# What does the modulo plus minus mean?

May someone please explain what the notations in the image means?

In general, for a modulus $$q$$, what does the $$+$$ in here $$\bmod^+ q$$ indicate? What does the $$\pm$$ in here $$\bmod^\pm q$$ mean?

• I don't think this is standard notation. Where did you find it? You should provide a reference. Guessing, it sounds like that defines the range of representatives, in the first line it would be $\{0,\ldots,q-1\}$ and in the second $\{-2^{d-1},\ldots,2^{d-1}-1\}$, but that's just a guess Aug 4 at 16:51
• It's in the Dilithium, round 3 specification documents. Aug 4 at 23:49

From the NIST Post-Quantum Cryptography Round 3 submission for Crystals-Kyber:

Modular reductions. For an even (resp. odd) positive integer α, we define $$r' = r\bmod^± α$$ to be the unique element $$r'$$in the range $$-\frac{α}{2} < r' \le \frac{α}{2}$$ (resp. $$-\frac{α-1}{2} \le r' \le \frac{α-1}{2}$$) such that $$r' = r\bmod α$$. For any positive integer α, we define $$r' = r\bmod^+ α$$ to be the unique element $$r'$$ in the range $$0 ≤ r' < α$$ such that $$r' = r\bmod α$$. When the exact representation is not important, we simply write $$r' = r\bmod α$$.

It is probably also available elsewhere, but this was my source.

• I think the proof of Gauss's lemma used in quadratic reciprocity had similar ideas of a "centered" mod p instead of the usual 0 thru p-1.
– qwr
Aug 5 at 5:48
• I wish this definition had used $r'\equiv r\pmod\alpha$ where there is $r'=r\bmod\alpha$. The former means that $r'-r$ is a multiple of $\alpha$. The later, unless otherwise specified, tends to mean what they write $r'=r\bmod^+\alpha$. See e.g. FIPS 186-4.
– fgrieu
Aug 5 at 6:08
• Since the OP was asking about Dilithium, I checked and the definition is the same, except they added a sentence "We will sometimes refer to this as a centered reduction modulo α" Aug 5 at 13:55
• I wish floating-point fmod() operator had been defined in this fashion, rather than as yielding a floating-point remainder in a manner that isn't a modular equivalence class. Aug 5 at 20:33
• @fgrieu arguably the notation should look something like $r' \stackrel{\operatorname{mod}\alpha}= r$ or perhaps $r' \in \{r|\operatorname{mod}\alpha\}$. I always found it weird to write $r' \equiv r\quad (\operatorname{mod}\alpha)$ because $r' \equiv r$ already looks like a strong equality between $r$ and $r'$, which it is not. Aug 5 at 20:58