1. MIRACL

2. Outline Why we use the libraries? Elliptic Curves Arithmetic
add, scalar multiplication, random point, …
ECC Protocols
ECDSA, ECIES, ECDH, …
Libraries with ECC implementation
MIRACL: Multiprecision Integer and Rational Arithmetic C/C++ Library

3. Why we use the libraries?
In C/C++
unsigned long long: 64 bits
RSA-1024
p and q are two primes of about 500 bits
How about RSA-32?
How about ECC-32?
RSA-1024 = ECC-160

4. Multi Precision Numbers
In Java:
BigInteger
In C/C++:
int bignum[100] ?
What are the operations, +, -, x, / ?
Efficient ?
Use Libraries!!!

5. Elliptic Curves Arithmetic
Elliptic curves over Zp (p>3)

6. Elliptic Curves Arithmetic
Elliptic curves over Zp (p>3)

7. Elliptic Curves Arithmetic
Elliptic curves over GF(2m)

8. Elliptic Curves Arithmetic
Elliptic curves over GF(2m)

9. Elliptic Curves Arithmetic
How we calculate kP
P+P+P+P+… ?
Double and Add !!!
How we choose a random point on an elliptic curve
Choose x, and compute the suitable y
Choose y, and compute the suitable x
Can I get more functions from the libraries?

10. ECC Protocols
ECDSA
ECDH/ECMQV
ECIES
ECIES-DEM
ECIES-KEM
ECIES-KEM-DEM

11. MIRACL
Multiprecision Integer and Rational Arithmetic Cryptographic Library

12. MIRACL

13. MIRACL File architecture To build the library
To build the library
Visual Studio
Unix-like
MAC

14. MIRACL Build in Ubuntu … unzip -j -aa -L master.zip –d miracl
-j ignores the directory structure inside the zip file
-aa converts all text files to Unix format
-L ensures that all filenames are lower-case
-d specify the directory to extract into
bash linux
bash linux64
bash liunx64_cpp

15. MIRACL Done -rwxr-xr-x 1 randy users 332452 May 22 08:03 schoof
-rw-r--r-- 1 randy users May 22 08:03 poly.o
-rw-r--r-- 1 randy users May 22 08:03 polymod.o
-rwxr-xr-x 1 randy users May 22 08:03 pk-demo
-rwxr-xr-x 1 randy users May 22 08:03 ecsver
-rwxr-xr-x 1 randy users May 22 08:03 ecsign
-rwxr-xr-x 1 randy users May 22 08:03 ecsgen
-rwxr-xr-x 1 randy users May 22 08:03 sample
-rwxr-xr-x 1 randy users May 22 08:03 brent
-rwxr-xr-x 1 randy users May 22 08:03 mersenne
-rwxr-xr-x 1 randy users May 22 08:03 fact
-rwxr-xr-x 1 randy users May 22 08:03 bmark
-rw-r--r-- 1 randy users May 22 08:03 miracl.a
-rw-r--r-- 1 randy users May 22 08:03 crt.o
-rw-r--r-- 1 randy users May 22 08:03 flash.o
-rw-r--r-- 1 randy users May 22 08:03 ec2.o
-rw-r--r-- 1 randy users May 22 08:03 ecn.o
-rw-r--r-- 1 randy users May 22 08:03 zzn.o
-rw-r--r-- 1 randy users May 22 08:03 big.o
-rw-r--r-- 1 randy users May 22 08:03 mrmuldv.c
-rw-r--r-- 1 randy users May 22 08:03 mirdef.h

16. MIRACL schoof schoof2 sea cm schoof -s 389712424879511 -3 238
mueller: generate modular polynomials
process: convert modular polynomials into those mod p
modpol: generate modular polynomials mod p
sea i p192.pol cm cm

17. MIRACL
genprime

18. MIRACL
ecp

19. MIRACL
ec2m