1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
Algorithm 1 – 5 Key generation in ECC [78]
Require: E is an EC over F, P is a point of order n on curve E.
Ensure: Pair of private key and public key.
1: Choose a random integer d, 0 < d < n
2: Q ⇐ dP
3: return Q, the public key
4: return d, the private key
Algorithm 1 – 6 Message signing in ECC [78]
Require: Message m with an arbitrary length, a hash value h(m) obtained from a
one-way function.
Ensure: Signature of the message m.
1: Choose random integer k, 0 < k < n
2: kP ⇐ (x1, y1) and r ⇐ x1 mod n (0 < x1 < q − 1)
3: if r = 0 then
4: Go back to the step 1.
5: end if
6: k −1 mod n
7: s ⇐ k −1 {h(m) + dr} mod n
8: if s = 0 then
9: Go back to the step 1.
10: end if
11: return (r, s)
1.4 Conclusions
In this section we have presented two nowadays most important public key cryp-
tosystems, namely RSA and ECC.
While RSA is massively applied by industry since several years, the ECC as
relatively new cryptographic algorithms just starts to win as better choice for im-
plementation of public-key algorithm especially for energy- and place-limited plat-
forms. The possibility to use much shorter key, and therefore less heavy arithmetical
operations makes from ECC an optimal algorithm for hardware implementation.
The description of both algorithms given in the thesis focuses on their most
intensively-used and heavy operation - the modular multiplication. This fact makes
from the multiplication an important target for our research as the improvements
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