RSA demo \302\2512007 by Mike May, S.J., Saint Louis University - maymk@slu.edu A fast demo of RSA cryptography- This worksheet uses components so that the Maple code is hidden.

Set a size for the primes used in terms of bits. (The size must be between 5 and 600 bits.) Size of p1use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. s1 := parse(GetProperty(Textp1size,'value')): if not((type(s1,integer)) and (s1 > 5)) then s1 := 5; end if; if (s1 > 600) then s1 := 600; end if; SetProperty(Textp1size,'value',s1); end use; in bits. Size of p2 use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. s2 := parse(GetProperty(Textp2size,'value')): if not((type(s1,integer)) and (s2 > 5)) then s2 := 5; end if; if (s2 > 600) then s1 := 600; end if; SetProperty(Textp2size,'value',s2); end use; in bits. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. s1 := parse(GetProperty(Textp1size,'value')); s2 := parse(GetProperty(Textp2size,'value')); p1 := nextprime(rand(2^s1)()); p2 := nextprime(rand(2^s2)()); SetProperty(Textp1,'value',p1); SetProperty(Textp2,'value',p2); n := p1*p2; n1 := (p1-1)*(p2-1); SetProperty(Textn, 'value', n); SetProperty(Textphin, 'value', n1); end use; p1 = , p2 = We also want to compute n = p1*p2 and phi(n) = (p1-1)*(p2-1). If we let Maple choose the primes, that was done by the last button. If we insert our own primes, the next button computes n and phi(n). use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. p1 := parse(GetProperty(Textp1,'value')); p2 := parse(GetProperty(Textp2,'value')); if (isprime(p1) and isprime(p2)) then n := p1*p2; n1 := (p1-1)*(p2-1); else n := "Either p1 or p2 is not prime"; n1 := n; end if: SetProperty(Textn, 'value', n); SetProperty(Textphin, 'value', n1); end use; n = phi(n) = Set the encryption key. The default is e = 2^16+1 = 65537. Recall that phi(n) and e should be relatively prime. e = . We now want to compute d, the inverse of e mod phi(n). use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Texte, 'value')); phin := parse(GetProperty(Textphin, 'value')); if (igcd(e,phin)=1) then; d := (1/e) mod phin; else; d := "Bad choice for e in relation to phi(n), try again"; end if; SetProperty(Textd,'value',d); end use; d = We would like to publish n and e, destroy p1 and p2, and keep d a secret.

We are ready to send numerical message, Recall that the message should be less than n. message = The ciphertext is message^e mod n. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. m := parse(GetProperty(Textmessage,'value')); e := parse(GetProperty(Texte,'value')); n := parse(GetProperty(Textn,'value')); if ((m<n) and type(m,integer)) then; c := Power(m,e) mod n; else; c := "Bad message, try again"; end if; SetProperty(Textciphertext,'value',c); end use; ciphertext = We can also check deciphering use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. c := parse(GetProperty(Textciphertext,'value')); SetProperty(Textdeciphermessage,'value',c); end use; deciphermessage= The decrypted message is ciphertext^d mod n. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. n := parse(GetProperty(Textn,'value')); d := parse(GetProperty(Textd,'value')); c := parse(GetProperty(Textdeciphermessage,'value')); m := Power(c,d) mod n; SetProperty(Textdecodedmessage,'value',m); end use; decodedmessage=

We are ready to look at sending and receiving ASCII messages. for encryption and decryption we need a modulus and an exponent. These can either be typed in or copied from above. modulus = use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. n := parse(GetProperty(Textn,'value')); SetProperty(Textmodulus,'value',n); end use; exponent = use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Textd,'value')); if not(type(e,integer)) then e := "bad value"; end if: SetProperty(Textexponent,'value',e); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Texte,'value')); if not(type(e, integer)) then e := "bad value"; end if; SetProperty(Textexponent,'value',e); end use; If we are encrypting, we start with the text of the message. text = use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. p1 := GetProperty(Texttext,'value'); p2 := convert(p1,bytes); num := 0: for count from 1 to nops(p2) do num := 256*num + op(count,p2); end do: SetProperty(Textnumber,'value',num); end use; If we are encrypting, we convert the text to a number, copy e into the exponent, and encrypt. In doing the conversion we consider the ASCII message as a number base 256, with characters representing their numerical equivalent in ASCII. Thus "a" converts to 97, "b" to 98, "A" to 65, "B" to 66, and so on. "abc" converts to 97*256^2+98*256^1+99*256^0. If we are decrypting, we copy the encrypted number, copy d into the exponent, and decrypt. number = use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num := parse(GetProperty(Textnumber,'value')); exponent := parse(GetProperty(Textexponent,'value')); modulus := parse(GetProperty(Textmodulus,'value')); if (type(num,integer)) then if (num < modulus) then num2 := Power(num, exponent) mod modulus; else num2 := "error, number too big for modulus"; end if: else num2 := "error, number must be an integer."; end if: (SetProperty(Textnumber2,'value', num2)); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Textd,'value')); if not(type(e,integer)) then e := "bad value"; end if: SetProperty(Textexponent,'value',e); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Texte,'value')); if not(type(e, integer)) then e := "bad value"; end if; SetProperty(Textexponent,'value',e); end use; number2 = If we know both d and e, we can check our work by raising to the other exponent to get back to our original number. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num := parse(GetProperty(Textnumber2,'value')); exponent := parse(GetProperty(Textexponent,'value')); modulus := parse(GetProperty(Textmodulus,'value')); if (type(num,integer)) then if (num < modulus) then num2 := Power(num, exponent) mod modulus; else num2 := "error, number too big for modulus"; end if: else num2 := "error, number must be an integer."; end if: (SetProperty(Textnumber3,'value', num2)); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Textd,'value')); if not(type(e,integer)) then e := "bad value"; end if: SetProperty(Textexponent,'value',e); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. e := parse(GetProperty(Texte,'value')); if not(type(e, integer)) then e := "bad value"; end if; SetProperty(Textexponent,'value',e); end use; number3 = Finally, we can convert numbers back to ASCII. If we have deciphered, this finally gives the message in readable form. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num1 := parse(GetProperty(Textnumber2,'value')); l1 := []: while num1 > 0 do; l1 := [op(l1), irem(num1, 256)]; num1 := iquo(num1, 256); end do: l1 := ListTools[Reverse](l1): p3 := convert(l1,bytes); SetProperty(Texttext2,'value',p3) end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num := parse(GetProperty(Textnumber2,'value')); SetProperty(Textnumber,'value',num); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num1 := parse(GetProperty(Textnumber3,'value')); l1 := []: while num1 > 0 do; l1 := [op(l1), irem(num1, 256)]; num1 := iquo(num1, 256); end do: l1 := ListTools[Reverse](l1): p3 := convert(l1,bytes); SetProperty(Texttext2,'value',p3) end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. num := parse(GetProperty(Textnumber3,'value')); SetProperty(Textnumber,'value',num); end use; text2 =

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