1. MIRACL & PBC
Yung-Hsiang Liu

2. Outline Why we use the libraries?
MIRACL (
Elliptic curves cryptography
ECDSA, ECIES, ECDH, ECMQV, …
PBC (
Pairing-based cryptography
IBE, IBNIKE, 3-party KE, IBS, BLS, PEKS, BE, ABE, …

3. Why we use the libraries?
Security
Big number and finite field arithmetic
Group law
Extension field arithmetic
Bilinear pairing: Miller’s algorithm, polynomial evaluation
Symmetric Key Size (bits)
RSA and Diffie-Hellman Key Size (bits)
Elliptic Curve Key Size (bits) 80
1024
160
112
2048
224
128
3072
256
192
7680
384
15360
521
Table 1: NIST Recommended Key Sizes

4. Libraries C/C++ Libraries MIRACL PBC
Multiprecision Integer and Rational Arithmetic Cryptographic Library
PBC
Pairing-Based Crypto library

5. MIRACL
Yung-Hsiang Liu
ECC 2014

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

7. ECDSA Parameters: 𝐸( 𝔽 𝑝 ) of prime order 𝑛, and a base point 𝑃
Key pair: private key 𝑑 𝐴 ∈ ℤ 𝑛 , public key 𝑄 𝐴 = 𝑑 𝐴 𝑃
Sign message 𝑚:
𝑧=𝐻𝑎𝑠ℎ 𝑚 mod 𝑛
𝑘 ∈ 𝑅 ℤ 𝑛 , and 𝑅=𝑘𝑃
𝑟=𝑥 𝑅 mod 𝑛 *
𝑠= 𝑘 −1 𝑧+𝑟 𝑑 𝐴 mod 𝑛 *
Signature: (𝑟,𝑠)
Verify the signature (𝑟,𝑠):
𝑧=𝐻𝑎𝑠ℎ 𝑚 mod 𝑛
𝑤= 𝑠 −1 mod 𝑛
𝑅=𝑧𝑤𝑃+𝑟𝑤 𝑄 𝐴
Check if 𝑥 𝑅 =𝑟

8. What MIRACL Provides – (1)
Big number
Finite field arithmetic
Elliptic curve group law
𝐸 𝔽 𝑝 : 𝑦 2 = 𝑥 3 +𝑎𝑥+𝑏
𝑃( 𝑥 1 , 𝑦 1 ), 𝑄( 𝑥 2 , 𝑦 2 )
𝑅 𝑥 3 , 𝑦 3 =𝑃+𝑄
𝑥 3 = 𝜆 2 − 𝑥 1 − 𝑥 2
𝑦 3 =𝜆 𝑥 1 − 𝑥 3 − 𝑦 1
𝜆= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 , if 𝑃≠𝑄 3 𝑥 1 2 +𝑎 2 𝑦 1 , if 𝑃=𝑄

9. What MIRACL Provides – (2)
Efficient scalar multiplication
Use “double and add” to calculate 𝑘𝑃
Random number generator
Pick a random big number
Random point on elliptic curves
Choose 𝑥, and test if there is a suitable 𝑦

10. Download MIRACL

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

12. Build MIRACL on Ubuntu Download from GitHub
Save as MIRACL-master.zip
unzip -j -aa -L MIRACL-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 linux64

13. Examples
genprime
schoof
ecdsa_sign
ecdsa_verify

14. Further Reading
Sample codes provided by MIRACL
*.cpp
Header files
*.h

15. PBC
Yung-Hsiang Liu
ECC 2014

16. Pairing-based Cryptography
Identity-based encryption
Identity-based non-interactive key distribution
3-party key distribution
Identity-based signature
BLS short signature

17. What PBC Provides Big number Finite field arithmetic
Extension field arithmetic
Elliptic curve group law
Bilinear pairings
Tate pairing
Miller’s algorithm
Polynomial evaluation

18. Download MIRACL

19. Build MIRACL To build the library Windows binaries and DLL
Windows binaries and DLL

20. Build MIRACL on Ubuntu Download from official website
Save as pbc tar.gz
tar xf pbc tar.gz
./configure
make
.libs/libpbc.*
make install

21. Examples BLS short signature
Identity-based non-interactive key distribution
3-party key distribution
Identity-based encryption
Identity-based signature

22. More Applications Searchable encryption Broadcast encryption
Attribute-based encryption

23. Further Reading http://crypto.stanford.edu/pbc/manual/
1. Installing PBC
2. Tutorial
3. Pairing functions
4. Element functions
5. Param functions
6. Other functions
7. Bundled programs
8. PBC internals
9. Security issues