Cryptography with Liberty BASIC : 103 RSA Algorithm

Onur Alver (CryptoMan)
Cryptography with Liberty BASIC : 103 RSA Algorithm

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RSA ALGORITHM


There is an excellent description about RSA here . https://www.di-mgt.com.au/rsa_alg.html

Step by Step RSA Description

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:

n =
11929413484016950905552721133125564964460656966152763801206748195494305685115033
38063159570377156202973050001186287708466899691128922122454571180605749959895170
80042105263427376322274266393116193517839570773505632231596681121927337473973220
312512599061231322250945506260066557538238517575390621262940383913963

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 "Which d multiplied by e divided by t have a remainder of 1?"

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

Simple RSA Example with Small Primes

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

Demonstration Program in Liberty Basic

dim stats(11)
dim SmallPrimes(1000)
 
[begin]
    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"
 
    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$("ms")
            print x;" Probably Prime. Generated in ";t2-t1;" milliseconds"
        end if
        if p then q=x else p=x
    next i
    print
    print "p=";dechex$(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."
 
[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]
    'two labels with no code!
 
    [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
 
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
 
 
 

Cryptography with Liberty BASIC : 103 RSA Algorithm