1. CS/COE 1501 Recitation RSA Encryption/Decryption
Extended Euclidean Algorithm Digital Signatures

2. RSA Encryption Say Alice wants to send a message to Bob
Looks up Bob’s public key
Convert the message into an integer: m
Compute the ciphertext c as:
c = me (mod n)
Send c to Bob

3. RSA Decryption Bob can simply: Compute m as: m = cd (mod n)
Convert m into Alice’s message

4. RSA Cryptosystem What are public/private keys? How messages encrypted?
How are messages decrypted?
How are keys generated?
Why is it secure?

5. RSA Cryptosystem What are public/private keys? Public Key = (e, n)
Private Key = (d, n)
How messages encrypted?
c = me (mod n)
How are messages decrypted?
m = cd (mod n)
How are keys generated?

6. How are keys generated? 4. Choose e such that
1. Choose two prime number p and q
2. Compute n = p * q
3. Compute φ(n)
φ(n) = φ(p) * φ(q) = (p - 1) * (q - 1)
4. Choose e such that
1 < e < φ(n), GCD(e, φ(n)) = 1
i.e., e and φ(n) are co-prime
5. Determine d as d ≡ e-1 mod(φ(n))

7. An Example 4. Choose e such that 1. Choose two prime number p and q
p=3, q=11
2. Compute n = p * q
n=3*11=33
3. Compute φ(n)
φ(n) = φ(p) * φ(q) = (p - 1) * (q - 1)
φ(n)=(3-1)*(11-1)=20
4. Choose e such that
1 < e < φ(n), GCD(e, φ(n)) = 1
i.e., e and φ(n) are co-prime
We can choose e=3, verify that 1<3< φ(n) =20, 3 and 20 are co-prime

8. An Example 5. Determine d as d ≡ e-1 (mod φ(n)) e * d ≡ 1 (mod φ(n))
Intuition, search from d=0, until e * d mod 20 = 1
We have already chosen e=3, here we choose d=7
We now get our Public Key and Private Key
Public Key=(e,n)=(3,33)
Private Key=(d,n)=(7,33)

9. An Example Encryption: c = me (mod n) Decryption: m = cd (mod n)
Public Key=(e,n)=(3,33)
Private Key=(d,n)=(7,33)
Alice said “hello” 7, 4, 11, 11, 14
Encrypt msg: 73 mod 33, 43 mod 33, 113 mod 33, 113 mod 33, 143 mod 33
Encrypted msg: 13, 31, 11, 11, 5

10. An Example Encryption: c = me (mod n) Decryption: m = cd (mod n)
Public Key=(e,n)=(3,33)
Private Key=(d,n)=(7,33)
Bob receive 13, 31, 11, 11, 5
Decrypt msg: 137 mod 33, 317 mod 33, 117 mod 33, 117 mod 33, 57 mod 33
Decrypt msg: 7, 4, 11, 11, 14 -> ‘hello’

11. Challenges Compute Exponentiation efficiently
137 mod 33, 317 mod 33, 117 mod 33, 117 mod 33, 57 mod 33
Determine d as d ≡ e-1 (mod φ(n)) efficiently
search from d=0, until e * d mod 20 = 1 Improve it?

12. Determine d d = e-1 mod(φ(n)) Means that d = 1/e mod(φ(n))
Means that e * d = 1 (mod φ(n))
Now, this can be equivalently stated as e * d = z * φ(n) + 1
For some z
Can further restate this as: e * d - z * φ(n) = 1
Or similarly: 1 = φ(n) * (-z) + e * d
How can we solve this?
Hint: recall that we know GCD(φ(n), e) = 1

13. Determine d GCD(a, b) = i = as + bt Let:
a = φ(n)
b = e
s = -z
t = d
i = 1
GCD(φ(n), e) = 1 = φ(n) * (-z) + e * d
We can compute d in linear time!

14. Extended Euclidean Algorithm
The extended Euclidean algorithm also computes the GCD of integers a and b, but also computes the Bézout numbers, s and t which satisfy the Bézout identity:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
We first walk through finding the GCD of a and b, then work backworks to find the Bézout numbers, s and t , for the original s and t.
We will fill out the following table, first following the pattern that the b in each step will become the a in the row below. The remainder of a/b becomes the b in the row below.

15. Extended Euclidean Algorithm
We know GCD(a,b)=GCD(b, a%b)
Suppose we are computing the GCD(a,b)
a*x + b*y = gcd
Suppose we already know the GCD(b, a %b), and we find x1 and y1
b*x1+(a%b)*y1=gcd
Associate the two formular
a%b=a-(a/b)*b
gcd= b*x1+(a-(a/b)*b)*y1
= b*x1+a*y1-(a/b)*b*y1
= a*y1+b*(x1-a/b*y1)
x=y1
y=x1-a/b*y1

16. Extended Euclidean Algorithm
public static int[] ExtendedEuclid(int a, int b) {
int[] ans = new int[3];
int q;
if (b == 0) { /* If b = 0, then we're done... */
ans[0] = a;
ans[1] = 1;
ans[2] = 0; }
else
{ /* Otherwise, make a recursive function call */
q = a/b;
ans = ExtendedEuclid (b, a % b);
int temp = ans[1] - ans[2]*q;
ans[1] = ans[2];
ans[2] = temp;
return ans;

17. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 2 3 4 5 6

18. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 2 3 4 5 6

19. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 4 5 6

20. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 4 5 6

21. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 4 5 6

22. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 4 5 6

23. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 4 5 6

24. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 4 5 6

25. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 4 5 6

26. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

27. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

28. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

29. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

30. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

31. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

32. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

33. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5

34. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5
NaN

35. Extended Euclidean Algorithm
We will now walk back up through the table, computing s and t for each step.
The s in each row is set to the t from the row below.
t is set according to the formula:
Remember s and t are defined such that:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

36. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

37. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

38. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 2 3 15 6 4 5
NaN
Because d is the GCD of the two numbers (i.e. the last non-zero remainder) and:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
We know that s and t will always be 1 and 0 in the bottom row (because a will be equal to d).
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

39. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

40. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

41. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

42. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

43. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

44. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
Verify:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
3= 6∗0 + 3∗1
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

45. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

46. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

47. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 -2 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

48. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 -2 5
NaN
Verify:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
3= 15∗1 + 6∗−2
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

49. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 4 -2 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

50. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

51. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

52. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 -2 4 5
NaN
Verify:
gcd 𝑎,𝑏 =𝑎∗𝑠+𝑏∗𝑡
3= 21∗−2 + 15∗3
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

53. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

54. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

55. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3 2 15
-11 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

56. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3
-11 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

57. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3
-11 14 2 15 6 -2 4 5
NaN
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

58. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3
-11 14 2 15 6 -2 4 5
NaN
To check your work, verify:
99∗ −11 +78∗14=3
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

59. Find the Bézout numbers and GCD of 99 and 78
Row a b
a/b
a%b d s t 1 99 78 21 3
-11 14 2 15 6 -2 4 5
NaN
To check your work, verify:
99∗ −11 +78∗14=3
𝑡= 𝑠 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 − 𝑎 𝑏 ∗ 𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠

60. Exercise 5. Determine d as d ≡ e-1 (mod φ(n)) e * d ≡ 1 (mod φ(n))
Intuition, search from d=0, until e * d mod 20 = 1
We have already chosen e=3, here we choose d=7
Now using the Extended Euclidean Algorithm
1 = 20*s+3*t t->d

61. Hash Functions

62. For Crypto Hash Functions, Output Should Appear Random

63. Digital Signatures – Public Key Cryptography

64. Creating a Digital Signature

65. Digital Signatures Often Use Commutative Operations
The order of Decryption and Encryption can be changed
Sending message m together with the decrypted data D(m, k^-1)
Using the public key k to verify message

66. Digital Signatures Often Use Commutative Operations

67. Digital Signatures Often Use Commutative Operations
Plaintext sent by sender

68. Digital Signatures Often Use Commutative Operations
Plaintext sent by sender
Cryptotext sent by sender using sender’s private key

69. Digital Signatures Often Use Commutative Operations
Plaintext sent by sender
Cryptotext sent by sender using sender’s private key
Sender’s public key

70. Digital Signatures Often Use Commutative Operations
Plaintext sent by sender =
Cryptotext sent by sender using sender’s private key
Sender’s public key

71. Digital Signatures Often Use Commutative Operations
Plaintext sent by sender =
Plaintext recovered matches
Cryptotext sent by sender using sender’s private key
Sender’s public key

72. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
We do not apply the algorithm on the plaintext, but on the hashed version

73. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
HASH ALGORITHM
Received:

74. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
HASH ALGORITHM
Received:

75. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
HASH ALGORITHM
Compute
Received:

76. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
HASH ALGORITHM
Compute =
Received:

77. Digital Signatures and Hashes
Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original:
Match.
Signature Verified.
HASH ALGORITHM
Compute =
Received:

78. Acknowledgements Adam J. Lee’s slides from CS 1653