What is the meaning of the notation $\langle a, b\rangle$ in certain cryptography papers?

Question

I'm trying to understand a paper from Brakerski and Vaikuntanathan. In it, they use the notation $\langle a,b \rangle$ but they don't explain what that means.

The answers on this question say that it's just another way to write a tuple, but I don't think that's right in this case since the authors use the regular $(a,b)$ notation in the rest of the paper to denote a tuple.

Does anyone know what else $\langle a,b \rangle$ could mean?

Answer 1

In mathematics, $\langle a,b \rangle$ represents the inner product of two vectors and this is a generalization of the dot product. It is one of the multiplications of the vectors in a Vector space and the result is a scalar.

An inner product satisfies these properties

1. $\langle c+a,b\rangle =\langle c,a\rangle +\langle a,b\rangle $.

2. $\langle \alpha a,b\rangle = \alpha\langle a,b\rangle$ where $\alpha$ is a scalar.

3. $\langle a,b\rangle =\langle b,a\rangle$.

4. $\langle a,a\rangle = 0 $ and equal if and only if $v=0$.

   • The fourth condition is the positive definite condition. Some authors define the inner product with only the first 3 conditions with a weaker 4th condition $\langle a, b \rangle =0$ for all $b$ then $a=0$.

   As noted in the comments by @levgeni, if the field is $\mathbb{F}_2$ then

   $$\langle (1,1),(1,1) \rangle = 1 \cdot 1 + 1 \cdot 1 = 2 = 0$$ and $$\langle (1,1),(1,0) \rangle = 1 \cdot 1 + 1 \cdot 0 = 1 = 1$$ therefore, the weaker condition is required.

The inner product definition changes according to the vector space. In the context of the paper, let

$$a =(a_1,\ldots,a_n)$$ $$b =(b_1,\ldots,b_n)$$

be two n-dimensional vectors then;

$$\langle a, b \rangle = a_1 \cdot b_1 + \cdots + a_n \cdot b_n $$