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Term

Safe Prime

Primes p where (p − 1)/2 is also prime (5, 7, 11, 23, 47, 59, 83, 107, 167, 179, …).

670 numbers tagged.

A safe prime is a prime \(p\) such that \((p-1)/2\) is also prime. That smaller companion is a Sophie Germain prime, so the two classes are mirror images of each other: \(p\) is safe exactly when \((p-1)/2\) is Sophie Germain.

The "safe" refers to cryptography. The multiplicative group modulo a safe prime has a large prime-order subgroup, which blocks small-subgroup attacks on Diffie–Hellman key exchange, and safe primes resist Pollard's \(p-1\) factoring method. Standardized DH groups (RFC 3526) all use safe primes with thousands of bits.

5 7 11 23 47 59 83 107 167 179 227 263 347 359 383 467 479 503 563 587 719 839 863 887 983 1,019 1,187 1,283 1,307 1,319 1,367 1,439 1,487 1,523 1,619 1,823 1,907 2,027 2,039 2,063 2,099 2,207 2,447 2,459 2,579 2,819 2,879 2,903 2,963 2,999 3,023 3,119 3,167 3,203 3,467 3,623 3,779 3,803 3,863 3,947 4,007 4,079 4,127 4,139 4,259 4,283 4,547 4,679 4,703 4,787 4,799 4,919 5,087 5,099 5,387 5,399 5,483 5,507 5,639 5,807 5,879 5,927 5,939 6,047 6,599 6,659 6,719 6,779 6,827 6,899 6,983 7,079 7,187 7,247 7,523 7,559 7,607 7,643 7,703 7,727 7,823 8,039 8,147 8,423 8,543 8,699 8,747 8,783 8,819 8,963 9,467 9,587 9,743 9,839 9,887 10,007 10,079 10,103 10,163 10,343 10,463 10,559 10,607 10,667 10,799 10,883 11,003 11,279 11,423 11,483 11,699 11,807 12,107 12,203 12,227 12,263 12,347 12,527 12,539 12,647 12,659 12,899 12,983 13,043 13,103 13,127 13,163 13,523 13,799 13,967 14,087 14,159 14,207 14,243 14,303 14,387 14,423 14,699 14,867 15,083 15,287 15,299 15,383 15,647 15,683 15,767 15,803 16,139 16,187 16,223 16,487 16,547 17,027 17,327 17,387 17,483 17,903 17,939 18,059 18,119 18,443 18,587 18,743 18,839 18,947 18,959 19,079 19,259 19,379 19,583 20,123 20,183 20,327 20,507 20,543 20,627 20,663 21,059 21,179 21,227