On Analysis of Recovering Short Generator Problems via Upper and Lower Bounds of Dirichlet L-Functions: Part 1

Abstract

This article is a survey on upper and lower bounds of Dirichlet L-functions $L(s,\chi )$ associated with Dirichlet characters $\chi$ at $s=1$. We give proofs of well-known upper and lower bounds of $L(1, \chi )$ to let the reader know the difficulty of giving (lower) bounds of $L(1,\chi )$. In the last part, we also review some explicit upper and lower bounds of Dirichlet L-functions which will be applied to security analysis of ideal lattice-based cryptography for cyclotomic fields explained in Part 2 (S. Okumura, On analysis of recovering short generator problems via upper and lower bounds of Dirichlet L-functions : Part 2 [20]).

Appendices

8 Appendix 1

We prepare several lemmas on complex analysis for Theorem 1.2.

Lemma 8.1

(The Schwartz lemma) Set $D=\{z \in \mathbb {C} \ | \ |z|<1 \}$. Let $f : D \rightarrow D$ be a holomorphic map such that $f(0)=0$. Then, $|f(z)|\le |z|$ for all $z \in D$ and $f'(0)\le 1$.

Moreover, suppose either that there exists $z_0 \in D$ such that $|f(z_0)|=|z_0|$, or that $f'(0)= 1$ holds. Then we have $f(z)=az$ for some $a \in \partial D$.

Proof

Set $g(z)= f(z)/z$ if $z \in D-\{0\}$ and $g(z)=f'(0)$ if $z=0$. Then, g is holomorphic on D. For each $r\in (0, 1)$ and the closed disc $D_r = \{z \in \mathbb {C} \ | \ |z|\le r \}$, by the maximum modulus principle, there exists $z_r \in \partial D_r$ such that $\max _{z \in D_r}|g(z)| = |g(z_r)|=\frac{|f(z_r)|}{|z_r|} \le 1/r$. As we take the limit $r\rightarrow 1-0$, we have the first assertion. As for the second assertion, we have only to verify that g is a constant, which is proved by the maximum modulus principle similarly.$\square$

Lemma 8.2

(The Borel–Carathéodory theorem) Let f(s) be a holomorphic function on $|s|\le R$ with $R>0$. For any $r \in (0, R)$, we have

$$\max _{|s|\le r} |f(s)| \ (= \max _{|s|=r}|f(s)|) \le \frac{2r}{R-r}\max _{|s|\le R} \mathfrak {R}(f(s)) +\frac{R+r}{R-r}|f(0)|.$$

Proof

First consider the case $f(0)=0$. Set $A=\max _{|s|\le R}\mathfrak {R}(f(s))$. Then $A \ge \mathfrak {R}(f(0))=0$. In what follows, we may assume $A>0$; indeed, if $A = 0$, the function $\mathfrak {R}(f(x+iy))$ in (x, y) with $x+iy \in D_R$ is a harmonic function and attains a maximum at the interior point 0 of $D_R$. By the maximum modulus principle, $\mathfrak {R}(f(s))$ must be a constant $\mathfrak {R}(f(0))=0$ on $D_R$. Thus f(s) is identically equal to zero on $D_R$.

Set $H_a=\{z \in \mathbb {C} \ | \ \mathfrak {R}(z)\le a\}$ for $a\in \mathbb {R}$. Let us define the functions g and h by $g(z)=z/A-1$ and $h(z)= \frac{z+1}{z-1}$, respectively. By applying the Schwartz lemma for the composite

$$D_R \xrightarrow {f} H_A \xrightarrow {g} H_0 \xrightarrow {h} D_R,$$

we have $|h\circ g\circ f(s)| \le |s/R|$ for all $s \in D_R$, that is,

$$\frac{|R \, f(s)|}{|f(s)-2A|} \le |s|,$$

by which we have $R |f(s)| \le r|f(s)-2A| \le r|f(s)|+2Ar$ on $D_r$. Hence we obtain $|f(s)|\le \frac{2r}{R-r}A$, which proves the assertion for f such that $f(0)=0$. For the case $f(0)\ne 0$, by applying the estimate above to $f(s)-f(0)$, we obtain

$$|f(s)| \le |f(s)-f(0)|+|f(0)| \le \frac{2r}{R-r}\max _{|s|\le R}\mathfrak {R}(f(s)-f(0)) + |f(0)|.$$

for $s\in D_r$. This completes the proof.$\square$

Lemma 8.3

([27, p. 56]) Suppose $r>0$ and let f(s) be a holomorphic function on $|s-s_0|\le r$ with $f(s_0)\ne 0$. For a constant $M>0$, suppose that $\left| \frac{f(s)}{f(s_0)}\right| <e^M$ on $|s-s_0|\le r$. Then, there exists an absolute constant $A>0$ such that

$$\left| \frac{f'(s)}{f(s)}-\sum _{\rho }\frac{1}{s-\rho }\right| <\frac{AM}{r},$$

where $\rho$ runs over the zeros of f(s) such that $|\rho -s_0|\le r/2$ (the set of zeros is regarded as a multi-set).

Furthermore, if we suppose $f(s)\ne 0$ for all $s\in \mathbb {C}$ such that $|s-s_0|\le r$ and $\mathfrak {R}(s_0)<\mathfrak {R}(s)$, then we have

$$-\mathfrak {R}\left( \frac{f'(s_0)}{f(s_0)}\right) <\frac{AM}{r}.$$

Proof

Set $g(s)=f(s)\prod _{\rho }(s-\rho )^{-1}$ and $h(s)=\log \frac{g(s)}{g(s_0)}$, where $\rho$ runs over the zeros of f(s) such that $|\rho -s_0|\le r/2$. Then, g(s) is holomorphic on $|s-s_0|\le r$ and non-zero for $|s-s_0| \le r/2$. For $s \in \mathbb {C}$ such that $|s-s_0|=r$, we have

$$\left| \frac{g(s)}{g(s_0)}\right| = \left| \frac{f(s)}{f(s_0)}\prod _{\rho }\frac{s_0-\rho }{s-\rho }\right| \le \left| \frac{f(s)}{f(s_0)}\right| <e^M$$

by $|s_0-\rho | \le r/2 \le |s-\rho |$. This inequality as above is still valid for $|s-s_0|\le r$ by the maximum modulus principle. From this, we have $\mathfrak {R}(h(s))<M$ for $|s-s_0|\le r$. Then, by the Borel–Carathéodory theorem, the estimate

$$|h(s)|< \frac{2\times 3r/8}{r/2-3r/8}M \le A_0 M$$

holds for $|s-s_0| \le 3r/8$ with $A_0>0$ being an absolute constant ($A_0=6$ is sufficient). Hence, for $|s-s_0| \le r/4$ we obtain

$$\left| \frac{f'(s)}{f(s)}-\sum _{\rho }\frac{1}{s-\rho }\right| = |h'(s)|= \left| \frac{1}{2\pi i}\oint _{|z-s|=r/4} \frac{h(z)}{(z-s)^2}dz \right| < \frac{4A_0M}{r}.$$

This completes the proof of the first assertion. The inequality as above gives us

$$-\mathfrak {R}\left( \frac{f'(s_0)}{f(s_0)}\right) < \frac{AM}{r} -\sum _{\rho }\mathfrak {R}\left( \frac{1}{s_0-\rho }\right) .$$

This completes the proof of the second assertion since $\mathfrak {R}\left( \frac{1}{s_0-\rho }\right) \ge 0$ holds under the assumption.$\square$

Proposition 8.4

([27, p. 57] Let f(s) be a holomorphic function on $|s-s_0|\le r$ such that $f(s_0)\ne 0$

$$\left| \frac{f(s)}{f(s_0)}\right|< e^M \quad \text {and} \quad \left| \frac{f'(s_0)}{f(s_0)}\right| < \frac{M}{r} \quad \text {on} \quad |s-s_0| \le r $$

for a constant $M>0$. Furthermore, suppose that $f(s)\ne 0$ for all $s=\sigma +it$ such that $\sigma \ge \sigma _0-2r'$ and $|s-s_0|\le r$, where we set $\sigma _0 = \mathfrak {R}(s_0)$ and $r'$ is some real number such that $0<r'<r/4$. Then, there exists an absolute constant $A'>0$ such that

$$\left| \frac{f'(s)}{f(s)}\right| < \frac{A'M}{r} \qquad |s-s_0|\le r'.$$

Proof

As in Lemma 8.3, we have $-\mathfrak {R}\left( \frac{f'(s)}{f(s)}\right) < \frac{AM}{r}$ for all $s=\sigma +it$ such that $|s-s_0|\le r/4$ and $\sigma \ge \sigma _0 -2r'$. Therefore, the Borel–Carathéodory theorem yields the inequalities

$$\max _{|s-s_0|\le r'}\left| -\frac{f'(s)}{f(s)}\right| \le \frac{2r'}{2r'-r'}\max _{|s-s_0|\le 2r'}\mathfrak {R}\left( \frac{-f'(s)}{f(s)}\right) + \frac{2r'+r'}{2r'-r'} \left| \frac{-f'(s_0)}{f(s_0)} \right| < \frac{(2A+3)M}{r}.$$

Thus we are done.$\square$

9 Appendix 2

Let $\chi$ be a primitive Dirichlet character modulo q. With the aid of a functional equation of $L(s,\chi )$ in Proposition 3.1 and the Hadamard factorization theorem of entire functions of finite order (cf. [3, Sects. 11 and 12] or [17, Sect. 10.2]), and by taking the logarithmic derivative, there exists a constant $B(\chi ) \in \mathbb {C}$ such that

$$\frac{L'(s,\chi )}{L(s,\chi )} = -\frac{a_\chi }{s-1}-\frac{1}{2}\log \frac{q}{\pi } -\frac{1}{2}\frac{\varGamma '(s/2+\delta _\chi /2)}{\varGamma (s/2+\delta _\chi /2)} +B(\chi ) + \sum _{\rho }\left( \frac{1}{s-\rho }-\frac{1}{\rho }\right) ,$$

where $\rho$ runs over all zeros of $L(s,\chi )$ in the region $0<\mathfrak {R}(s)<1$, and we put $a_\chi =1$ if $\chi$ is trivial and $a_\chi =0$ otherwise. Furthermore, taking the real part yields $-\mathfrak {R}\left( \frac{L'(s,\chi )}{L(s,\chi )}\right) = \frac{1}{2}\log \frac{q}{\pi } +\frac{1}{2}\mathfrak {R}\left( \frac{\varGamma '(s/2+\delta _\chi /2)}{\varGamma (s/2+\delta _\chi /2)}\right) -\mathfrak {R}(B(\chi )) - \mathfrak {R}\sum _{\rho }\left( \frac{1}{s-\rho }-\frac{1}{\rho }\right) .$ With the aid of $\mathfrak {R}(B(\chi )) = -\sum _{\rho }\mathfrak {R}(\frac{1}{\rho })$ (cf. [3, p. 83, (18)]) and $\mathfrak {R}\left( \frac{\varGamma '(s/2+\delta _\chi /2)}{\varGamma (s/2+\delta _\chi /2)}\right) =\mathscr {O}(\log (3+|t|))$ from Stirling’s formula, we have the following.

Lemma 9.1

There exists an absolute constant $C>0$ such that

$$\begin{aligned} -\mathfrak {R}\left( \frac{L'(s,\chi )}{L(s,\chi )}\right) < \mathfrak {R}\left( \frac{a_\chi }{s-1}\right) + C \log \{q(3+|t|)\} -\sum _{\rho }\mathfrak {R}\left( \frac{1}{s-\rho }\right) \quad \mathfrak {R}(s)>1. \end{aligned}$$

This inequality is often used in the proof of Proposition 6.1 on zero-free regions. Here we notice that the inequality in Proposition 8.4 holds on a closed disc but Lemma 9.1 holds on a half plane.

9 Appendix 3

For application to security analysis of ideal lattice-based cryptography for cyclotomic fields, we review explicit upper and lower bounds of $L(1,\chi )$ (cf. [20, 22]). Such bounds of Dirichlet L-functions are related to the size of a dual basis of a log-unit lattice for a cyclotomic field, and moreover the size of such a dual basis is concerned with the RSG attack¹ as in [2].

Let $\delta _{a,b}$ denote the Kronecker delta.

Theorem 10.1

([24, Corollaries 1 and 3] (see also [12, Corollary 1.2]) For any non-trivial primitive Dirichlet character modulo q with $\chi (-1)=(-1)^\varepsilon$ for $\varepsilon \in \{0,1\}$, we have

$$| L(1,\chi ) |\le \frac{1}{2}\log q+\delta _{\varepsilon , 1} \left( \frac{5}{2}-\log 6\right) .$$

Furthermore, if 2|q, then we have

$$| L(1,\chi ) | \le \frac{1}{4}\log q +\delta _{\varepsilon , 0}\left( \frac{1}{2}\log 2\right) + \delta _{\varepsilon ,1}\left( \frac{5}{4}-\frac{1}{2}\log 3\right) .$$

Theorem 10.2

([4, Theorem 1.1]) For any primitive Dirichlet character modulo q such that 3|q and $\chi (-1)=(-1)^\varepsilon$ for $\varepsilon \in \{0,1\}$, we have

$$ | L(1,\chi ) |\le \frac{1}{3}\log q + \delta _{\varepsilon ,0} \, 0.368296+\delta _{\varepsilon , 1} \, 0.838374. $$

By combining Theorems 10.1 and 10.2, we have the following estimates.

Corollary 10.1

Let $\chi$ be a non-trivial even primitive Dirichlet character modulo $q=p^k$, where p is prime and $k \in \mathbb {N}$. Let $k_\chi$ be the integer such that $f_\chi =p^{k_\chi }$. Then, we have

$$\displaystyle |L(1,\chi )|\le {\left\{ \begin{array}{ll} \frac{1}{4}(\log f_\chi + 2\log 2) &{} (p=2),\\ \delta _{k_\chi ,2}\left( \frac{1}{2}\log f_\chi \right) +(\sum _{m=3}^{\infty }\delta _{k_\chi ,m})\,\frac{1}{3}(\log f_\chi +1.104888) &{} (p=3),\\ \frac{1}{2}\log f_\chi &{} (p\ge 5). \end{array}\right. }$$

Here we note $k_\chi \ge 2$ if $p=3$ by the fact that there exists no even non-trivial Dirichlet characters modulo 3.

Theorem 10.3

([14, Corollary 2]) For any non-quadratic primitive Dirichlet character modulo $q>1$, we have

$$|L(1, \chi )|\ge \frac{1}{10 \log (q/\pi )}.$$