By Keijo Ruohonen
(translation by means of Jussi Kangas and Paul Coughlan)
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Additional info for Mathematical cryptology
4 (iii)). Hence, if the number m has primitive roots at all, there are φ(φ(m)) of them.
If the signature is all that matters and not keeping the message safe, it is enough for A to send B the pair (w, dA(w)). This simplest version of verification/signature is vulnerable and there are better protocols, see Chapter 13. 3 Rise and Fall of Knapsack Cryptosystems An example of the effects of the preceding section’s complexity considerations is the fate of the well-known public-key system KNAPSACK3 or the knapsack system. The knapsack system is based on the so-called knapsack problem. Its input is (a, m) where a = (a1 , a2 , .
M), and 1 ≤ g < m then g is a so-called primitive root of m or a primitive root modulo m. Of course, in this case necessarily gcd(g, m) = 1. Since then the powers 1, g, g 2, . . , g φ(m)−1 are not congruent—otherwise the smaller power could be divided out from the congruence and a lower order for g would be obtained—and there are φ(m) of them, they actually form a reduced residue system. 1 . 5. A number m ≥ 2 has primitive roots if and only if it is either 2 or 4 or of the form pk or 2pk where p is an odd prime.