite number below:

<pre class="text">n = 11929413484016950905552721133125564964460656966152763801206748195494305685115033 38063159570377156202973050001186287708466899691128922122454571180605749959895170 80042105263427376322274266393116193517839570773505632231596681121927337473973220 312512599061231322250945506260066557538238517575390621262940383913963</pre>
 
Can you guess what is p and q from this n? 
 
Not so easy, this is the idea. 
 
Now, we have to generate two large prime numbers p and q to determine 
 
n = p.q called common modulus n. 
 
t = (p-1).(q-1) called Euler-Totient number t. 
 
Start by chosing an e such as 3 an easy to use small prime, and d related to e 
such that following formulas to calculate c enciphered data from a clear data and 
recover the clear data a from c by using exponent d. 
 
You can find d iteratively from the equality e.d ≡ 1 mod t which can be rephrased 
as follows &quot;Which d multiplied by e divided by t have a remainder of 1?&quot; 
 
We will call (e,n) our Public Key and (d,n) the secret key or private key. 
 
Alice can publish or send her Public Key (e,n) to Bob to send her his private 
messages by email and Evil Corp's intelligence officer Eve reading every mail 
from Evil Corp's POP3/SNMP traffic should not be able to make any sense from 
the garbled text. 
 
Bob will encipher the his message with the following encryption formula: 
 
c = a e mod n 
 
 
 
Finally, Alice will decipher the encrypted message c using the following 
decryption formula: 
 
a = c d mod n 
 
<h2>Simple RSA Example with Small Primes</h2>
 
 
 Let's choose two small primes p=3 and q=11 
 
 
 Therefore, n = p . q = 3 x 11 = 33, the common modulus n=33. 
 
 
 Next, let's find the Euler-Totient number t=(p-1)(q-1)=(3-1)(11-1)=20. 
 
 
 We now choose the smallest possible odd prime number 3 as e and therefore 
 Alice's public key is (3,33) and Alice sends this to Bob with WhatsApp as 
 a different channel where both Alice and Bob trusts. 
 
 
 Alice than sets out the find d from e.d = 1 mod t equality and asks the 
 question which d multiplied by 3 and divided by 20 gives a remainder of 1? 
 This is not very hard to find, 3 x 7 = 21 and 21 divided by 20 gives a 
 remainder of 1. 
 
 
 Therefore, Alice's secret key is ( 7, 33 ) and she keeps this private 
 only to be used by Alice to decipher secret message coming to her with 
 her Public Key ( 3,33 ) 
 
 
 
 Alice sends Bob a suggested coding table as follows with WhatsApp. 
 
 
 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 
 
 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z . , ? ! _ ( ) 
 
 
 Bob sends the following message to Alice: 
 
 
 SZZNYSZYANYDAXC,Y. MYNZXSCEALYNWIYIEYI.NYECEZYANYEIIE 
 
 Eve intercepts SZZNYSZYANYDAXC,Y. MYNZXSCEALYNWIYIEYI.NYECEZYANYEIIE but she can not 
 make any sense out of this garbled text because she does not have the secret key. 
 
 
 Alice converts this message according to above coding table: 
 
 19 26 26 14 25 19 26 25 01 14 25 04 01 24 03 28 25 27 31 13 25 14 26 24 19 03 05 01 12 25 14 23 09 25 09 05 25 09 27 14 25 05 03 05 26 25 01 14 25 05 09 09 05 
 
 
 Finally Alice deciphers each number with her secret key (7,33) using the formula : 
 
 
 a = c 7 mod 33 
 
 like 
 a = 19 7 mod 33 = 893871739 mod 33 = 13 which corresponds to letter M 
 
 When we calculate all the numbers in the above message we will get: 
 
 
 13 05 05 20 31 13 05 31 01 20 31 16 01 18 09 19 31 03 04 07 31 20 05 18 13 09 14 01 12 31 20 23 15 31 15 14 31 15 03 20 31 14 09 14 05 31 01 20 31 14 15 15 14 
 
 
 When we look at our coding table we obtain the following text message: 
 
 
 M E E T _M E_ AT_ P A R I S_ C D G_ T E R M I N A L_ T W O_ O N_ O C T_ N I N E_ A T_ N O O N 
 
<h2>CONCLUSION</h2>
 
Obviously when you use small primes and obtain a modulus of 33 we are doing a 6 bit RSA encryption which is effectively a simple substition cipher which 
goes back in history to the Ceaser's age, so simple substition ciphers are called Ceaser's Ciphers and these are easily breakable with Frequency Analysis. 
All of those super XOR 'unbreakable' ciphers are substitution ciphers and obviously they are very easily breakable. 
 
Therefore, we have to have a key size of 2048 bits and above as of 2017 for a secure RSA encryption implementation. We only gave the small primes 
example to make the idea easily illustrated. You can even test this with an EXCEL worksheet. The aritmetic is the same regardless of the size of prime 
numbers. Formula is the same and even with high school mathematics knowledge you can understand RSA. 
 
However, you can try much larger prime numbers with the following Liberty Basic example for about up to 128 bits modulus sizes for reasonable computational 
times with Liberty Basic. If you want to use keys sizes such as 1024 or 2048 bits, you must call an external DLL such as OpenSSL. 
 
On the other hand, even 128 bit RSA with Liberty Basic is pretty good for casual encryption purposes. Liberty Basic has an undocumented feature for 
making huge integer arithmetic which is not possible with other programming languages. It was possible to make this RSA demonstration with Liberty Basic 
due to this unusual capability. 
 
Remember, with RSA the block of data you are encrypting must be the same size with the modulus bit size, For example for a modulus size of 128 bits, 
the block size must be 128 bits or 16 bytes. If the data you are encrypting is less than the modulus size, you must pad your data block with a random 
number after a delimiter to make it equal to modulus size. 2048 bits will be 256 bytes data blocks. Please, note that you can do 2048 bits RSA encryption 
with exponent 3 using Liberty Basic but it will be too slow to decrypt 2048 bits blocks with native Liberty Basic. 
 
One of the challenges of the RSA algorithm is the TRUST CHAIN. How will you trust somebody on Internet who sends his Public Key to you and invite 
you to use this to send him/her encrypted messages such as trusting a web site who ask you to send your credit card number using an allegedly secure 
public key? 
 
This is the reason for the need for CA Certificate Authorities. CAs act like Notaries by signing Public Keys of individuals and companies so that we 
can trust those public keys. These are called Public Key Certificates which are effectively Encrypted Public Keys which are enciphered with the 
Secret Root Key of the CA. So, instead of Plain Public Key of an individual or company we request a Certificate signed by a trusted CA which can be 
company wide internal CA or globally accepted CA like VeriSign, Comodo, Thawte, etc. Assuming that we have the Public Keys of the trusted CA 
we can decrypt CA CERTIFICATE using the CA PUBLIC KEY and we can recover the PUBLIC KEY of the party we are going to send a secret 
message. Arithmetic is the same. However, for this tutorial there is no need to make it more complicated with CA CERTIFICATES. Once the basics 
are understood, it is not hard to make this extension. 
 
 
 
<h2>Demonstration Program in Liberty Basic</h2>
 
<pre class="lb">dim stats(11) dim SmallPrimes(1000) [begin] print &quot;Liberty Basic RSA Demonstration&quot; print &quot;Loading Small Primes&quot; for i=1 to 1000 read x SmallPrimes(i)=x next NoOfSmallPrimes=1000 print NoOfSmallPrimes;&quot; Primes Loaded&quot; print&quot;Generating Random Primes&quot; for i=1 to 2 t1=time$(&quot;ms&quot;) [TryAnother] print print &quot;Prime No &quot;;i if i=1 then x=Random(30) else x=Random(30) iterations=0 [Loop] iterations=iterations+1 if MillerRabin(x,7)=1 then 'print &quot;Composite&quot; x=x+2 goto [Loop] else t2=time$(&quot;ms&quot;) print x;&quot; Probably Prime. Generated in &quot;;t2-t1;&quot; milliseconds&quot; end if if p then q=x else p=x next i print print &quot;p=&quot;;dechex$(p) [Retry] restore print &quot;q=&quot;;dechex$(q) 'Common modulus N=(p)(q) n=p*q print &quot;Key Length &quot;;len(dechex$(n))*4;&quot; bits &quot; print 'Euler Totient Number M=(p-1)(q-1) m=(p-1)*(q-1) 'Choose a suitable prime E relatively prime to M for i=1 to 12 read e if (GCD(e,m)=1) then goto [Start] next i [Start] print &quot;Common Modulus, n=&quot;;dechex$(n) print &quot;Euler-Totient No, m=&quot;;dechex$(m) print &quot;Public Exponent, e=&quot;;dechex$(e) d=ExtBinEuclid( e, m ) print &quot;Secret Exponent, d=&quot;;dechex$(d) DIM TEST(10) DIM ENCR(10) DIM DECR(10) TEST(1)=TEXT2DEC(&quot;LIBERTY BASIC IS THE BEST&quot;) TEST(2)=TEXT2DEC(&quot;WHICH BASIC CAN DO THIS &quot;) TEST(3)=TEXT2DEC(&quot;WITHOUT CALLING EXT DLL ?&quot;) TEST(4)=TEXT2DEC(&quot;LB CAN DO BIG INTEGERS ! &quot;) TEST(5)=TEXT2DEC(&quot;UNDOCUMENTED LB FEATURE. &quot;) print print &quot;RSA ENCRYPTION DEMO&quot; for i=1 to 5 t1=time$(&quot;ms&quot;) ENCR(i)=FastExp(TEST(i), e, n) t2=time$(&quot;ms&quot;) print TEST(i); print &quot; &quot;;ENCR(i); print &quot; &quot;;t2-t1;&quot; ms&quot; print DEC2TEXT$( TEST(i) );&quot; --&gt; &quot;;DEC2TEXT$( ENCR(i) ) print next i print print &quot;&quot; print print &quot;RSA DECRYPTION DEMO&quot; for i=1 to 5 t1=time$(&quot;ms&quot;) DECR(i)=FastExp(ENCR(i), d, n) t2=time$(&quot;ms&quot;) print ENCR(i); print &quot; &quot;;DECR(i); print &quot; &quot;;t2-t1;&quot; ms&quot; print DEC2TEXT$( ENCR(i) );&quot; --&gt; &quot;;DEC2TEXT$( DECR(i) ) print next i print &quot; &quot; print print &quot;RSA Demo Finished.&quot; [stop] END Function GCD( m,n ) ' Find greatest common divisor with Extend Euclidian Algorithm ' Knuth Vol 1 P.13 Algorithm E ap =1 :b =1 :a =0 :bp =0: c =m :d =n [StepE2] q = int(c/d) :r = c-q*d if r&lt;&gt;0 then c=d :d=r :t=ap :ap=a :a=t-q*a :t=bp :bp=b :b=t-q*b 'print ap;&quot; &quot;;b;&quot; &quot;;a;&quot; &quot;;bp;&quot; &quot;;c;&quot; &quot;;d;&quot; &quot;;t;&quot; &quot;;q goto [StepE2] end if GCD=a*m+b*n 'print ap;&quot; &quot;;b;&quot; &quot;;a;&quot; &quot;;bp;&quot; &quot;;c;&quot; &quot;;d;&quot; &quot;;t;&quot; &quot;;q End Function 'Extended Euclidian GCD Function ExtBinEuclid( u, v ) k=0 :t1=0 :t2=0 :t3=0 if u&lt;v then temp=u u=v v=temp end if while (IsEven( u ) and IsEven( v )) k = k+1 u = int(u/2) v = int(v/2) wend u1 = 1: u2 = 0: u3 =u: t1 =v: t2 =u-1: t3 =v [Loop1] 'two labels with no code! [Loop2] ' print &quot;*&quot; if (IsEven(u3)) then if IsOdd(u1) or IsOdd(u2) then u1=u1+v u2=u2+u end if u1=int(u1/2) u2=int(u2/2) u3=int(u3/2) end if if IsEven(t3) or (u3&lt;t3) then temp=u1: u1=t1: t1=temp temp=u2: u2=t2: t2=temp temp=u3: u3=t3: t3=temp end if if IsEven(u3) then goto [Loop2] end if while u1&lt;t1 OR u2&lt;t2 u1=u1+v: u2=u2+u wend u1=u1-t1: u2=u2-t2: u3=u3-t3 if (t3&gt;0) then goto [Loop1] end if while u1&gt;=v AND u2&gt;=u u1=ul-v: u2=u2-u wend ExtBinEuclid=u-u2 End Function function IsEven( x ) if ( x MOD 2 )=0 then IsEven=1 else IsEven=0 end if end function function IsOdd( x ) if ( x MOD 2 )=0 then IsOdd=0 else IsOdd=1 end if end function Function FastExp(x, y, N) if (y=1) then 'MOD(x,N) FastExp=x-int(x/N)*N goto [ExitFunction] end if if ( y and 1) = 0 then dum1=y/2 dum2=y-int(y/2)*2 'MOD(y,2) temp=FastExp(x,dum1,N) z=temp*temp FastExp=z-int(z/N)*N 'MOD(temp*temp,N) goto [ExitFunction] else dum1=y-1 dum1=dum1/2 temp=FastExp(x,dum1,N) dum2=temp*temp temp=dum2-int(dum2/N)*N 'MOD(dum2,N) z=temp*x FastExp=z-int(z/N)*N 'MOD(temp*x,N) goto [ExitFunction] end if [ExitFunction] end function Function PowMod( a, n, m) r = 1 while (n &gt; 0) if (n AND 1) then '/* test lowest bit */ r = MulMod(r, a, m) '/* multiply (mod m) */ end if a = MulMod(a, a, m) '/* square */ n = int(n/2) '/* divided by 2 */ wend PowMod=r End Function Function MulMod( a, b, m) if (m = 0) then MulMod=a * b ' /* (mod 0) */ Else r = 0 while (a &gt; 0) if (a AND 1) then ' /* test lowest bit */ r= r+b if (r &gt; m) then r = (r MOD m) ' /* add (mod m) */ end if end if a = int(a/2) ' /* divided by 2 */ b = b*2 if (b &gt; m) then b = (b MOD m) ' /* times 2 (mod m) */ end if wend MulMod=r End If End Function Function rand( x ) x=x*5 x=x+1 rand=x End Function Function MillerRabin(n,b) 'print &quot;Miller Rabin&quot; 't1=time$(&quot;ms&quot;) if IsEven(n) then MillerRabin=1 goto [ExtFn] end if i=0 [Loop] i=i+1 if i&gt;1000 then goto [Continue] if ( n MOD SmallPrimes(i) )=0 then MillerRabin=1 goto [ExtFn] end if goto [Loop] [Continue] if GCD(n,b)&gt;1 then MillerRabin=1 goto [ExtFn] end if q=n-1 t=0 while (int(q) AND 1 )=0 t=t+1 q=int(q/2) wend r=FastExp(b, q, n) if ( r &lt;&gt; 1 ) then e=0 while ( e &lt; (t-1) ) if ( r &lt;&gt; (n-1) ) then r=FastExp(r, r, n) else Exit While end if e=e+1 wend [ExitLoop] end if if ( (r=1) OR (r=(n-1)) ) then MillerRabin=0 else MillerRabin=1 end if [ExtFn] End Function Function Random( Digits ) ' x=INT(RND(1)*TIME$(&quot;ms&quot;)*9912812828239112219) * INT(RND(1)*9912166437771297131373) * ' INT(RND(1)*71777126181142123) * INT(RND(1)*7119119672435637981) * ' INT(RND(1)*991216643912127789) * INT(RND(1)*79126181142123) * ' INT(RND(1)*711911128376332417) * INT(RND(1)*991216643123129) * ' INT(RND(1)*79126181142123) * INT(RND(1)*6661912727312317) ' Random=INT(VAL(RIGHT$(STR$(x,1))) x=INT(RND(1)*TIME$(&quot;ms&quot;)*9912812828239112219) * INT(RND(1)*9912166437771297131373) *_ INT(RND(1)*71777126181142123) * INT(RND(1)*7119119672435637981) *_ INT(RND(1)*991216643912127789) * INT(RND(1)*79126181142123) *_ INT(RND(1)*711911128376332417) x=x*x+x+41 y$=mid$(str$(x),INT(rnd(1)*30+1),Digits ) ldg=val(right$(y$,1)) z=0 if ldg=0 then z=1 if ldg=2 then z=1 if ldg=4 then z=1 if ldg=6 then z=1 if ldg=8 then z=1 Random=val(y$)+z End Function FUNCTION TEXT2DEC( x$ ) a$=UPPER$(x$) y$=&quot;&quot; FOR i=1 TO LEN(a$) y$=y$+STR$(ASC(MID$(a$,i,1))) NEXT TEXT2DEC=VAL(y$) END FUNCTION FUNCTION DEC2TEXT$( n ) a$=STR$(n) y$=&quot;&quot; FOR i=1 TO LEN(a$)-1 STEP 2 m=VAL(MID$(a$,i,2)) if m&gt;30 and m&lt;99 then y$=y$+CHR$(m) else y$=y$+&quot;.&quot; NEXT DEC2TEXT$=y$ END FUNCTION data 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 data 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 data 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 data 127, 131, 137, 139, 149, 151, 157, 163, 167, 173 data 179, 181, 191, 193, 197, 199, 211, 223, 227, 229 data 233, 239, 241, 251, 257, 263, 269, 271, 277, 281 data 283, 293, 307, 311, 313, 317, 331, 337, 347, 349 data 353, 359, 367, 373, 379, 383, 389, 397, 401, 409 data 419, 421, 431, 433, 439, 443, 449, 457, 461, 463 data 467, 479, 487, 491, 499, 503, 509, 521, 523, 541 data 547, 557, 563, 569, 571, 577, 587, 593, 599, 601 data 607, 613, 617, 619, 631, 641, 643, 647, 653, 659 data 661, 673, 677, 683, 691, 701, 709, 719, 727, 733 data 739, 743, 751, 757, 761, 769, 773, 787, 797, 809 data 811, 821, 823, 827, 829, 839, 853, 857, 859, 863 data 877, 881, 883, 887, 907, 911, 919, 929, 937, 941 data 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013 data 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069 data 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151 data 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223 data 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291 data 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373 data 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451 data 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511 data 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583 data 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657 data 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733 data 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811 data 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889 data 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987 data 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053 data 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129 data 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213 data 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287 data 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357 data 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423 data 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531 data 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617 data 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687 data 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741 data 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819 data 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903 data 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999 data 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079 data 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181 data 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257 data 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331 data 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413 data 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511 data 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571 data 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643 data 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727 data 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821 data 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907 data 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989 data 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057 data 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139 data 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231 data 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297 data 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409 data 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493 data 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583 data 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657 data 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751 data 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831 data 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937 data 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003 data 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087 data 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179 data 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279 data 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387 data 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443 data 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521 data 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639 data 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693 data 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791 data 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857 data 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939 data 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053 data 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133 data 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221 data 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301 data 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367 data 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473 data 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571 data 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673 data 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761 data 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833 data 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917 data 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997 data 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103 data 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207 data 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297 data 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411 data 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499 data 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561 data 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643 data 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723 data 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829 data 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919 </pre>
 
 
 <img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/>