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Lecture 8, DSA

The document outlines the Digital Signature Algorithm (DSA), detailing the key generation process, signature creation, and verification steps. It provides specific examples with calculations for clarity, including the selection of prime numbers and the computation of keys and signatures. Additionally, it includes practice problems for further understanding of the algorithm.

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ilya.siraev.2018
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21 views8 pages

Lecture 8, DSA

The document outlines the Digital Signature Algorithm (DSA), detailing the key generation process, signature creation, and verification steps. It provides specific examples with calculations for clarity, including the selection of prime numbers and the computation of keys and signatures. Additionally, it includes practice problems for further understanding of the algorithm.

Uploaded by

ilya.siraev.2018
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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DSA (Digital Signature Algorithm)

Al-Tarazi Assaubay
Senior Lecturer
Department of Computational and Data Sciences
Department of Intelligent Systems and Cybersecurity
Algorithm, part 1: keys

Global keys:
1 p - prime number
2 q - prime divisor of p − 1
3 h - any integer 2 ≤ h ≤ p − 2 (usually h = 2 is used)
4 g = h(p−1)/q (mod p) ̸= 1
User keys:
1 private key x : 1 ≤ x ≤ q − 1
2 public key y = gx (mod p)

DSA (Digital Signature Algorithm) 2/8


Algorithm, part 2: signature

1 k - random integer 1 ≤ k ≤ q − 1
2 (Hash(M)) (mod q), but we use M : 0 ≤ M ≤ q − 1
3 s1 = (gk (mod p)) (mod q), (s1 ̸= 0)
4 s2 = k−1 (M + x · s1 ) (mod q) (s2 ̸= 0)

DSA (Digital Signature Algorithm) 3/8


Algorithm, part 3: verification

1 1 ≤ s1 ≤ q − 1, 1 ≤ s2 ≤ q − 1
2 u1 = M · s−1
2 (mod q)
−1
3 u2 = s1 · s2 (mod q)
gu1 · yu2

4 v= (mod p) (mod q)
5 v = s1 means the signature is authentic.

DSA (Digital Signature Algorithm) 4/8


Example, part 1

Choose a prime p = 53
Since q is a prime divisor of p − 1, we factorize 53 − 1:

53 − 1 = 52 = 4 · 13 = 22 · 13, thus, let q = 13

Choose standard h = 2 and compute g:

g = 252/13 (mod 53) = 16

Choose x: 1 ≤ x ≤ 12. Say, x = 7


Compute y
y = 167 (mod 53) = 49

DSA (Digital Signature Algorithm) 5/8


Example, part 2

Let M = 5, k = 5
Then k−1 (mod q) = 5−1 (mod 13) = 8
Compute s1 = 165 (mod 53) (mod 13) = 24 (mod 13) = 11


Compute s2 = 8(5 + 7 · 11) (mod 13) = 6

DSA (Digital Signature Algorithm) 6/8


Example, part 3

Compute s−1
2 (mod q) = 6
−1 (mod 13) = 11

Compute u1 = M · s−1
2 (mod q) = 5 · 11 (mod 13) = 3

Compute u2 = s1 · s−1
2 (mod 13) = 11 · 11 (mod 13) = 4

Compute v = gu1 · yu2 (mod 53) (mod 13):




v = 163 · 494 (mod 53) (mod 13) = 15 · 44 (mod 53) (mod 13)
 

= 660 (mod 53) (mod 13) = 24 (mod 13) = 11.

Since v = s1 , the signature is authentic.

DSA (Digital Signature Algorithm) 7/8


Practice

1 Let p = 23, q = 11, h = 2, compute g.


Then let x = 9, compute y.
Compute the signature (s1 , s2 ) for M = 10, k = 7.
Verify the signature.
2 Let p = 47, q = 23, h = 2, compute g.
Then let x = 17, compute y.
Compute the signature (s1 , s2 ) for M = 21, k = 6.
Verify the signature.

DSA (Digital Signature Algorithm) 8/8

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