Am Dienstag, 15. August 2017, 00:21:05 CEST schrieb Theodore Ts'o: Hi Theodore, > Have you looked at section 3.1.1 of the above cited paper? > > http://eprint.iacr.org/2012/251.pdf Thanks for the hint, but that does not seem to solve the mystery either. When I use magma with GF(2^32), I see that all polynomials are neither primitive nor irreducible: F:=GF(4294967296); F; P<x>:=PolynomialRing(F); P; print "Old polynomials:"; P<x>:=x^128 + x^103 + x^76 + x^51 +x^25 + x + 1; P; print "is irreducible: "; IsIrreducible(P); print "is primitive: "; IsPrimitive(P); P<x>:=x^32 + x^26 + x^20 + x^14 + x^7 + x + 1; P; print "is irreducible: "; IsIrreducible(P); print "is primitive: "; IsPrimitive(P); print "New polynomials:"; P<x>:=x^128 + x^104 + x^76 + x^51 +x^25 + x + 1; P; print "is irreducible: "; IsIrreducible(P); print "is primitive: "; IsPrimitive(P); P<x>:=x^32 + x^26 + x^19 + x^14 + x^7 + x + 1; P; print "is irreducible: "; IsIrreducible(P); print "is primitive: "; IsPrimitive(P); The output is: Finite field of size 2^32 Univariate Polynomial Ring in x over GF(2^32) Old polynomials: x^128 + x^103 + x^76 + x^51 + x^25 + x + 1 is irreducible: false is primitive: false x^32 + x^26 + x^20 + x^14 + x^7 + x + 1 is irreducible: false is primitive: false New polynomials: x^128 + x^104 + x^76 + x^51 + x^25 + x + 1 is irreducible: false is primitive: false x^32 + x^26 + x^19 + x^14 + x^7 + x + 1 is irreducible: false is primitive: false Thus, I am unsure how the referenced document concludes that the new polynomials are irreducible over GF(2^32). Ciao Stephan
