rs
Solution
How RSA works (sorry if I use different variables from your class; the writing on the slide you
included is too small to see):
1) Choose two distinct primes p and q. Let pq = n.
2) Calculate (n) = (pq) = (p - 1)(q - 1)
(where is Euler\'s totient function, i.e., (n) = the number of positive integers less than or equal to
n that are coprime to n)
3) Find an e such that gcd((n), e) = 1
4) Compute d such that de = 1 (mod (n)) (i.e., d the modular multiplicative inverse of e mod
(pq))
e and n will be made public for people to encrypt messages with. d should remain private and
will decrypt messages. (Obviously, (n) should also remain private as it would allow people to
find d. For strong cryptography, n=pq should be sufficiently large that it is hard to factor and
thus it is hard to guess (n).)
To encrypt a message m (where m < n):
1) Compute c = me (mod n). This is the encrypted message.
To decrypt c:
1) Compute cd (mod n) = (me)d (mod n) = (med) (mod n) =* m1 (mod n) = m.
*this equality works because of Euler\'s Theorem, a^((p)) = 1 mod p (and here we have de = 1
(mod (n)))
Your problem:
Encrypt \"HOW ARE YOU\". First convert to an integer string as described.
A = 00, B = 01, C = 02, D = 03, E = 04, F = 05, G = 06, H = 07, I = 08, J = 09, K = 10, L = 11,
M = 12, N = 13, O = 14, P = 15, Q = 16, R = 17, S = 18, T = 19, U = 20, V = 21, W = 22, X =
23, Y = 24, Z = 25, space = 26.
HOW ARE YOU = 07 14 22 26 00 17 04 26 24 14 20
= 0714222600170426241420
To encrypt, we need the message integer to be less than our chosen n, and n = 100. So we\'ll keep
the message broken up into chunks of two. (Interestingly, n should be a product of two primes,
and they\'ve given you an n that isn\'t. But we\'ll apply the RSA encryption algorithm anyway.)
To encrypt each message chunk:
c = me (mod n)
where c is the encrypted message chunk, m is the original message chunk, e is the given value of
13, and n is the given value of 100. Encrypt each chunk:
m = 07
c = 713 (mod 100)
c = 7
m = 14
c = 1413 (mod 100)
c = 44
m = 22
c = 2213 (mod 100)
c = 52
m = 26
c = 2613 (mod 100)
c = 76
m = 00
c = 013 (mod 100)
c = 0
m = 17
c = 1713 (mod 100)
c = 37
m = 04
c = 413 (mod 100)
c = 64
m = 26
c = 2613 (mod 100)
c = 76
m = 24
c = 2413 (mod 100)
c = 24
m = 14
c = 1413 (mod 100)
c = 44
m = 20
c = 2013 (mod 100)
c = 0
Combine all the c\'s together into an encrypted message. For each m -> c chunk we have:
07 -> 07
14 -> 44
22 -> 52
26 -> 76
00 -> 00
17 -> 37
04 -> 64
26 -> 76
24 -> 24
14 -> 44
20 -> 00
So the whole encrypted message c is:
07 44 52 76 00 37 64 76 24 44 00
or
0744527600376476244400.
c = 2613 (mod 100)
c = 76.
rsSolutionHow RSA works (sorry if I use differ.pdf
1. rs
Solution
How RSA works (sorry if I use different variables from your class; the writing on the slide you
included is too small to see):
1) Choose two distinct primes p and q. Let pq = n.
2) Calculate (n) = (pq) = (p - 1)(q - 1)
(where is Euler's totient function, i.e., (n) = the number of positive integers less than or equal to
n that are coprime to n)
3) Find an e such that gcd((n), e) = 1
4) Compute d such that de = 1 (mod (n)) (i.e., d the modular multiplicative inverse of e mod
(pq))
e and n will be made public for people to encrypt messages with. d should remain private and
will decrypt messages. (Obviously, (n) should also remain private as it would allow people to
find d. For strong cryptography, n=pq should be sufficiently large that it is hard to factor and
thus it is hard to guess (n).)
To encrypt a message m (where m < n):
1) Compute c = me (mod n). This is the encrypted message.
To decrypt c:
1) Compute cd (mod n) = (me)d (mod n) = (med) (mod n) =* m1 (mod n) = m.
2. *this equality works because of Euler's Theorem, a^((p)) = 1 mod p (and here we have de = 1
(mod (n)))
Your problem:
Encrypt "HOW ARE YOU". First convert to an integer string as described.
A = 00, B = 01, C = 02, D = 03, E = 04, F = 05, G = 06, H = 07, I = 08, J = 09, K = 10, L = 11,
M = 12, N = 13, O = 14, P = 15, Q = 16, R = 17, S = 18, T = 19, U = 20, V = 21, W = 22, X =
23, Y = 24, Z = 25, space = 26.
HOW ARE YOU = 07 14 22 26 00 17 04 26 24 14 20
= 0714222600170426241420
To encrypt, we need the message integer to be less than our chosen n, and n = 100. So we'll keep
the message broken up into chunks of two. (Interestingly, n should be a product of two primes,
and they've given you an n that isn't. But we'll apply the RSA encryption algorithm anyway.)
To encrypt each message chunk:
c = me (mod n)
where c is the encrypted message chunk, m is the original message chunk, e is the given value of
13, and n is the given value of 100. Encrypt each chunk:
m = 07
c = 713 (mod 100)
c = 7
m = 14
c = 1413 (mod 100)
c = 44
m = 22
c = 2213 (mod 100)
c = 52
m = 26
c = 2613 (mod 100)
c = 76
m = 00
c = 013 (mod 100)
c = 0
m = 17
c = 1713 (mod 100)
c = 37
m = 04
3. c = 413 (mod 100)
c = 64
m = 26
c = 2613 (mod 100)
c = 76
m = 24
c = 2413 (mod 100)
c = 24
m = 14
c = 1413 (mod 100)
c = 44
m = 20
c = 2013 (mod 100)
c = 0
Combine all the c's together into an encrypted message. For each m -> c chunk we have:
07 -> 07
14 -> 44
22 -> 52
26 -> 76
00 -> 00
17 -> 37
04 -> 64
26 -> 76
24 -> 24
14 -> 44
20 -> 00
So the whole encrypted message c is:
07 44 52 76 00 37 64 76 24 44 00
or
0744527600376476244400.
c = 2613 (mod 100)
c = 76
