under wraps

under wraps (kept a secret, not told to others) — покрытый мраком тайны, засекреченный

keep sth under wraps (keep something hidden, not show or talk about; kept confidential, ready to be revealed when the right time comes.) — держать в тайне, держать в секрете; не разглашать (до поры до времени); утаивать что-л.; замалчивать что-л.; умалчивать о чем-л.

Example 1: Let's keep this under wraps. We don't want the press finding out.

Example 2: I can’t keep my feelings under wraps any longer. I think I should be the one to tell you this.

Example 3: She wanted to keep it (the identity of the baby’s father) under wraps for a while longer.

remain under wraps — оставаться загадкой, тайной

keep sth quiet and keep sth still ( to keep something a secret) -- утаивать что-л.; замалчиватьth что-л.; умалчивать о чем-л.
keep quiet (about sth)

Example 1: I'm quitting my job, but my boss doesn't know yet. Please keep it quiet. Okay. I'll keep it still.

Example 2:The Vixen B tests at Maralinga were kept quiet for decades: why did no one ask what was going on?

Example 3:The subsequent investigation and court martial were kept quiet, with much of the information not released until the 1990s.

Example 4: The majority of the crashes occurred on landing and take-off. These crashes were kept quiet from us but they don’t surprise me. The Hermes was a very difficult, unresponsive platform.

Example 5:Landmark events such as children murdering their parents occurred in previous decades, but those events were far less likely to receive the barrage of media coverage they get today. Many of those events
were kept quiet because those crimes were committed by juveniles. Today, these events are widely reported in the media, with one consequence being that adults characterize the whole generation of adolescents by the actions of a few.

see also
[keep under one's hat]
[sweep sth under the carpet]
[behind closed doors]
[behind the scenes]
Just as a package is wrapped in paper, so a secret can be kept under cover