The order of a finite group is the number of elements it contains. Subgroups can have different orders, so order tells you how large the available mathematical space is. section 6.3.1 section 5.2.1
A small modular example
Modulo 5, repeatedly multiplying 4 gives 4, then 1, then 4 again. The generated subgroup is {1, 4}, so its order is 2. Other subgroups modulo 5 have different numbers of elements. section 5.2.1
Order matters in cryptographic calculations:
- For a prime modulus p, the multiplicative group has order p − 1. section 6.3.1
- RSA derives the private exponent d as the modular inverse of the public exponent e modulo the group order. section 6.3.1
- In Diffie–Hellman settings, subgroup sizes and parameter choices affect the cryptographic problem and its security. section 5.2.1 section 5.2.3
