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CryptographyWithLB103
LIST ARTICLES HERE

• ArduinoAsSlaveBoard - - mmiscool Edit
• Debugging Techniques - - JackKelly6 6 … Cryptography with Liberty Basic 103: Public Key Cryptography: RSA Algorithm By Onur Alver ( CryptoMan ) INTRODUCTION Table of Contents Cryptography with Liberty Basic 103: Public Key Cryptography: R SA Algorithm We will continue in our series of articles about Cryptography with Public Key Cryptography. RSA ALGORITHM RSA Algorithm is based on the assumption on the difficulty of factoring large composite numbers. Code: dim stats(11) dim SmallPrimes(1000) 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 playwave "ding.wav", async print "Liberty Basic RSA Demonstration" print "Loading Small Primes" for i=1 to 1000: read x : SmallPrimes(i)=x :next :NoOfSmallPrimes=1000 print NoOfSmallPrimes;" Primes Loaded" print"Generating Random Primes" 'goto [Jump] for i=1 to 2 t1=time$("ms") [TryAnother] print print "Prime No ";i if i=1 then x=Random(30) else x=Random(30) iterations=0 [Loop] iterations=iterations+1 if MillerRabin(x,7)=1 then 'print "Composite" x=x+2 goto [Loop] else t2=time$("ms") print x;" Probably Prime. Generated in ";t2-t1;" milliseconds" playwave "tada.wav", async end if if p then q=x else p=x next i print playwave "tada.wav", async print print "p=";dechex$(p) [Retry] restore print "q=";dechex$(q) 'Common modulus N=(p)(q) n=p*q print "Key Length ";len(dechex$(n))*4;" bits " print 'Euler Totient Number M=(p-1)(q-1) m=(p-1)*(q-1) 'Choose a suitable prime E relatively prime to M for i=1 to 12 read e if (GCD(e,m)=1) then goto [Start] next i [Start] print "Common Modulus, n=";dechex$(n) print "Euler-Totient No, m=";dechex$(m) print "Public Exponent, e=";dechex$(e) d=ExtBinEuclid( e, m ) print "Secret Exponent, d=";dechex$(d) DIM TEST(10) DIM ENCR(10) DIM DECR(10) TEST(1)=TEXT2DEC("LIBERTY BASIC IS THE BEST") TEST(2)=TEXT2DEC("WHICH BASIC CAN DO THIS ") TEST(3)=TEXT2DEC("WITHOUT CALLING EXT DLL ?") TEST(4)=TEXT2DEC("LB CAN DO BIG INTEGERS ! ") TEST(5)=TEXT2DEC("UNDOCUMENTED LB FEATURE. ") print print "RSA ENCRYPTION DEMO" for i=1 to 5 t1=time$("ms") ENCR(i)=FastExp(TEST(i), e, n) t2=time$("ms") print TEST(i); print " ";ENCR(i); print " ";t2-t1;" ms" print DEC2TEXT$( TEST(i) );" --> ";DEC2TEXT$( ENCR(i) ) print next i print print "" print print "RSA DECRYPTION DEMO" for i=1 to 5 t1=time$("ms") DECR(i)=FastExp(ENCR(i), d, n) t2=time$("ms") print ENCR(i); print " ";DECR(i); print " ";t2-t1;" ms" print DEC2TEXT$( ENCR(i) );" --> ";DEC2TEXT$( DECR(i) ) print next i print " " print print "RSA Demo Finished." playwave "ding.wav", async [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<>0 then c=d :d=r :t=ap :ap=a :a=t-q*a :t=bp :bp=b :b=t-q*b 'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q goto [StepE2] end if GCD=a*m+b*n 'print ap;" ";b;" ";a;" ";bp;" ";c;" ";d;" ";t;" ";q End Function 'Extended Euclidian GCD Function ExtBinEuclid( u, v ) k=0 :t1=0 :t2=0 :t3=0 if u<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] [Loop2] ' print "*" 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<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<t1 OR u2<t2 u1=u1+v: u2=u2+u wend u1=u1-t1: u2=u2-t2: u3=u3-t3 if (t3>0) then goto [Loop1] end if while u1>=v AND u2>=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 > 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 > 0) if (a AND 1) then ' /* test lowest bit */ r= r+b if (r > 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 > 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 "Miller Rabin" 't1=time$("ms") if IsEven(n) then MillerRabin=1 goto [ExtFn] end if i=0 [Loop] i=i+1 if i>1000 then goto [Continue] if ( n MOD SmallPrimes(i) )=0 then MillerRabin=1 goto [ExtFn] end if goto [Loop] [Continue] if GCD(n,b)>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 <> 1 ) then e=0 while ( e < (t-1) ) if ( r <> (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$("ms")*9912812828239112219) * INT(RND(1)*9912166437771297131373) * INT(RND(1)*71777126181142123) * INT(RND(1)*7119119672435637981) * INT(RND(1)*991216643912127789) * INT(RND(1)*79126181142123) * INT(RND(1)*711911128376332417) * INT(RND(1)*991216643123129) * INT(RND(1)*79126181142123) * INT(RND(1)*6661912727312317) ' Random=INT(VAL(RIGHT$(STR$(x,1))) x=INT(RND(1)*TIME$("ms")*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$) y$="" 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$="" FOR i=1 TO LEN(a$)-1 STEP 2 m=VAL(MID$(a$,i,2)) if m>30 and m<99 then y$=y$+CHR$(m) else y$=y$+"." NEXT DEC2TEXT$=y$ END FUNCTION