"The adversary" is standard terminology in analyzing cryptographic security.
It would be interesting to figure out who introduced it. I looked briefly on Google Scholar, and I see it's used in the modern sense as early as Yao (1982), but it might have been used much earlier. It doesn't appear in the Diffie-Hellman (1976) or RSA (1977) papers.
"Adversary" is simply a standard English word. It is neither religious or jargon. My assumption is that the reason it is often used in cryptography papers is because "attacker" rings a bit harshly when you repeat it over and over again.
Attacker rings harsh, and very limp. "The Adversary" is a common occult, religious and Biblical term for the Devil, Satan. The Hebrew word Satan means "the adversary". That's why it's funny and appropriate, lol!
It would be interesting to figure out who introduced it. I looked briefly on Google Scholar, and I see it's used in the modern sense as early as Yao (1982), but it might have been used much earlier. It doesn't appear in the Diffie-Hellman (1976) or RSA (1977) papers.