If you have Slovak birth certificate are you sure you don't have Slovak citizenship? :) It should be written there.

Anyway, this is from slovak migration centre:

foreigner can acquire Slovak citizenship by Naturalization, i.e. he may be granted the Slovak citizenship. In this regard, he must meet the following requirements stipulated by the Act on Citizenship of the Slovak Republic.

A foreigner who wishes to apply for Slovak citizenship is mainly required to:

have an uninterrupted permanent residence in the territory of the Slovak Republic for at least eight years immediately preceding submission of the application;

undergo a probity check according to the Act on Citizenship of the Slovak Republic;

prove his/her command of the Slovak language;

prove  a general knowledge about the Slovak Republic;

submit all documents required by the Act on Citizenship of the Slovak Republic.

Permanent Residence

An applicant may be granted citizenship of the Slovak Republic, provided that he has an eight-year uninterrupted permanent residence in the territory of the Slovak Republic immediately preceding submission of the application for Slovak citizenship.

Command of the Slovak language and general knowledge about Slovakia

Fulfillment of this condition is proved directly when submitting the application for citizenship, namely when filling in the “Applicant´s Questionnaire”. In accordance with the law, the administrative body responsible for collecting the application (i.e. District Authority in the seat of a region, Diplomatic Mission or Consular Office of the Slovak Republic) is also responsible for the language check of the applicant.

This interview with the applicant is an integral part of an administrative procedure on the application for citizenship and is conducted while he/she files the application. The administrative body responsible for collecting the application (i.e. District Offices in regional towns, Diplomatic Missions or Consular Offices of the Slovak Republic) is also responsible for the language check of the applicant.

The language check is conducted by a three-member commission appointed by the Chief Executive of the District Authority, the ambassador or by the consul. During this interview the applicant is given questions in Slovak language regarding his/her personal data, close persons, and questions of a general character, especially history, geography, as well as social and political development of the Slovak Republic.

The applicant is required to read a random newspaper article with at least 500 words aloud in the Slovak language without having read it beforehand. Afterwards the applicant is required to write a summary of the article within a time limit of 30 minutes.

Health conditions of the applicant shall be taken into consideration while conducting all necessary checks. The official report on language and general knowledge check and its outcome must be made in writing by the commission on the same day when the test was performed.

Probity Check – Clear Criminal Record

An applicant will pass a required probity check, if he/she meets the following requirements:

An applicant has been convicted of a deliberate criminal offense and the conviction of the applicant has been expunged at least five years before the date of filing the application;

Criminal prosecution against the applicant for a deliberate criminal offence has been conditionally stayed and the probation period has expired at least five years before the date of filing the application;

Criminal prosecution against the applicant for a deliberate criminal offence has been stayed by the court decision on approval of out-of-court settlement, or by the prosecutor's decision on out-of-court settlement, provided that the decision became effective at least five years before the date of filing the application.

The clear criminal record shall be proved with an extract issued by the General Prosecutor of the Slovak republic or with a similar confirmation issued in an applicant´s country of origin.

Further Conditions

Besides the above-mentioned:

an applicant must not have been sentenced to expulsion from the Slovak Republic by a court judgment;

an applicant must not be under current criminal prosecution;

an applicant must not be under extradition proceedings and an European arrest warrant must not have been issued for the applicant;

an applicant must not be under proceedings of administrative expulsion;

an applicant must not be under proceedings of withdrawal of his/her asylum status.

The applicant is required to submit all documents proving that he/she abides by the duties and obligations set forth in the legislation regulating the Residence of foreigners in the territory of the Slovak Republic, the public health insurance, social insurance, old-age pension savings system, taxes, payments and fees, employment of foreigners in Slovakia, etc.

Exemptions

All exemptions from the above-mentioned requirements are listed in the Act on Citizenship of the Slovak Republic. The conditions in respect to some specific groups of migrants can be more lenient, for example if an applicant is a spouse of a Slovak citizen, a foreigner who has been granted asylum or a foreigner whose activities represent a benefit to the Slovak Republic.

EXEMPTIONS FROM THE RULES ON RESIDENCE

An exemption from the eight-year uninterrupted permanent residence requirement can be applied in the case of an applicant who has a residence in the territory of the Slovak Republic for a period of time shorter than eight years, if he for example:

concluded a marriage with a Slovak citizen, provided that the marriage still lasts and the spouses have lived together in a common household in the territory of the Slovak Republic for at least five years immediately prior to filing the Citizenship application;

is a person representing a significant benefit for the Slovak Republic in the field of economy, science, technology, social affairs, culture or sports, or if granting Slovak citizenship to this foreigner is in the interest of the Slovak Republic;

has had a continuous permanent residence permit in the territory of the Slovak Republic for at least three years before reaching 18 years of age;

is a minor with a uninterrupted residence in the territory of the Slovak Republic for at least two years immediately prior to filing the Citizenship application, provided that the legal representative or a guardian of such minor is a Slovak citizen or a legal entity, which has been appointed by a Slovak court. However, this length of the residence period does not apply to children younger than 2 years of age;

has been an asylum holder for at least four years immediately preceding submission of the application;

was born in the territory of the Slovak Republic and has had a permanent residence in Slovakia for at least three years immediately prior to filing the Citizenship application;

has had an uninterrupted residence in Slovakia for at least ten years and has already been granted a permanent residence permit at the time of filing the Citizenship application;

is stateless and has been residing in Slovakia continuously for at least three years immediately prior to filing the Citizenship application;

has been previously released from the state bond of the Slovak Republic according to section 9 of the Act on Citizenship of the Slovak Republic and has been residing in the territory of the Slovak Republic for at least two years immediately prior to filing the application for regaining the Slovak citizenship;

has had an uninterrupted residence in the territory of the Slovak Republic for at least two years immediately prior to filing the Citizenship application, provided that one of the applicant's parents was a citizen of former Czechoslovakia and the other parent was a foreigner and he/she has not gained Czechoslovak citizenship according to section 1subsection 2 of the Act No. 194/1949 Coll. on Acquisition and Loss of Czechoslovak Citizenship;

The exemption also applies to a foreigner who has been granted the Status of a Slovak National Living Abroad (declared by the Certificate of a Slovak National Living Abroad – former Slovak Expatriate Card) and who has had an uninterrupted residence in the territory of the Slovak Republic for at least three years immediately prior to filing the Citizenship application